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5-1997

Fourth SIAM Conference on Applications of Dynamical Systems

SIAM Activity Group on Dynamical Systems

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Recommended Citation Final program and abstracts, May 18-22, 1997

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cJ May 18-22, 1997 Snowbird Ski and Summer Resort • Snowbird, Utah Sponsored by SIAM Activity Group on Dynamical Systems Conference Themes The themes of the 1997 conference will include the following topics. Principal Themes: • Dynamics in undergraduate education • Experimental studies of nonlinear phenomena • Hamiltonian systems and transport • Mathematical biology • Noise in dynamical systems • Patterns and spatio-temporal chaos Applications in • Synchronization • Aerospace engineering • Biology • Condensed matter physics • Control • Fluids • Manufacturing • Me;h~~~~nograPhY 19970915 120 • Lasers and o~ • Quantum UldU) • 51a m.@ Society for Industrial and Applied Mathematics http://www.siam.org/meetingslds97/ds97home.htm 2 " DYNAMICAL SYSTEMS Conference Prl

Contents A Message from the Conference Chairs ...

Get-Togethers 2 Dear Colleagues: Welcoming Message 2 Welcome to Snowbird for the Fourth SIAM Conference on Applications of Dynamica Systems. Organizing Committee 2 This highly interdisciplinary meeting brings together a diverse group of mathematicians Audiovisual Notice 2 scientists, and engineers, all working on dynamical systems and their applications. Th< Program Overview 3 themes were chosen to highlight those areas in which dynamical systems has recently mad, Program-at-a-Glance 4-7 exciting progress and to provide a forum for discussion of future directions. Conference Program 8-34 The response to our Call for Papers was extraordinary. As a result, the program is packed ful of fascinating sessions. Besides the 11 Invited Presentations, we have 51 Minisymposia, 3' Speaker Index 35-37 sessions of Contributed Papers, and 2 Poster Sessions. Although it was combinatoriall: Abstracts 38-130 impossible, we have tried hard to avoid conflicts. We hope you'll find this conference, and your visit to Snowbird, to be stimulating Funding Agency educational, and fun. Mary Silber and Steve Strogatz SIAM thanks the National Science Foun­ Co-Chairs, Organizing Committee dation and the Office of Naval Research for their support in conducting this con­ ference. Audiovisual Notice Two standard overhead projectors and two screens will be provided in every meeting ro( Get-Togethers Speakers with special audiovisual equipment needs should inform SIAM of their spec requirements by April 25, 1997. If we do not hear from speakers by that date, it will be underst( Welcoming Reception that a standard overhead projector is all that is needed. Saturday, May 17,1997 If a speaker sends a request for special audiovisual equipment and decides not to use 6:00 PM - 8:00 PM requested equipment after it has been installed, the speaker will be responsible for paying Poster Session I and Reception rental fee. Tuesday, May 20, 1997 Some examples of special audiovisual equipment and rental fees are 7:30 PM - 9:30 PM LCD Panel ...... $250 26" Data Monitor ...... $2 Poster Session " and Reception 35mm Slide Projector ...... $38 Shure Mixer ...... S Wednesday, May 21, 1997 Video Projector ...... $500 IBM PC Computer ...... $2 7:30 PM - 9:30 PM 112" VHS VCR ...... 55 Xenon 35mm Projector ...... $2 Organizing Committee Important Notice to all Meeting Participants Mary Silber ICaCornell University MS = Minisymposium (30 minutes) Peter Bates Brigham Young University There are two Poster Sessions. Poster I will take place on Tuesday evening from 7 :30 PM-9 PM. Poster II will take place at the same time on Wednesday evening. There will be receptif Charles R. Doering during these sessions. Poster presenters are asked to post their materials at 6:30 PM on eitl University of Michigan, Ann Arbor day that they are scheduled. By 9:30 PM on Tuesday or Wednesday, all posted materi should be removed from the poster boards. Any materials left on the board after that ti Christopher K. R. T. Jones will be removed and discarded. SIAM cannot be responsible for any discarded pos Brown University displays that are left at the end of the session. William L. Kath For more information about Poster presentations, please refer to "Guidelines for Meet Northwestern University Participants" at http://www.siam.org/meetings/guidhome.htm. Yakov Pesin For papers will multiple authors, the speaker is shown in italics if known at press time. Pennsylvania State University The conference organizers expect every speaker of a scheduled presentation to register and attt Vered Rom-Kedar the conference. If it becomes necessary for a speaker to cancel the presentation, the speakeJ Weivnann Institute of Science, Israel expected to find an alternate presenter, or one of the speaker's co-authors should give presentation. The speaker should inform the SIAM conference department of any change to I Michael Tabor her scheduled presentation. University of Arizona A "no-show" or canceled presentation can cause serious inconvenience to the attendees! conference organizers. September 10, 1997

Defense Technical Information Center 8725 John J. Kingman Road STE 0944 Fort Belvoir, VA 22060-6218

REF: Grant #N00014-97-1-0404

Gentlemen:

Enclosed is one (1) copy of the Final Program and Abstract book for the SIAM Conference on Dynamical Systems held in Snowbird, UT on May 19-12, 1997. No proceedings will be published for this conference.

Sincerely,

A. Robert Bellace Director, Finance & Administration

~ociety for Q§7ndustrial and A.pplied C>lfathernatics

3600 University City Science Center + Philadelphia. PA + 19104-2688 (215)382-9800 + [email protected] + Fax: (215)386-7999 + http://IIVIIVIIV.siarn.org Conference Program DYNAMICAL SYSTEMS " 3

PROGRAM OVERVIEW Following are subject classifications for the sessions. The codes in parentheses designate session type and number. The session types are contributed presentations (CP), invited presentations (IP), mini symposia (MS), and poster presentations. The poster presentations are in two sessions - Poster I and Poster II, Tuesday and Wednesday evenings, respectively.

Principal Themes Model Reduction, Analysis, and Control of Scaling Exponents in Turbulence (IP4) Spatio-Temporal Dynamics Using KL Lasers and Optical Fibers Dynamics in Undergraduate Education Methods (MS3I) Dynamics of Laser Arrays: Coherence, Chaos, Teaching Differential Equations Dynamically Nonlinear Dynamics and Pattern Formation and Control (MS45) (MS24) in Combustion (MS50) Fundamental Nonlinear Dynamics of Diode Teaching Dynamical Systems to Undergradu­ Pattern Formation and Singular Perturbations Lasers (MS37) ates (IP3) (MSI8) Lasers (CPI) Teaching Dynamics (CPI7) Reaction-Diffusion Equations (CP26) Nonlinear Optics and Solitons (CP4), (CP7) Hamiltonian Systems and Transport Spatio-Temporal Chaos (CP27) Periodically and Randomly Driven Dynamics Hamiltonian Systems (CP8) Spatio-Temporal Chaos: Characterization and in Nonlinear Optical Fibers ( MS33) Integrable Systems Methods for Curve Evolu­ Control (MS6) Manufacturing tion (MS22) Spiral-Wave Domains and Filaments: Experi­ Bucket Brigade Production Lines (IP6) Slow Evolution in Conservative Systems (MS 17) ments, Numerics, Theory (MSII) Mechanics Transport in Hamiltonian Systems (MS36) Universal Spatio-Temporal Chaos in Large Applied Mechanics (CP3) Assemblies of Simple Dynamical Units (IPlO) Mathematical Biology Oceanography Biological Microswimming (MS28) Waves and Ginzburg-Landau Equations (CP24) Dynamical System Methods for Oceanic and Biology (CP30) Synchronization Atmospheric Flows - Parts I and II (MS2) and Biomolecular Motors (IP7) Applications of Synchronized Chaos and (MS7) Hyperchaos (MS38) Bursting and Biochemical Oscillations (CP23) Interplay of Fluid Dynamics and Biology of Continuum Models of Biological Macromol­ Communicating with Chaos (MS5) Plankton Population Models (MS13) ecules (MS32) Nonlinear Oscillators (MS9) Lagrangian Transport in Mesoscale Ocean Dynamical and Statistical Modeling of Bio­ Phase Dynamics and Synchronization in Cha­ Structures (IPI) logical Systems (MS42) otic Systems (MSI6) Low-Frequency Variability in the Double­ Dynamics of Cortical Neural Networks - Parts Riddling in Chaotic Systems (MS51) Gyre Circulation (MS30) I and II (MS 17), (MS23) Synchronization I and II (CP9), (CP21) Ouantum Chaos Dynamics of Curves and Filaments (MS25) New Directions in Quantum Chaos (MS43) Mathematics and Medicine: From the Labora­ Applications in: Small Electronics and Quantum Chaos (IP8) tory to the Clinic (MS 10) Aerospace Engineering Mathematical Methods Molecular Motors (MS35) Nonlinear Phenomena in Aeroengine Dynam­ Applications of the Geometric Phase (MS41) Neural and Excitable Systems (CP18), (CP20) ics and Control (MS4) Attractors, Fractals, and Dimensions (CPlO) Noise in Dynamical Systems Condensed Matter Physics Bifurcation and Symmetry (CP33) New Results for Stochastic PDEs (MS12) Applications in Physics (CP13) Bifurcation Theory and Systems of Nonlinear Noise and Heteroclinic Phenomena (MS48) Dynamical Systems Theory and Non-equilib­ Conservation Laws ( MS29) Nonlinear Dynamics of Large Fluctuations rium Statistical Mechanics (MS44) Computing Invariant Manifolds (MS20) (MSI9) Josephson Junction Arrays: Progress and Energy Transfer in Nonlinear Partial Differ­ Processing Signals from Noisy Chaotic Sys­ Puzzles (MS21) ential Equations (MS46) tems (MS8) Control Invariant Measures and Ergodicity (CP12) Random Dynamical Systems (MS26) Control and Shadowing (MS47) Maps (CP6) Stochastic Resonance (CP25 ) Controlling Chaos (CP19) Melnikov Theory (CP22) Stochastic Resonance in Medicine and Biol­ Engineering and Control (CP37) Melnikov Methods for Partial Differential ogy (MS3) Nonlinear Control of Lagrangian Systems (lP9) Equations (MS49) Stochastic Resonance in Sensory Biology (IP5) Periodic Orbits in Chaotic Systems (MS40) Numerical Methods (CP28) Patterns and Spatio-Temporal Chaos Unstable Periodic Orbits (CP5) Partial Differential Equations (CP36) Complex Ginzburg-Landau Equations as Per­ Fluids Perturbation Methods and ODEs (CP2) turbations of Nonlinear Schrodinger Equa­ Chaotic Advection, Turbulence, and Trans­ Singular Parturbations (CP15) tions (IPll) - port (CP32), (CP35) Spatially Discrete Dynamical Systems - Experiments on Spatio-Temporal Chaos (MS 1) Convection and Hexagonal Patterns (CP14) Theory and Applications ( MS39) Intricate Interfaces and Modulated Membranes Diffusion and Turbulence on Water Surfaces Time Series and Signal Processing (CP34) -Their Geometry and Nonlinear Dynamics (lP2) (MS15) Topology in Dynamics (MS34) Karhunen-Loeve Methods (CP3I) Fluids (CP16), (CP29) Ulam's Conjecture and the Approximation of Nonlinear Waves (CPII) Invariant Measures (MS14) 4 DYNAMICAL SYSTEMS Conference Program PROGRAM·AT-A·GI.ANCE

Saturday, May 17 Sunday, May 18 Evening MS3 Stochastic Resonance 3:00 PM-5:00 PM 7:00 PM-9:00 PM in Medicine and Biology Concurrent Sessions Concurrent Sessions 6:00 PM-8:00 PM Organizer: James J. MS6 Spatio-Temporal Registration Collins, Boston University MSll Spiral-Wave Chaos: Characterization Domains and Filaments: Ballroom III - Level B Ballroom Lobby - Level B and Control Experiments, Numerics, 6:00 PM-8:00 PM MS4 Nonlinear Phenomena Organizer: Hermann Theory Welcoming Reception in Aeroengine Dynamics Riecke, Northwestern Organizer: Gregory Huber, and Control University University of Chicago Golden Cliff Room - Level B (This session has been Ballroom I - Level B Ballroom I - Level B cancelled) Sunday, May 18 MS7 Dynamical System MS12 New Results for Organizer: Mark R. Myers, Methods for Oceanic and Stochastic PDES United Technologies Morning Atmospheric Flows (Part II Research Center Organizer: Charles R. of II) Doering, University of 7:30 AM-4:00 PM Magpie A & B - Level B Organizer: Roberto Michigan, Ann Arbor Registration MSS Communicating with Camassa, Los Alamos Ballroom II - Level B Ballroom Lobby - Level B National Laboratory Chaos MS13 Interplay of Fluid 8:15 AM Organizer: Epaminondas Ballroom II - Level B Rosa, Jr., University of Dynamics and Biology of MSS Processing Signals Welcome Remarks and Maryland, College Park Plankton Population Models Announcements from Noisy Chaotic Systems Organizer: John Brindley, Wasatch A & B - Level C Organizers: Mark Muldoon, University of Leeds, United Mary Silber and Steven CPl Lasers University of Manchester Kingdom Strogatz Chair: Peter A. Braza, Institute of Science and Ballroom III - Level B University of North Florida Technology, United 8:30 AM-9:30 AM Kingdom; and Holger MS14 Ulam's Conjecture IPI Lagrangian Transport Maybird - Level C Kantz, Max Planck Institute and the Approximation of in Mesoscale Ocean CP2 Perturbation Methods for Physics of Complex Invariant Measures Structures and ODEs Systems, Germany (This session will run until Christopher K. R. T. Jones, Chair: Ferdinand Verhulst, Magpie A & B - Level B 9:30 PM) Brown University University of Utrecht, The Organizer: Fern Y. Hunt, Netherlands MS9 Nonlinear Oscillators Chair: Roberto Camassa, Organizer: Richard H. National Institute of Los Alamos National Superior B - Level C Rand, Cornell University Standards and Technology Laboratory CP3 Applied Mechanics Wasatch A & B - Level C Superior A - Level C Ballroom I, II, III - Level B Chair: Marian Wiercigroch, MSIO Mathematics and CP7 Nonlinear Optics and University of Aberdeen, Solitons II 9:30 AM-IO:00 AM Kings College, United Medicine: From the Coffee Laboratory to the Clinic Chair: William L. Kath, Kingdom Northwestern University Organizer: Michael C. Ballroom Lobby - Level B Superior A - Level C and Mezzanine Lobby - Mackey, McGill University, Magpie A & B - Level B Level C Afternoon Canada CPS Hamiltonian Systems Ballroom III - Level B Chair: M. C. Depassier, 10:00 AM-12:00 PM 12:00 PM-l:30 PM CP4 Nonlinear Optics and Universidad Catolica de Concurrent Sessions Lunch Chile, Chile Solitons I MSI Experiments on 1:30 PM-2:30 PM Chair: Anne Niculae, Superior B - Level C Spatio-Temporal Chaos Northwestern University CP9 Synchronization I Organizer: Eberhard IP2 Intricate Interfaces Chair: Reggie Brown, Bodenschatz, Cornell and Modulated Membranes Superior B - Level C College of William & Mary University - Their Geometry and CPS Unstable Periodic Ballroom I '- Level B Nonlinear Dynamics Orbits Wasatch A & B - Level C Raymond E. Goldstein, Chair: Steven J. Schiff, CPIO Attractors, Fractals, MS2 Dynamical System University of Arizona Children's National Methods for Oceanic and and Dimensions Chair: Michael Tabor, Medical Center Chair: Brian Hunt, University Atmospheric Flows (Part I University of Arizona Maybird - Level C of Maryland, College Park ofII) Organizer: Roberto Ballroom I, II, III - Level B CP6Maps Maybird - Level C Camassa, Los Alamos 2:30 PM-3:00 PM Chair: Colin Sparrow, National Laboratory University of Cambridge, Coffee United Kingdom Ballroom II - Level B Ballroom Lobby - Level B Superior A - Level C and Mezzanine Lobby - Level C Evening 5:00 PM-7:00 PM NOTE: Please refer Dinner (attendees will be on to the Important Notice their own) to all Meeting Participants. Conference Program DYNAMICAL SYSTEMS 5

PROGRAM·AT-A·GI.ANCE

Monday, May 19 Tuesday, May 20 Morning CPU Nonlinear Waves MS22 Integrable Systems Morning Chair: Manfred F. Goz, Methods for Curve 7:30 AM-4:00 PM Princeton University Evolution 7:30 AM-4:00 PM Registration Maybird - Level C Organizers: Annalisa Registration Calini, University of Ballroom Lobby - Level B Ballroom Lobby - Level B CP12 Invariant Measures Charleston; and Thomas 8:30 AM-9:30 AM and Ergodicity Ivey, Case 8:30 AM-9:30 AM Chair: Yuri Latushkin, IP3 Teaching Dynamical University of Missouri, Western Reserve University IP6 Bucket Brigade Systems to Undergraduates Columbia Wasatch A & B - Level C Production Lines Robert L. Devaney, Boston Leonid A. Bunimovich, Superior A - Level C University MS23 Dynamics of Georgia Institute of Cortical Neural Networks Chair: Robert Borrelli, CP13 Applications in Technology Physics (Part II of II) Harvey Mudd College Organizers: Xiao-Jing Chair: Kurt Wiesenfeld, Chair: Adam S. Landsberg, Georgia Institute of Ballroom I, II, III - Level B Haverford College Wang, Brandeis University; David Terman, Ohio State Technology 9:30 AM-1O:00 AM Superior B - Level C University, Columbus; and Ballroom I, II, III - Level B Coffee John Rinzel, National Afternoon 9:30 AM-lO:OO AM Ballroom Lobby - Level B Institutes of Health Coffee and Mezzanine Lobby - 12:00 PM-l:30 PM Ballroom II - Level B Level C Lunch Ballroom Lobby - Level B MS24 Teaching and Mezzanine Lobby - 10:00 AM-12:00 PM 1:30 PM-2:30 PM Differential Equations Level C Concurrent Sessions Dynamically IP4 Scaling Exponents in Organizer: Beverly H. 10:00 AM-12:00 PM MS15 Diffusion and Turbulence Concurrent Sessions Turbulence on Water West, Harvey Mudd College Peter S. Constantin, and Cornell University Surfaces University of Chicago MS25 Dynamics of Curves Ballroom III - Level B and Filaments Organizer: Preben Alstrom, Chair: Peter Bates, . Niels Bohr Institute, Denmark CP14 Convection and Organizer: Isaac Klapper, Brigham Young University Montana State University Magpie A & B - Level B Hexagonal Patterns Ballroom I, II, III - Level B Ballroom I - Level B MS16 Phase Dynamics and Chair: Anne C. Skeldon, 2:30 PM-3:30 PM City University, United MS26 Random Dynamical Synchronization in Chaotic Kingdom Systems IP5 Stochastic Resonance Systems Organizer: Jurgen Kurths, in Sensory Biology Maybird - Level C (This session will run until Universitiit Potsdam, Frank Moss, University of CP15 Pattern Formation 12:30 PM) Germany Missouri, St. Louis and Singular Perturbations Organizer: N. Sri Ballroom I - Level B Chair: John Rinzel, National II Namachchivaya, University of Illinois, Urbana MS17 Dynamics of Institutes of Health Chair: Tasso Kaper, Boston University Cortical Neural Networks Ballroom I, II, III - Level B Ballroom II - Level B Superior B - Level C (Part I of II) 3:30 PM-4:00 PM MS27 Slow Evolution in Organizers: Xiao-Jing Coffee CP16 Fluids I Conservative Systems Wang, Brandeis University; Chair: Brian F. Farrell, Organizer: Norman R. David Terman, Ohio State Ballroom Lobby - Level B Harvard University Lebovitz, University of University, Columbus; and and Mezzanine Lobby - Chicago John Rinzel, National Level C Superior A - Level C Wasatch A & B - Level C Institutes of Health 4:00 PM-6:00 PM Evening MS28 Biological Ballroom II - Level B Concurrent Sessions 6:00 PM-7:00 PM Microswimming MS18 Pattern Formation MS20 Computing Business Meeting Organizers: Richard W. and Singular Perturbations Invariant Manifolds Montgomery, University of Organizers: Arjen Doelman, Organizer: John SIAM Activity Group on California, Santa Cruz; and University of Utrecht, The Guckenheimer, Cornell Dynamical Systems Jair Koiller, Laboratorio Netherlands; Tasso Kaper, University Nacional de Computacao Boston University; and Golden Cliff Room - Level B Cientijica, Brazil Todd Kapitula, University Magpie A & B - Level B Ballroom III - Level B of New Mexico, Albuquerque MS21 Josephson Junction Ballroom III - Level B Arrays: Progress and MS29 Bifurcation Theory Puzzles and Systems of Nonlinear MS19 Nonlinear Dynamics Conservation Laws of Large Fluctuations Organizer: Kurt A. Wiesenfeld, Georgia Organizer: Stephen Organizer: Mark I. Institute of Technology Schecter, North Carolina Dykman, Michigan State State University University Ballroom I - Level B Maybird - Level C Wasatch A & B - Level C 6 DYNAMICAL SYSTEMS Conference Program

PROGRAM-AT-A-GLANCE

Tuesday, May 20 Wednesday, May 21 CP17 Teaching Dynamics MS33 Periodically and Morning CP24 Waves and Chair: James A. Walsh, Randomly Driven Ginzburg-Landau Oberlin College Dynamics in Nonlinear 7:30 AM-4:00 PM Equations Superior A- Level C Optical Fibers Registration Chair: Rebecca Hoyle, University of Cambridge, Organizers: J. Nathan Kutz, Ballroom Lobby - Level B CP18 Neural and Excitable Princeton University; and United Kingdom Systems I William L. Kath, Northwest­ 8:30 AM-9:30 AM Superior B - Level C Chair: Tomas Gedeon, ern University Montana State University IP8 Small Electronics and CP25 Stochastic Resonance Ballroom II - Level B Quantum Chaos Superior B - Level C Chair: Bruce J. Gluckman, MS34 Topology in Charles M. Marcus, Naval Surface Waifare CP19 Controlling Chaos Dynamics Stanford University Center and the Children's Chair: Elbert E. N. Macau, Organizer: James Yorke, Chair: Boris L. Altshuler, Research Institute of University of Maryland, University of Maryland, NEC Research, Inc. George Washington College Park University College Park Ballroom I, II, III - Level B Magpie A & B - Level B Ballroom III - Level B Maybird - Level C 9:30 AM-1O:00 AM Afternoon CP20 Neural and Excitable Coffee Afternoon Systems II 12:00 PM-I:30 PM Chair: Amitabha Bose, New Ballroom Lobby - Level B 12:00 PM-l:30 PM Lunch Jersey Institute of Technology and Mezzanine Lobby - Lunch Level C 1:30 PM-2:30 PM Superior B - Level C 1:30 PM-2:30 PM 10:00 AM-12:00 PM IP7 Biomolecular Motors CP21 Synchronization II Concurrent Sessions IP9 Nonlinear Control of Charles S. Peskin, Courant Chair: Peter Smereka, Lagrangian Systems Institute of Mathematical University of Michigan, Ann MS35 Molecular Motors Richard M. Murray, Sciences, New York Arbor Organizer: George F. California Institute of University Maybird - Level C Oster, University of Technology California, Berkeley Chair: Charles R. Doering, CP22 Melnikov Theory Chair: Jerrold E. Marsden, Magpie A & B - Level B University of Michigan, Ann Chair: Timothy Whalen, California Institute of Arbor National Institute of MS36 Transport in Technology Ballroom I, II, III - Level B Standards and Technology Hamiltonian Systems Ballroom I, II, III - Level B Organizer: Vered Rom­ 2:30 PM-3:00 PM Superior A - Level C Kedar, Weizmann Institute 2:30 PM-3:00 PM Coffee Evening of Science, Israel Coffee Ballroom Lobby - Level B Ballroom I - Level B Ballroom Lobby - Level B and Mezzanine Lobby - 5:30 PM-7:00 PM MS37 Fundamental and Mezzanine Lobby - Level C Dinner (attendees are on Level C their own) Nonlinear Dynamics of 3:00 PM-5:00 PM Diode Lasers 3:00 PM-5:00 PM Concurrent Sessions 6:30 PM-7:30 PM Poster setup Organizer: Bernd Concurrent Sessions MS30 Low-Frequency Ballroom I, II, III - LevelB Krauskopf, Vrije Universiteit, The Netherlands MS40 Periodic Orbits in Variability in the Double­ 7:30 PM-9:30 PM Chaotic Systems Gyre Circulation Ballroom II - Level B Poster Session Organizers: Ernest Barreto Organizers: Balu T. Nadiga MS38 Applications of and Edward Ott, University and Darryl D. Holm, Los Dessert and Coffee will be Synchronized Chaos and of Maryland, College Park Alamos National Labora­ served Hyperchaos Ballroom I - Level B tory Poster Session I Organizer: Louis M. MS41 Applications of the Magpie A & B - Level B Ballroom I, II, III - Level B Pecora, Naval Research Laboratory Geometric Phase MS31 Model Reduction, Organizers: Jerrold E. Analysis, and Control of Ballroom III - Level B Marsden, California Spatio-Temporal Dynamics MS39 Spatially Discrete Institute of Technology; and Using KL Methods Dynamical Systems - Paul K. Newton, University Organizer: Ira B. Schwartz, Theory and Applications of Southern California U.S. Naval Research Organizers: John Mallet­ Ballroom II - Level B Laboratory Paret, Brown University; MS42 Dynamical and Ballroom I - Level B and Shui-Nee Chow, Statistical Modeling of MS32 Continuum Models Georgia Institute of Technology Biological Systems of Biological Organizer: Yuhai Tu, IBM Macromolecules Wasatch A & B - Level B T. J. Watson Research Organizer: John H. CP23 Bursting and Center Maddocks, University of Biochemical Oscillations Ballroom III - Level B Maryland, College Park Chair: William R. Derrick, Wasatch A & B - Level C University of Montana Superior A - Level C Conference Program DYNAMiCAl SYSTEMS 7 PROGRAM-AT-A-GLANCE

Wednesday, May 21 Thursday, May 22 MS43 Quantum Chaos and Morning CP32 Chaotic Advection, MS51 Riddling in Chaotic Mesoscopic Physics Turbulence, and Transport I Systems Organizer: Charles M. 7:30 AM-1O:00 AM Chair: Patrick D. Miller, Organizer: Ying-Cheng Lai, Marcus, Stanford University Registration Brown University University of Kansas Magpie A & B- Level B Ballroom Lobby - Level B Wasatch A & B - Level C Ballroom III - Level B CP26 Reaction-Diffusion 8:30 AM-9:30 AM CP33 Bifurcation and CP34 Time Series and Equations Symmetry Signal Processing Chair: Jack Dockery, IPI0 Universal Spatio­ Chair: Sue Ann Campbell, Chair: Michael E. Davies, Montana State University Temporal Chaos in Large University of Waterloo, University College London, Assemblies of Canada United Kingdom Wasatch A & B - Level C Simple Dynamical Units Maybird - Level C Maybird - Level C CP27 Spatio-Temporal Yoshiki Kuramoto, Kyoto Chaos University, Japan Afternoon CP35 Chaotic Advection, Turbulence, and Transport Chair: David Peak, Utah Chair: Rajarshi Roy, State University Georgia Institute of 12:00 PM-1:30 PM II Chair: Diego del Castillo­ Maybird - Level C Technology Lunch Negrete, University of CP28 Numerical Methods Ballroom I, II, III - Level B 1:30 PM-2:30 PM California, San Diego Chair: Robert L. Warnock, 9:30 AM-1O:00 AM IPll Complex Ginzburg­ Wasatch A & B- Level C Stanford University Coffee Landau Equations as CP36 Partial Differential Superior A - Level C Ballroom Lobby - Level B Perturbations of Nonlinear Equations CP29 Fluids II and Mezzanine Lobby - Schrodinger Equations Chair: Jinqiao Duan, Chair: Robert Ghrist, Level C C. David Levermore, Clemson University University of Texas, Austin University of Arizona 10:00 AM-12:00 PM Superior B - Level C Superior B- Level C Chair: Vered Rom-Kedar, Concurrent Sessions Weizmann Institute of CP37 Engineering and Evening MS44 Dynamical Systems Science, Israel Control Chair: Hans True, Techni­ Theory and Ballroom I, II, III - Level B 5:30 PM-7:00 PM cal University of Denmark, Nonequilibrium Dinner (attendees are on 2:30 PM-3:00 PM Denmark Statistical Mechanics their own) Organizer: Robert Dorfman, Coffee Superior A - Level C 6:30 PM-7:30 PM University of Maryland, Ballroom Lobby - Level B 5:20 Conference adjourns College Park Poster setup starts and Mezzanine Lobby - Ballroom I - Level B Level C Ballroom I, II, III - Level B MS45 Dynamics of Laser 3:00 PM-5:00 PM 7:30 PM-9:30 PM Arrays: Coherence, Chaos Concurrent Sessions Poster Session opens and Control MS48 Noise and Dessert and Coffee will be Organizer: Rajarshi Roy, Heteroclinic Phenomena served Georgia Institute of Organizers: Vivien Kirk, Technology Poster Session II University of Auckland, Ballroom I, II, III - Level B Ballroom II - Level B New Zealand; and Emily MS46 Energy Transfer in Stone, Nonlinear Partial Utah State University Differential Equations Ballroom I - Level B Organizer: Michael I. Weinstein, University of MS49 Melnikov Methods Michigan, Ann Arbor for Partial Differential Equations Magpie A & B - Level B Organizers: Constance M. MS47 Control and Schober, Old Dominion Shadowing University; and Gregor Organizer: Eric J. Kostelich, Kovacic, Arizona State University Rensselaer Polytechnic Ballroom III - Level B Institute CP30 Biology Ballroom II - Level B Chair: Sharon R. Lubkin, MS50 Nonlinear Dynamics University of Washington and Pattern Formation in Superior B - Level C Combustion CP31 Karhunen-Loeve Organizer: Vladimir A. Volpert, Northwestern Methods University Chair: Thomas W. Carr, Southern Methodist Magpie A & B - Level B University Superior A - Level C 8 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Sunday, May 18 8:30 AM-9:30AM 10:00 AM-12:00 PM play for certain phenomena. To gain some insight it is still useful to study simpler models Ballroom I - Level B that target a specific mechanism expected to Chair: Roberto Camas sa, Los be important. It is in this context that dynami­ Alamos National Laboratory MSl cal systems methods can be effective. The Ballroom I, II, III - Level B Experiments on Spatio-Temporal minisymposium's aim is to provide several IPI Chaos examples in this direction. lagrangian Transport The minisymposium brings together four ex­ Organizer: Roberto Camassa in Mesoscale Ocean Structures perimentalists working on spatio-temporal Los Alamos National Laboratory disorder/chaos (STC) in vastly different sys­ 10:00 Dynamical Systems Theory and Much activity in the ocean is mediated tems. This minisymposium will address STC Dynamical Oceanography by mesoscale structures, such as mean­ in chemical oscillations, in the aggregation of Roger M. Samelson, Woods Hole dering jets and vortex rings. How these slime mold amoebae, in optical resonators, Oceanographic Institution structures share fluid, and thus fluid prop­ and in thermal convection. Spatio-temporal 10:30 Advection Induced by the Breaking of erties, with the ambient ocean, and how chaos appears to be a generic state occurring Vorticity Conservation via Viscous they transport fluid between different in dynamical systems when the system size Dissipation parts of their own internal structure, is (aspect-ratio) is increased and spatial degrees Sanjeeva Balasuriya, University of Peradeniya, Sri Lanka; Christopher K. R. T. crucial in estimating and understanding of freedom become important. To date, no their impact on ocean properties at all Jones and Bjorn Sandstede, Brown clear understanding of spatio temporal chaos University scales. Quantifying such transport is a has emerged. This minisymposium is set up to problem in dynamical systems, but one attract the attention of the "dynamical" com­ 11 :00 Mathematical Problems in the Theory of Shallow Water that poses many new challenges because munity to the problem of spatio-temporal Marcel Oliver, University of California, Irvine simple analytic models are far too crude chaos. It will also show that we have well­ to offer us any real understanding. Re­ defined and controllable experimental sys­ II :30 Inverse Scattering Analysis of Internal cent progress on the development of tech­ tems that allow quantitative comparison with Waves in the Andaman Sea A. R. Osborne, University of Torino, Italy niques for transport studies within realis­ theories. tic numerical models will be discussed. Organizer: Eberhard Bodenschatz Comparisons will be given between the Cornell University 10:00 AM-12:00 PM extent of Lagrangian transport predicted Ballroom III - Level B by these models for structures such as the 10:00 Transitions from Spiral Waves to Gulf Stream and transport across the Chemical Turbulence in a Reaction­ MS3 Gulf Stream by other mechanisms, such Diffusion System Oi Ouyang, NEC Research Institute, Inc. as ring detachment. Recent work on con­ Stochastic Resonance in Medicine siderations of viscosity-induced trans­ 10:30 Competition Between Spiral Waves and and Biology port suggests some surprising conclu­ Targets in Dictyostelium Discoideum Raymond E. Goldstein, University of Arizona Stochastic resonance (SR) is a phenomenon sions as to the dominant mode of trans­ wherein the response of a nonlinear system to port out of meandering jets. This latter II :00 A Primary Bifurcation to Spatio­ Temporal Chaos a weak input signal is optimized by the pres­ work involves a Melinkov calculation on ence of a particular level of noise. SR has been the perturbed velocity field, about which Michael Dennin, University of California, Irvine examined theoretically and experimentally in little is known other than it satisfies a a wide variety of systems, including biologi­ certain PDE. II :30 Spatio-Temporal Chaos in Rayleigh­ Benard Convection cal systems. This minisymposium will focus Christopher K. R. T. Jones Eberhard Bodenschatz, Organizer on recent SR studies that deal with neuro­ Division of Applied Mathematics physiological sensory systems, including the Brown University theoretical and experimental techniques that 10:00 AM-12:00 PM are used to characterize SR in sensory neu­ Ballroom II - Level B rons. Possible physiological and clinical ap­ plications of SR (e,g" noise-enhanced sen­ MS2 sory function in humans) will also be ad­ Dynamical System Methods for dressed, along with the biophysical and Oceanic and Atmospheric Flows bioengineering challenges surrounding these applications. {Part I of II) Organizer: James J. Collins Numerical simulations of atmospheric and Boston University oceanic flows are reaching a stage where 10:00 Noise, Hair Cells, and the Leopard Frog quantitative comparisons with real data are Kurt Wiesenfeld and Peter Jung, Georgia possible. While this progress constitutes a Institute of Technology; and Fernan striking confirmation of the validity of the Jaramillo, Emory University basic models, like the Primitive Equations, 10:30 Noise-Enhanced Sensory Function sometimes it falls short of enhancing our James J. Collins, Organizer understanding of the basic mechanisms at Conference Program DYNAMICAL SYSTEMS " 9

CONFERENCE PROGRAM Sunday, May 18 11 :00 Stochastic Resonance in Human Muscle 11 :00 Higher Dimensional Symbolic Encoding 10:20 Evolution Towards Symmetry Spindles-A Potential Mechanism for Ying-Chen Lai, University of Kansas Ferdinand Verhulst University of Utrecht Fusimotor Gain Control 11 :30 Encoding Information in Chemical The Netherlands Paul J. Cordo, R. S. Dow Neurological Sciences Chaos 10:40 The Moving Singularities of the Institute; Sabine M. P. Verschueren, Catholic Erik M. BollI. United States Military Academy Perturbation Expansion of the Classical University of Leuven, Belgium:J Timothy Inglis, Kepler Problem University of British Columbia, Canada; Frank Mohammad Tajdari, University of New Moss, University of Missouri, St Louis; Daniel 10:00 AM-12:00 PM England M. Merfeld, R. S. Dow Neurological Sciences Chair: Peter A. Braza, University of Institute; and James J Collins, Organizer 11 :00 Necessary and Sufficient Conditions for North Florida Finite-Time Blow-Up in ODE's 11 :30 Augmentation of Sensory Nerve Action Maybird - Level C Craig Hyde and Alain Goriely, University of Potentials During Muscle Contraction Arizona Faye Y. Chiou-Tan, Kevin N. Magee, Stephen CPI Tuel. lawrence Robinson, Thomas 11 :20 Simple Periodic Orbits in 1-1-1 Krouskop, Maureen R. Nelson, and Frank lasers Resonance: Cubic and Quartic Moss, Baylor College of Medicine Potentials 10:00 Phase Jumps of pi in a Laser with a S. Ferrer, Universidad de Zaragoza, Spain; Periodically Forced Injected Signal M. Lara, Real Instituto y Observatorio de la 10:00 AM-12:00 PM Peter A. Braza, University of North Florida Armada, Spain; J. Palacian, Universidad 10:20 Homoclinic Orbits for the Second Publica de Navarra, Spain; J F. San Juan, This session has been Harmonic Generation of Light in an Universidad de la Rioja, Spain; and P. Optical Cavity Yanguas, Universidad Publica de Navarra, cancelled. Alejandro B. Aceves, University of New Spain Mexico, Albuquerque; Darryl D. Holm, Los 11 :40 Fast Resonance Shifting as a Mechanism MS4 Alamos National laboratory; Gregor Kovacic of Instability in Dynamics Illustrated by Nonlinear Phenomena in and liya Timofeyev. Rensselaer Polytechnic Comets Institute Edward Belbruno, The Geometry Center Aeroengine Dynamics and Control 10:40 Synchronized Patterns in Coupled Organizer: Mark R. Myers Lasers 10:00 AM-12:00 PM United Technologies Research Center Gerhard Oangelmayr, Colorado State University; and Michael Wegelin, University Chair: Marian Wiercigroch, University of of Tubingen, Germany 10:00 AM-12:00 PM Aberdeen, Kings College, United Kingdom 11 :00 Spatio-Temporal Dynamics of Broad­ Superior A - Level C Wasatch A & B - Level C Area Semiconductor Lasers with Optical Feedback CP3 MSS George R. Gray and Ming-Wei Pan, Applied Mechanics Communicating with Chaos University of Utah; and David H. DeTienne, Nichols Corporation, Salt lake City 10:00 Forced Oscillations of a Discretely The realization that chaos can be controlled 11 :20 Bifurcation Diagram of Two Lasers with Supported Nonlinear String Using by small perturbations leads to the idea that Injected Field Nonsmooth Transformations chaotic systems can be used to produce a A. I. Khibnik, Cornell University; Y. Braiman, Gary Salenger and Alexander F. Vakakis, signal bearing desired information. In fact, it Emory University; T. A. B. Kennedy and Kurt University of Illinois, Urbana has been demonstrated experimentally that Wiesenfeld, Georgia Institute of Technology 10:20 Homoclinic Orbits and Localized the symbolic dynamics of a chaotic attractor 11 :40 Nonlinear Dynamics of Semiconductor Buckling of Cylindrical Shells can be made to follow a prescribed symbol Laser Arrays Gabriel J. Lord and Alan R. Champneys, sequence, so any desired message can be Luis F. Romero, Emilio L Zapata, and J L University of Bristol, United Kingdom; and encoded and transmitted. In this Ramos, University of Malaga, Spain Giles W. Hunt, University of Bath, United minisymposium the basic ideas of chaos in Kingdom communication and the experiments that dem­ 10:40 Stochastic Effect on Chaotic Dynamics onstrate their implementation will be pre­ 10:00 AM-12:00 PM of Cutting Process sented. Recent developments in higher di­ Chair: Ferdinand Verhulst, University of Marian Wiercigroch, University of Aberdeen, mensional encoding and filtering of in-band Utrecht, The Netherlands Kings College, United Kingdom; and noise will also be introduced. Superior B - Level C Alexander H-D. Cheng, University of Delaware Organizer: Epaminondas Rosa, Jr. University of Maryland, College Park CP2 11 :00 Dynamical Friction Modeling Perturbation Methods and ODEs Harry Dankowicz, Royal Institute of 10:00 Noise Filtering in Communication with Technology, Sweden Chaos 10:00 Hamiltonian G-space Normal Forms as a 11 :20 Critical Behaviour in the Torsional Epaminondas Rosa, Jr, Organizer Geometric Framework for Perturbation Buckling of Anisotropic Rods 10:30 Experimental Control of Chaos for Theory G. H M van der Heyden and J M. T. Communication Anthony Blaom, California Institute of Thompson, University College London, Scott Hayes, U.S. Army Research laboratory Technology United Kingdom 10 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Sunday, May 18 11 :40 Graph Theoretic Implications for low-dimensional dynamical systems are of 3:00 PM-5:00 PM Piecewise Linear Systems of Arbitrary limited use in these high-dimensional sys­ Dimension tems. This minisymposium explores new ap­ Magpie A & B - Level B John Hogan and Martin Homer, University proaches for characterizing and controlling of Bristol, United Kingdom the dynamics of such systems.The talks will M58 focus on large-scale properties (phase diffu­ Processing Signals from Noisy 1:30 PM-2:30 PM sion), emerging localized objects (defects, Chaotic Systems spirals, ... ) and their dynamical relevance, as Experimental observations are always subject well as the stabilization of embedded periodic Chair: Michael Tabor, University of to noise, yet most methods for their interpre­ states by time-delay feedback. Arizona tation in terms of deterministic chaos were Ballroom I, II, III - Level B Organizer: Hermann Riecke developed to work near the noise free limit; Northwestern University techniques based on Takens' s embedding theo­ IP2 3:00 Predicting Spiral Chaos in Rayleigh­ rem and the various minimization criteria were Intricate Interfaces and Benard Convection: A Phase Dynamics designed to determine the underlying dynam­ Modulated Membranes - Their Approach to the Onset and Properties ics treat noise as an afterthought. Here we of Spatiotemporal Chaos discuss new results for time series, consider­ Geometry and Nonlinear Michael C Cross, California Institute of ing large amplitude measurement (additive) Technology Dynamics noise as well as dynamical (multiplicative) Many problems in pattern formation in­ 3:30 Using Finite-Time lyapunov Dimensions noise. Talks will range from theory to applica­ volve the conformations and dynamics to Measure the Dynamical Complexity tion including an embedding theorem fornoisy of Topological Defects systems, a novel look at optimal modeling and of filaments, interfaces, and surfaces, David Egolf. Cornell University Theoretical descriptions of phenomena prediction, and signal separation applied to such as the fingering of flux domains in 4:00 Phase Diffusion in localized Spatio­ noisy ECG data. Temporal Amplitude Chaos of Para­ superconductors, chemical front motion metrically Excited Waves Organizers: Mark Muldoon, University in gel reactors, and propagating shape Glen D. Granzow, Northwestern University; of Manchester Institute of Science and transformations in lipid vesicles all share and Hermann Riecke, Organizer Technology, United Kingdom; and common structures of global conserva­ Holger Kantz, Max Planck Institute for 4:30 Suppressing Spatio-Temporal Chaos Physics of Complex Systems, Germany tion laws, nontrivial variational prin­ Using Time-Delayed Feedback ciples, complex surface geometry, and Michael E. Bleich and Joshua E. S. Socolar, 3:00 Embedding in the Presence of strong nonlinearity and nonlocality, The Duke University; David Hochheiser and Jerome Dynamical Noise speaker will give an overview of theo­ V. Moloney, University of Arizona Mark Muldoon, Organizer; 1. P. Huke and D. retical and experimental work on these S. Broomhead, University of Manchester various systems, highlighting someemerg­ Institute of Science and Technology, United 3:00 PM-5:00 PM Kingdom ing unifying principles as well as important open problems in theory, experiment, and Ballroom II - Level B 3:30 Markov Chain Monte Carlo Methods in computation. Particular attention is given Nonlinear Signal Processing to conceptual and computational lessons M57 Michael E. Davies, University College London, United Kingdom learned from the connections between Dynamical System Methods for integrable systems and the differential ge­ Oceanic and Atmospheric Flows 4:00 Modeling Noisy Chaotic Data ometry of curve motion. Holger Kantz, Organizer; and Lars Jaeger, (Part II of II) Max Planck Institute for Physics of Complex Raymond E. Goldstein Systems, Germany Department of Physics, University of (For description see Part I, MS2.) Organizer: Roberto Camassa 4:30 Processing of Noisy Nonlinear SignalS: Arizona Los Alamos National Laboratory The Fetal ECG Thomas Schreiber and Marcus Richter, 3:00 A leading-Order Singular Hamiltonian University of Wuppertal, Germany; and Perturbation Method in Fluid Dynamics Daniel T. Kaplan, Macalester College 3:00 PM-5:00 PM On no Bokhove, Woods Hole Ballroom I - Level B Oceanographic Institution 3:30 On Hamiltonian Balanced Models of 3:00 PM-5:00 PM M56 Atmosphere-Ocean Eddy Dynamics Wasatch A & B - Level C Spatio-Temporal Chaos: Character­ Ian Roulstone, UK Meteorological Office, ization and Control United Kingdom M59 4:00 Statistical-Dynamical Methods in Nonlinear Oscillators Many spatially extended physical systems Atmospheric Prediction exhibit complex behavior in time as well as Joe Tribbia, National Center for Atmospheric The purpose of this mini symposium is to space. Such spatio-temporal chaos has been Research review recent advances in the theory of non­ observed in fluid convection, chemical reac­ 4:30 Averaging and Reduced Dynamics in linear oscillators. This will include systems of tions, wide-area lasers, biological aggrega­ Simple Atmospheric Models coupled oscillators as well as oscillators with tion patterns, etc. and can be described by Ojoko Wirosoetisno and Theodore G. quasiperiodic forcing. These topics have ap­ deterministic PDE, Concepts appropriate for Shepherd, University of Toronto, Canada plications to many areas, including laser de- Conference Program DYNAMICAL SYSTEMS " 11

CONFERENCE PROGRAM un ay, ay 18 sign, biological processes involving rhythmi­ 3:00 Use of Mathematical Methods for 3:00 PM-5:00 PM cal phenomenon, such as locomotion and heart­ Protocol Design in Cancer beat, as well as traditional mechanics applica­ Zvia Agur, Tel Aviv University, Israel Chair: Steven J. Schiff, Children's tions, for example to forced pendula.The math­ 3:30 Clinical and Mathematical Aspects of National Medical Center ematical methods involved in these presenta­ Resetting and Entraining Reentrant Maybird - Level C tions will include asymptotic analysis as well Tachycardia as numerical simulation. The dynamical phe­ Leon Glass, McGill University, Canada CPS nomena will include periodic, quasiperiodic, 4:00 Modeling the Pupil Light Reflex with Unstable Periodic Orbits and chaotic behavior. The intended audience Delay Differential Equations John G. Milton, University of Chicago; and 3:00 Characterization of Blowout Bifurcation will include researchers interested in dynami­ by Unstable Periodic Orbits cal systems and applications. Jacques Belair, Universite de Montreal, Canada Yoshihiko Nagai and Ying-Cheng Lai, Organizer: Richard H. Rand University of Kansas 4:30 Spectral Properties of the Cornell University Tubuloglomerular Feedback System 3:20 Hopf's last Hope: Spatia-Temporal 3:00 Symmetric and Asymmetric Dynamics of Harold E. LByton, Duke University; and E. B. Chaos in Terms of Unstable Recurrent Linearly Coupled van der Pol Oscillators Pitman and L. C. Moore, State University of Patterns Duane W Storti, Per G. Reinhall, and David New York, Stony Brook F. Christiansen, P. Cvitanoviac, and V. M. Sliger, University of Washington Putkaradze, Niels Bohr Institute, Denmark 3:30 New Developments in Forced and 3:00 PM-5:00 PM 3:40 Detecting Unstable Periodic Orbit in Coupled Relaxation Oscillations Experimental Data Immersed·Boundary Method Chair: Anne Niculae, Northwestern Paul T. So, Children's National Medical Mark Levi, Rensselaer Polytechnic Institute University Center; Edward Ott, University of Maryland, College Park; Steven J. Schiff, Children's 4:00 On the Jump Phenomenon in Coupled Superior B - Level C National Medical Center; Tim Sauer, George Oscillators Mason University; and Celso Grebogi, . Adriaan H. P van der Burgh and Mark CP4 University of Maryland, College Park Huiskes, Delft University of Technology, The Nonlinear Optics and Solitons I Netherlands 4:00 From Billiards to Brains - Applying Periodic Orbit Theory to Mammalian 4:30 Transition Curves in the Quasiperiodic 3:00 Phase Dynamics in Optical Communica- Neuronal Networks Mathieu Equation tions Steven Schiff, Children's National Medical Randolf S. Zounes, Cornell University; and Gregory G. Luther, University of Notre J. Center Richard H. Rand, Organizer Dame and BRIMS, Hewlett-Packard Laboratories, United Kingdom; Mark S. Alber, University of Notre Dame; Jerrold E. 3:00 PM-5:00 PM Marsden, California Institute of Technology; 3:00 PM-5:00 PM and Jonathan Robbins, BRIMS, Hewlett­ Chair: Colin Sparrow, University of Ballroom III - Level B Packard Laboratories, United Kingdom and Cambridge, United Kingdom MS10 Bmtol University, United Kingdom Superior A - Level C 3:20 Timing Jitter Reduction in a Fiber laser Mathematics and Medicine: From Mode-locked by an Input Bit Stream CP6 the Laboratory to the Clinic Anne Niculae and William L. Kath Maps Northwestern University , This minisymposium will illustrate the power 3:00 The Dynamics of Some Digital Filters of combining mathematics and modeling in 3:40 Pulse Dynamics in Fiber lasers J. Nathan Kutz and Philip Holmes, Princeton Charles Tresser, IBM T. J. Watson Research the biological/medical sciences by examining Center four major areas: cardiology, neurology, kid­ University; Michael Weinstein, University of Michigan, Ann Arbor; and Keren Bergman, ney dynamics, and treatment strategies in 3:20 Dynamic Period·Doubling Bifurcations Princeton University of a Unimodal Map cancer. Each of these is of importance experi­ 4:00 Optical Pulse Dynamics in Dispersion Huw G. Davies and Krishna Rangavajhula, mentally and clinically. Current modeling University of New Brunswick, Canada work in these areas ranges from the use of Managed Fibers Tian-Shiang Yang and William L Kath, 3:40 Collapsing of Chaos (nonlinear) finite difference equations through Northwestern University Guocheng Yuan and James A. Yorke, ordinary differential equations and differen­ University of Maryland, College Park tial delay (functional) equations. Unfortunately 4:20 Switching-Induced Timing Jitter in Nonlinear Optical loop Mirrors in almost every specific modeling situation, 4:00 Nonanalytic Perturbation of Complex Michael J. Mills and William L Kath, Analytic Maps little is known about the mathematical proper­ Northwestern University James Montaldi, Institut NonUneaire de ties of the model and thus (fortunately) there Nice, France; and Bruce B. Peckham is usually the opportunity for breaking new 4:40 Nonlinear Polarization-Mode Dispersion in Optical Fibers with Randomly University of Minnesota, Duluth ' ground in mathematical research. In the course Varying Birefringence 4:20 Finding All Periodic Orbits of Maps of their talks the speakers will illustrate some William L. Kath, Northwestern University; P. Using Newton Methods: Sizes of Basins of these possibilities. K. A. Wai, The Hong Kong Polytechnic Jacob R. Miller, Shippensburg University; Organizer: Michael C. Mackey University, Hong Kong; Curtis R. Menyuk and James A. Yorke, University of Maryland, McGill University, Canada and J. W. Zhang, University of Maryland, College Park Baltimore County 12 " DYNAMICAl SYSTEMS Conference Program

CONFERENCE PROGRAM Sunday, May 18 4:40 Dynamics of Non-Expanding Maps and 7:00 PM-9:00 PM much simpler 3 or even 2 component models, Applications in Discrete Event Theory designed to explore crucial processes in the Colin Sparrow, University of Cambridge, Ballroom II - Level B simplest context.The interplay of this light United Kingdom sensitive biology with fluid flow has been MS12 little explored as yet and the minisymposium 7:00 PM-9:00 PM New Results for Stochastic PDEs will bring together some of the existing ap­ Ballroom I - Level B Recent developments in the analysis of sto­ proaches. chastic PDEs are highlighted by the speakers Organizer: John Brindley MSll in this mini symposium. Rigorous connections University of Leeds, United Kingdom Spiral-Wave Domains and Fila­ between interacting particle systems and mea­ 7:00 Diurnal Vertical Migration, A Mecha- ments: Experiments, Numerics, sured valued process formally described by nism for Patchiness in Shear Flows parabolic stochastic PDEs with multiplicative John Brindley, Organizer; and Louise Theory noise are discussed, as well as issues of Matthews, University of Leeds, United The problems of spiral-wave stability and wavefront propagation in such models. Com­ Kingdom dynamics in two and three dimensions benefit plimentary results for wavefront propagation 7:30 Vertical Migration in Phytoplankton­ from the interaction of different approaches in hydrodynamic equations, in particular the Zooplankton Interactions and on the feedback from its diverse applica­ Burgers equation, with random data are also Kathleen Crowe, University of California, tions. The speakers in this mini symposium presented, and a study ofthe effect of noise on Davis bring to bear a variety of methods on these bifurcations in spatially distributed systems is 8:00 Chaotic Advection and Plankton problems. Methods employed include experi­ described. Patchiness mental studies of chemical reactions and of Organizer: Charles R. Doering Igor Mezic. University of California, Santa specially designed excitable media, numeri­ University of Michigan, Ann Arbor Barbara 8:30 Spatial Structure in Oceanic Plankton cal work on amplitude and few-species reac­ 7:00 Measure-Valued Processes and Populations tion-diffusion equations, and in one case, com­ Stochastic Modeling of Interacting parison of numerics with biological patterns. Particle Systems Horst Malchow, University of Osnabrueck, Germany Applications are a main feature of this Donald Dawson, Fields Institute, Canada minisymposium. At least three of the speakers 7:30 Width of Wavefronts for Random present experimental data, and two speakers Travelling Waves 7:00 PM-9:30 PM will present numerical simulations of three­ Carl Mueller, University of Rochester; and Superior A - Level C dimensional patterns. Since the methods vary Roger Tribe, University of WaNVick, England wildly among the speakers, we are interested 8:00 Front Propagation in Noisy Burgers MS14 in the spectrum of problems encountered and Type Equations Ulams Conjecture and the Ap­ surmounted. Jan Wehr and Jack Xin, University of Arizona Organizer: Gregory Huber 8:30 Effect of Noise on Bifurcations in proximation of Invariant Measures University of Chicago Spatially Extended Systems Continued successful interaction between com­ 7:00 Scaling Relations in Chemical Spirals: Grant Lythe, Los Alamos National Laboratory putational observations of random-like be­ Selection and Dispersion havior and mathematically rigorous descrip­ Andrew Belmonte, University of Pittsburgh; 7:00 PM-9:00 PM tion of dynamical systems will depend on the Jean-Marc Flesselles, Laboratoire PMMH, development of methods for computing the ESPCI, France; Vilmos Gaspar, Kossuth Lajos Ballroom III - Level B invariant measures of invariant sets whose University, Hungary; and Oi Ouyang, accuracy can be assessed. This minisymposium Institute Non-Linea ire de Nice, France MS13 will be devoted to a discussion of a method 7:30 Control of Spiral Territories in Interplay of Fluid Dynamics and first proposed by Stanislaw UJam for approxi­ Dictyostelium Biology of Plankton Population mating the absolutely continuous invariant Herbert Levine, University of California, San Models measures of finite-dimensional maps. There Diego will be a discussion of convergence rates in 8:00 The Dynamics of Scroll Wave Filaments Plankton constitutes the largest component of the absolutely continuous case, and presenta­ in the Complex Ginzburg-landau the world's biomass, exerting a vital influence tion of recent results on approximation of Equation on global fluxes of C02 and other important measures and attractors where the Birkhoff Michael Gabbay. Edward Ott, and PaNez climatological gases, such as dimethyol sul­ Ergodic theorem holds almost everywhere Guzdar, University of Maryland, College Park phate, as well as forming the basis of marine (Lebesgue). Our purpose is to present an eas­ 8:30 Optical Tomography of Chemical Waves food webs. Mathematical models describing ily implemented alternative to the computa­ in Three Dimensions the complex and highly nonlinear population tion of invariant measures (histograms) by Arthur T. Winfree, University of Arizona dynamics of plant like phytoplankton and her­ box counting-a procedure for which there are bivorous (and sometimes carnivorous) zoop­ to our knowledge no general estimates. We lankton vary from many-component systems show how the method can be used to calculate of PDEs, designed for accurate simulation, to Lyapunov exponents. Highlighted will be ex- Conference Program DYNAMICAL SYSTEMS " 13

CONFERENCE PROGRAM unday, May 18 tensions of the original method to multi-di­ 8:20 Mass Exchanges Among Korteweg-de 7:20 Signal Masking Using Synchronized mensional maps and set-valued maps. Con­ Vries Solitons Chaos-Proof and Analysis by Perturba· vergence rates of the method will be discussed Peter D Miller, The Australian National tion Theory in the absolutely continuous case, and results University, Australia; and Peter L Mohamed A. Ali, Cornell University Christiansen, The Technical University of on the approximation of attractors and invari­ 7:40 Determining Coupling that Guarantees Denmark, Denmark ant measures will be presented for the singular Synchronization Between Identical case. Chaotic System Method Reggie Brown, College of William and Mary Organizer: Fern Y. Hunt 7:00 PM-9:00 PM National Institute of Standards and Chair: M. C. Depassier, Universidad 8:00 Automatic Gain Control using SynChro­ nized Chaos Technology Catolica de Chile, Chile Ned J Corron, Dynetics, Inc.. Huntsville 7:00 Computing Physical Measures of Mixing Superior B - Level C 8:20 On Generalized Synchronization of Multidimensional Systems with an Chaos in Mutually Coupled Systems Application to Lyapunov Exponents CPS Nikolai F. Rulkov, Mikhail M. Sushchik Jr., Gary Froyland, The University of Western Hamiltonian Systems and Henry D. I. Abarbanel, University of Australia, Australia 7: 00 Instabilities and Degeneracies of the 4- California, San Diego 7:30 A Finite Element Method for the Dimensional Stochastic Web 8:40 Stick-Slip Dynamics as an Elastic Frobenius-Perron Operator Equation Sergey Pekarsky. The Weizmann Institute of Excitable Media Jiu Ding, University of Southern Mississippi; Science, Israel and California Institute of Julyan H. E. Cartwright Emilio Hernandez and Aihui Zhou, Academia Sinica, People's Technology; and Vered Rom-Kedar, The Garcia, and Oreste Piro, Universitat de les Republic of China Weizmann Institute of Science, Israel Illes Balears, Spain 8:00 Approximating Attractors and Chain 7:20 On Smooth Hamiltonian Flows Limiting Transitive Invariant Sets with Ulam's to Hyperbolic Billiards Method 7:00 PM-9:00 PM Dmitry Turaev and Vered Rom-Kedar, The Fern Y. Hunt Organizer Weizmann Institute of Science, Israel Chair: Brian R. Hunt, University of 8:30 Markov Finite Approximation of Maryland, College Park 7:40 Variational Principle for Bifurcations in Frobenius-Perron Operators and One Dimensional Hamiltonian Systems Maybird - Level C Invariant Measures for Set-Valued R. D. Benguria and M C Depassier, Dynamical Systems CP10 Walter Miller, Howard University Universidad Catolica de Chile, Chile 8:00 The Hamiltonian and Lagrangian Attractors, Fractals, and Dimen­ 9:00 Cone Conditions and Error Bounds for Ulam's Method Approaches to the Dynamics of sions Nonholonomic Systems Rua D. A. Murray, University of Cambridge, Wang-Sang Koon, University of California, 7:00 Nondifferentiable Attractors, Data United Kingdom Berkeley; and Jerrold E. Marsden, California Filtering, and Fractal Dimension Institute of Technology Increase 7:00 PM-9:00 PM Louis M Pecora and Thomas L Carroll, 8:20 Approximation of KAM Tori and Aubry­ Naval Research Laboratory Chair: William L. Kath, Northwestern Mather Sets by Periodic Solutions of a 7:20 Crisis in Ouasiperiodically Forced University Perturbed Burgers' Equation Vadim Zharnitsky, Los Alamos National Systems Magpie A & B - Level B Laboratory Ulrike Feudel, Universitat Potsdam, Germany CP7 8:40 A Leading-Order Singular Hamiltonian 7:40 Invariant Graphs And Strange Perturbation Method in Fluid Dynamics Nonchaotic Attractors for Nonlinear Optics and Solitons II Onno Bokhove, Woods Hole Ouasiperiodically Forced Systems Jaroslav Stark, University College London, 7:00 Noisy Ouas~Periodicity in the Simula- Oceanographic Institution United Kingdom tion of the Discrete Nonlinear Schrodinger Equation 7:00 PM-9:00 PM 8:00 Blowout Bifurcation Route to Strange Makoto Umeki, University of Tokyo, Japan Nonchaotic Attractors 7:20 Soliton Stability in Optical Transmission Chair: Reggie Brown, College of William Tolga Yalccinkaya and Ying-Cheng Lai, University of Kansas Lines using Semiconductor Amplifiers and Mary and Fast Saturable Absorbers Wasatch A & B - Level C 8:20 Which Dimension of Fractal Measures S. K. Turitsyn, Heinrich Heine Universitat are Preserved by Typical Projections? Dusseldorf, Germany CP9 Vadim Yu. Kaloshin. Princeton University; 7:40 Vector Solitons and Their Internal Synchronization I and Brian R. Hunt. University of Maryland, College Park Oscillations 7:00 New Bifurcations from an Invariant Jianke Yang, University of Vermont 8:40 Fractal Dimensions of Chaotic Saddles Subspace of Dynamical Systems 8:00 The Application of the Direct Scattering Philip} Aston and Peter Ashwin, University Brian R. Hunt. Edward Ott, and James Transform to the NRZ-to-Soliton Data A. of Surrey, United Kingdom; and Matthew Yorke, University of Maryland, College Park Conversion Problem Nicol, University of Manchester Institute of S. Burtsevand Roberto Camassa, Los Alamos Science and Technology, United Kingdom 9:00 Determining General Fractal Dimen­ National Laboratory; and P. Mamyshev, Bell sions from Small Data Sets Laboratories, Lucent Technologies A. J Roberts, University of Southern Queensland, Australia 14 " DYNAMICAl SYSTEMS Conference Program

CONFERENCE PROGRAM Monday, May 19 8:30 AM-9:30 AM where turbulent surface waves are formed and 10:00 AM-12:00 PM sustained by external forcing. Also here mea­ surements of self-diffusivity and relative Ballroom II - Level B Chair: Robert Borrelli, Harvey Mudd diffusivity of floating particles reveal a non­ MS17 College Brownian motion.We wish to compare the results from upper-ocean studies with those Dynamics of Cortical Neural Ballroom I, II, III - Level B from laboratory experiments and with related Networks {Part I of "I theoretical results for wave turbulence. IP3 Recent theoretical studies have been actively Teaching Dynamical Systems Organizer: Preben Alstrom engaged in understanding collective phenom­ to Undergraduates Niels Bohr Institute, Denmark ena oflarge neural networks, that are involved 10:00 Drifter Trajectories in Geophysical in different behavioral states and sensory in­ For the past several years, we have of­ Flows formation processing. In this mini symposium, fered a two-part course in dynamical Antonello Provenza Ie, Istituto di eight talks in two related sessions will be systems to lower-level undergraduate Cosmogeofisica, Italy focused on neurodynamical processes such as students at Boston University. The first 10:30 Motion of Floating Particles on a synchronous oscillations and wave propaga­ course is a dynamical systems oriented Turbulently Driven Fluid tion. We would like to illustrate how a novel version ofthe sophomore-level differen­ Walter I. Goldburg and Cecil Cheung, network behavior may arise due to particular tial equations course. In this course, many University of Pittsburgh aspects of complex single neuron dynamics of the exact methods of solution of ODES 11 :00 Particle Motion in Capillary Surface and/or synaptic circuits, such as neural firing have been replaced by more numerical Waves patterns (e.g. spike adaptation, bursting), syn­ and qualitative techniques. The second Preben Aistrom, Organizer; Mogens T. aptic kinetics and depression, cortical local course is asophomore-junior-level course Levinsen and Elsebeth Schroder, Niels Bohr circuit, random sparse connectivity, and re­ Institute, Denmark in discrete dynamics. We view this course current inhibition. as a prelude to the standard undergradu­ ate real analysis course. Students are Organizers: Xiao-Jing Wang, Brandeis 10:00 AM-12:00 PM University; David Terman, Ohio State introduced to such topics as abstract Ballroom I - Level B University, Columbus; and John Rinzel, metric spaces, dense sets, continuity, etc. National Institutes of Health in the fairly concrete arena of iteration of MS16 real and complex functions. In this pre­ 10:00 Rhythms and "Lurching" Waves in the Phase Dynamics and Synchroniza­ "Sleeping" Thalamic Slice sentation, the speaker will discuss some John Rinzel and Xiao-Jing Wang, of the issues involved in organizing such tion in Chaotic Systems Organizers; and David Golomb, Ben-Gurian courses, including relations with client Since the time of Huygens' pioneering work University, Israel disciplines. on synchronization, there has been much 10:30 SynChronous and AsynChronous States Robert L. Devaney progress in synchronization phenomena, also in a Network of Spiking Neurons Department of Mathematics of chaotic systems. The speakers in this Wulfram Gerstner, Swiss Federal Institute of Boston University minisymposium will present new phenomena Technology. Switzerland of phase synchronization and their relation to 11 :00 Image Segmentation Based on complete synchronization. The emphasis is also Oscillatory Correlation 10:00 AM-12:00 PM on phase dynamics in complex spatio-tempo­ David Terman, Organizer; and DeUang ral structure formation. Wang, Ohio State University Magpie A & B - Level B Organizer: Jiirgen Kurths 11 :30 Saddle-Node Bifurcations and Wave Universitiit Postdam, Germany Propagation in Model Cortical Column MS15 Structures Diffusion and Turbulence on 10:00 Phase Synchronization of Chaotic Frank C. Hoppensteadt, Arizona State Systems University Water Surfaces JOrgen Kurths, Organizer The purpose of the mini symposium is to focus 10:30 Synchronization Transitions in Coupled on the motion of particles, floats, and drifters Chaotic Oscillators: From Phase to 10:00 AM-12:00 PM on a turbulent water surface. Results obtained Compete Synchronization Ballroom III - Level B for the self-diffusivity by tracking drifters in Michael Rosenblum, Universitat Potsdam, the ocean and for the relative diffusivity by Germany and Russian Academy of Sciences, MS18 tracking pairs of drifters all show that the Russia Pattern Formation and Singular motion is far from being Brownian. In the 11 :00 Long-Range Rotating Order in Perturbations geophysical dynamical regime, at time scales Extensively-Chaotic Systems between roughly 1 day and 10 days and length Hugues Chate, Centre d'Etudes de Sa clay, Many physical systems exhibit dynamics on scales between roughl y 10 km and 100 km, the France multiple time scales and/or multiple length drifter motion possesses significant persis­ 11 :30 Spiral Waves in Chaotic Systems scales. Examples arise in nonlinear fiber op­ tence ('memory'), which is not present in Raymond Kapral, University of Toronto, tics, multi-degree-of-freedom Hamiltonian ordinary Brownian motion. A similar persis­ Canada systems, chemical reactions, fluid dynamics, tence is observed in laboratory experiments, and neurophysiology. In this minisymposium, the speakers will address important current Conference Program DYNAMICAl SYSTEMS " 15

CONFERENCE PROGRAM Monday, May 19 nonlinear dynamics questions in problems optimal paths plays a role similar to that of the 10:00 AM-12:00 PM from several of these areas. For systems mod­ phase portrait in nonlinear dynamics. For eled by reaction diffusion equations, spatial nonequilibrium physical and chemical sys­ Chair: Yuri Latushkin, University of patterns evolving in complex time-dependent tems this pattern generically has singular fea­ Missouri, Columbia ways will be discussed. In addition, a number tures. Their analysis and applications of the Superior A - Level C of speakers will focus on the mathematical results to the problems of escape from a meta­ theory and applications of the existence and stable state and signal enhancement by fluctu­ (P12 stability of time asymptotic stationary pat­ ating systems will be discussed. Invariant Measures and Ergodicity terns. These states include traveling waves, Organizer: Mark I_ Dykman 10:00 An Exact Formula for the Essential their concatenations, and spatially periodic Michigan State University Spectral Radius of the Matrix Ruelle states. In those systems modeled by Hamilton's 10:00 The Stochastic Manifestations of Chaos Operator on Spaces of Hoelder and equations, speakers will focus on the exist­ Linda E. Reichl. University of Texas, Austin Differentiable Vector-Functions ence and stability of a diverse array of mul­ Yuri Latushkin, University of Missouri, tiple pulse orbits and where they arise in 10:30 Signal Enhancement by Large Columbia Fluctuations applications. Mathematically, the methods Peter V, E. McClintock, Lancaster University, 10:20 Transfer Operators in Conformal used in these studies range from variational United Kingdom Dynamics techniques and matched asymptotic expan­ Marius Urbansku, University of North Texas sions to analytical and geometrical methods in 11 :00 Periodic Modulation of the Rate of Noise-Induced Escape through an 10:40 On the Lax-Phillips Scattering Theory of stability theory and singular perturbation Unstable Limit Cycle Uniformly Hyperbolic Chaotic Dynamics theory. Theoretical tools from complex dy­ Robert S, Maier, University of Arizona Thomas J. Taylor, Arizona State University; namics and electromagnetism theory also play and Duk-Hyung Lee, Indiana Wesleyan 11 :30 Critical Behavior of the Distribution of University significant roles. Fluctuational Paths Organizers: Arjen Doelman, University Mark I, Dykman, Organizer; and Vadim N, 11 :00 Convergence of the Transfer Operator of Utrecht, The Netherlands; Tasso Smelyanskiy, Michigan State University for Rational Maps Kaper, Boston University; and Todd Nicolai Haydn, University of Southern Kapitula, University of New Mexico, California Albuquerque 10:00 AM-12:00 PM 10:00 Singular Limits of Phase Equations for Chair: Manfred F. Goz, Princeton 10:00 AM-12:00 PM Patterns Far from Threshold University Nicholas M. Ercolani and Robert Indik, Maybird - Level C Chair: Adam S. Landsberg, Haverford University of Arizona; Alan Newell, College University of Warwick, England; and Thierry (Pll Superior B - Level C Passot Observatoire de Nice, France Nonlinear Waves 10:30 Self-Replicating Spots (P13 john Pearson, Los Alamos National 10:00 Recent Progress in Mult~Dimensional Applications in Physics Laboratory Mode Conversion Allan N. Kaufman, Alain 1. Brizard, and jim 10:00 Relaxation of Magnetic Knots to 11 :00 Stability Analysis of Pulse Solutions of Minimal Braids the Gray-Scott Model Morehead, Lawrence Berkeley National Laboratory; and Eugene R, Tracy, College Renzo L. Ricca, University College London, Robert Gardner, University of United Kingdom Massachusetts, Amherst of William and Mary 10:20 Bifurcations and Instabilities in Moving 11 :30 Oscillatory Instabilities of Labyrinthine 10:20 Local Singularities in Breaking Waves Daniel p, Lathrop, Emory University Vortex Lattice Fronts Igor Alanson and Valerii Vinokur, Argonne David Muraki, Courant Institute of 10:40 A Robust Heteroclinic Cycle in an 0121x National Laboratory Mathematical Sciences, New York University Zz Steady-State Mode Interaction p, Hirschberg and Edgar Knobloch, 10:40 Flux Creep in 2-D Josephson Junction University of California, Berkeley Arrays 10:00 AM-I 2:00 PM Adam S. Landsberg, Haverford College; and 11 :00 Two-Period Ouasiperiodic Attractors of Kurt Wiesenfeld, Georgia Institute of Wasatch A & B - Level C Weakly Nonlinear Gas Dynamics Equations Technology MS19 Michael G. Shefter and Rodolfo R, Rosales, 11 :00 Geometrical Phase in Twisted Pairs Massachusetts Institute of Technology used in Communication Networks Nonlinear Dynamics of Large Mayank Sharma and D. Subbarao, Indian Fluctuations 11 :20 The Strongly Attracting Character of Institute of Technology, India Large Amplitude Nonlinear Resonant Substantial progress has been made recently Acoustic Waves Without Shocks 11 :20 Transonic Solutions for a Ouantum in understanding large fluctuations in nonlin­ Dimitri D. Vaynblat and Rodolfo R, Rosales, Hydrodynamic Semiconductor Model ear systems. It came through application of Massachusetts Institute of Technology Peter Szmolyan, Technische Universitiit Wien, Austria the methods of nonlinear dynamics and catas­ 11 :40 A Nonlinear Wave Equation Arising in trophe theory. An important concept in de­ Two-Phase Flow 11 :40 Synchronization by Disorder in Parallel scribing large fluctuations is the most prob­ Manfred F, Goz, Princeton University Arrays of Josephson Junctions able, or optimal, fluctuational path to a given Yuri Braiman, Emory University state. For large fluctuations, the pattern of 16 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Monday, May 19 1:30 PM-2:30 PM 4:00 PM-6:00 PM have attracted increasing attention in the dy­ namics community. At the same time, experi­ Chair: Peter Bates, Brigham Young Univer­ Magpie A & B - Level B mental progress on these superconducting cir­ sity MS20 cuits has blossomed. These two lines of in­ Ballroom I, II, III - Level B quiry are beginning to overlap in an essential Computing Invariant Manifolds way. This minisymposium features three lead­ IP4 Geometric desciptions of dynamical systems ing experimentalists in the field to report both Scaling Exponents in Turbulence include varied types of invariant manifolds, recent results and open questions. The pur­ pose is to focus attention on issues that arise in The speaker will discuss some of the current including stable and unstable manifolds of the study of real Josephson junction arrays, as issues regarding scaling exponents in equilibrium points and periodic orbits, nor­ described by the foremost applied practitio­ hydrodynamical equations, including passive mally hyperbolic invariant manifolds, mani­ ners in the field. (These are speakers the scalars, active scalars and two dimensional folds of equilibria in families and bifurcation dynamics community is rarely exposed to.) turbulence. sets in multiparameter families. This mini symposium will highlight computational The intended audience is anyone interested in Peter S. Constantin methods to determine invariant manifolds of real world applications of coupled oscillator Department of Mathematics dimension larger than one. The algorithmic systems. University of Chicago basis for such computations is not firmly Organizer: Kurt A. Wiesenfeld established and is undergoing rapid evolution. Georgia Institute of Technology 2:30 PM-3:30 PM The problems entail significantly greater geo­ 4:00 Generation of Submillimeter Wave metric complexity than is encountered in the Radiation Using Unear Phase-locked Chair: John Rinzel, National computation of invariant curves. The speak­ Josephson Junction Arrays Institutes of Health ers will describe their experiences in imple­ James Lukens, State University of New York, Stony Brook Ballroom I, II, III - Level B menting calculations of two dimensional in­ variant manifolds on example problems. 4:30 Josephson Junction Arrays: Practical IPS Organizer: John Guckenheimer Devices and Nonlinear Dynamics Cornell University Terry P. Orlando, Massachusetts Institute of Stochastic Resonance in Technology Sensory Biology 4:00 Computation of Heteroclinic Two­ Dimensional Invariant Manifold 5:00 Dynamical Properties of Two-Dimen­ In certain nonlinear systems, of which Interactions sional Josephson-Junction Arrays neurons are an example, the addition of Mark E. Johnson, Princeton University; A. B. Cawthorne, P. Barbara, and Chris J. random fluctuations, or "noise", can en­ Michael S. Jolly, Indiana University, Lobb, University of Maryland, College Park hance the detection and transmission ef­ Bloomington; loannis G. Kevrekidis, ficiency of the neural networks designed Princeton University; and John Lowengrub, 4:00 PM-6:00 PM to perceive weak stimuli. This University of Minnesota. Minneapolis Wasatch A & B - Level C counterintuitive statistical process, called 4:30 Computing Invariant Manifolds of "stochastic resonance" (SR), is well es­ Saddle-Type MS22 tablished in a variety of physical sys­ Hinke Osinga, University of Minnesota, tems. Recently it has been observed in Minneapolis Integrable Systems Methods for the sensory nervous systems of two 5:00 Computing Stable Sets of Noninvertible Curve Evolution arthropods, crayfish and crickets, and Mappings This minisymposium proposes to discuss dif­ Frederick J. Wicklin and Chia-Hsing Nien, may be deeply linked to the evolution of ferent aspects of the interaction between soliton University of Minnesota, Minneapolis all sensory networks. equations and the properties and dynamics of This talk will focus on the experimental 5:30 Stable and Unstable Manifolds of Halo space curves. Interest in modeling curves and Orbits in the Circular Restricted Three­ knots by means of completely integrable equa­ observations at two levels: the individual body Problem tions has revived in recent years with applica­ sensory neuron, and the next higher net­ Robert Thurman and Patrick Worfolk, work level, the terminal ganglion,and University of Minnesota, Minneapolis tions to topological fluid dynamics, the evolu­ introduce human perception of SR in tion of vortex filaments and vortex patches, noisy visual images. A simple statistical and to the modeling of DNA and macromol­ theory will be used throughout to inter­ 4:00 PM-6:00 PM ecules. This minisymposium is directed to an pret the experimental results. Ballroom I - Level B audience whose main interests are nonlinear Frank Moss dynamics and completely integrable equa­ Center for Neurodynamics and MS21 tions, knot theory and its applications in biol­ Department of Physics and As­ Josephson Junction Arrays: ogy and fluid dynamics, applied dynamical tronomy Progress and Puzzles systems and their connections to differential University of Missouri, St. Louis geometry and topology of curves and sur­ Josephson junction arrays are superconduct­ faces. Its principal purpose is to encourage ing nonlinear circuits which hold promise for applications of well-established techniques of a variety of applications in extreme high fre­ integrable systems to a variety of problems in quency electronics. As a paradigm of coupled which curve dynamics and knots arise. It also oscillator systems, Josephson junction arrays advocates that, on the one hand, a geometric Conference Program DYNAMICAL SYSTEMS " 17

CONFERENCE PROGRAM Monday, May 19 interpretation of soliton equations can shed 4:00 PM-6:00 PM 4:00 PM-6:00 PM new light on their structure; on the other hand, their large families of special solutions, such Ballroom III - Level B Chair: Anne C. Skeldon, City University, as the solitons and the multi -phase solutions, United Kingdom can provide an important tool for the study of MS24 Maybird - Level C the topological properties of related curves. Teaching Differential Equations CP14 Organizers: Annalisa Calini, University Dynamically Convection and Hexagonal of Charleston; and Thomas Ivey, Case The advent in the early 80s of widely acces­ Western Reserve University sible computer graphics revolutionized dif­ Patterns 4:00 Geometric Realizations of Fordy-Kulish ferential equations, and therefore their teach­ 4:00 Takens-Bogdanov Bifurcation on the Integrable Systems, Sym-Pohlmeyer ing. Now we easily see many solutions simul­ Hexagonal Lattice for Double-Layer Curves, and Related Variation Formulas taneously, even for equations that do not have (part I) Convection Joel Langer, Case Western Reserve solutions in closed form. Visual perspective Yuriko Y. Renardy and Michael Renardy, University; and Ron Perline, Drexel and geometric content extend our capabilities; Virginia Polytechnic Institute and State University we can handle more complicated equations University; and Kaoru Fujimura, Japan and more general dynamical systems. Bifur­ Atomic Energy Research Institute, Japan 4:30 Geometric Realizations of Fordy-Kulish Integrable Systems, Sym-Pohlmeyer cation and chaos are no longer relegated to 4:20 Non-Potential Effects in Pattern Curves, and Related Variation Formulas graduate school; these topics are common­ Formation (part II) place in introductory courses. Far beyond our Alexander Golovin, Northwestern University; Joel Langer, Case Western Reserve classrooms, the new approach helps scientists Asaf Hari, Alexander Nepomnyaschchy, University; and Ron Periine, Drexel in numerous programs not traditionally math­ Alexander NUl, and Leonid Pismen Technion-Israel Institute of Technol~gy, University ematical (e.g., FDA, NIH, U.S. Geological Israel 5:00 Homotopies and Knot Types of Elastic Survey) to make models with dynamical sys­ Rods tems and experiment with parameters. This 4:40 Multiplication of Defects in Hexagonal Thomas Ivey, Organizer minisymposium will be of interest to nonaca­ Patterms A. A. Nepomnyashchy, Technion-Israel 5:30 Backlund Transformations of Knots of demic scientists who want to know what is Institute of Technology, Israel; P. Colinet Constant Torsion going on in the classroom, and how they or and J. C. Legros, Universite Libre de Annalisa Calini and Thomas Ivey, Organizers their COlleagues might access it, as well as our Bruxelles, Belgium university colleagues. 5:00 Patterns in Long Wavelength Convec­ 4:00 PM-6:00 PM Organizer: Beverly H. West tion Harvey Mudd College and Cornell Anne C. Skeldon, City University, United Ballroom II - Level B University Kingdom; and Mary Silber, Northwestern MS23 4:00 ODE Architect: CODEE'S Multimedia University Package 5:20 Ginzburg-Landau Equations for Dynamics of Cortical Neural Robert L. Borrelli, Harvey Mudd College Hexagonal Patterns Networks (Part II of II) 4:30 How to Make a Pendulum Do Anything Bias Echebarria and Carlos Perez-Garcia Universidad de Navarra, Spain ' (For description, see Part I, MSI7.) You Want Organizers: Xiao-Jing Wang, Brandeis John H. Hubbard, Cornell University 5:40 The Takens-Bogdanov Bifurcation in University; David Terman, Ohio State 5:00 Teaching ODE's with the WWW and A Long-Wave Theory University, Columbus; and John Rinzel, Calculator-Based Laboratory Jean-Luc Thiffeault, University of Texas, National Institutes of Health Frank Wattenberg, Carroll College and Austin; and Neil J. Balmforth, University of Nottingham, United Kingdom 4:00 Synaptically Induced Bistability and Montana State University Transition to Repetitive Activity 5:30 Visualization and Verification, with Bard Ermentrout and Jim Uschock, Mathematica, of Solutions to Differen­ 4:00 PM-6:00 PM University of Pittsburgh tial Equations Chair: Tasso Kaper, Boston University 4:30 Modeling Cortical 40 Hz Oscillations: Stan Wagon, Macalester College Superior B - Level C Pacemaker Neurons and Synaptic Synchronization CP15 Xiao-Jing Wang, Organizer 5:00 Pattern Generation by Two Coupled Pattern Formation and Singular Time-Discrete Neural Networks with Perturbations II SynaptiC Depression Walter Senn and Jurg Streit, University of 4:00 An Accurate Description of Slow-Motion Bern, Switzerland Dynamics in Singularly Perturbed Reaction-Diffusion Systems 5:30 The Role of Spike Adaptation in Bjorn Sandstede, Weierstrass Institute for Shaping Spatiotemporal Patterns of Applied Analysis and Stochastics, Germany Activity in Cortex David Hansel, Centre de Physique 4:20 IrregUlar Jumping in the Perturbed Theorique, Ecole Polytechnique, France Nonlinear Schrodinger Equation George Haller, Brown University 18 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Monday, May 19 Tuesday, May 20 4:40 Singularly Perturbed and Non-Local 10:00 Title to be determined Modulation Equations 8:30 AM-9:30 AM L. Mahadevan, Massachusetts Institute of V Rottschafer and A. Doelman, University of Technology Utrecht, The Netherlands Chair: Kurt Wiesenfeld, Georgia 10:30 Nonlinear Dynamics of Elastic Filament 5:00 Multiple-Pulse Homoclinic and Periodic Institute of Technology Alain Goriely and Michael Tabor, University Orbits in Singularly Perturbed Systems Ballroom I, /I, /II - Level B of Arizona Cristina Soto-Trevino, Boston University 11 :00 Curve Dynamics and Gene Regulation 5:20 Homotopy Classes for Stable Connec­ IP6 Gadi Fibich and Danny Petrasek, University tions between Hamiltonian Saddle­ Bucket Brigade Production lines of California, Los Angeles; and Isaac Focus Equilibria Klapper, Organizer Bucket brigade manufacturing is a way of R. VanderVorst, Georgia Institute of 11 :30 Removing the Stiffness of Curvature in Technology and University of Leiden, The organizing workers on a production line, in which there are fewer workers than stations 3-D Filament Calculations Netherlands; W. D. Kalies, California Thomas Y. Hou, California Institute of and each worker carries his work from Polytechnic State University, San Luis Technology; Isaac Klapper, Organizer; and Obispo; and J Kwapisz, Georgia Institute of station to station. The order on a line is Hui Si, California Institute of Technology Technology fIxed, We address the problem of the opti­ 5:40 Existence and Stability of Singular mal sequencing of workers on a line so that Heteroclinic Orbits for the Ginzburg­ the production rate of such line would be 10:00 AM-12:30 PM Landau Equation maximal. The performance of such pro­ Ballroom /I - Level B Todd Kapitula, University of New Mexico, duction lines can be effectively described Albuquerque by the dynamical system defIned on a MS26 simplex. The application of ideas of dy­ Random Dynamical Systems namical systems theory resulted in increased 4:00 PM-6:00 PM This minisymposium will focus on both theory production rate of such lines in the apparel and applications of random dynamical sys­ Chair: Brian F. Farrell, Harvard Univer­ industry. The randomized version of this sity tems. During the past ten years there has been model proved to be effIcient for coordinat­ Superior A - Level C a great interest and real progress in stochastic ing pickers in the warehouses that resulted bifurcation theory, stochastic normal form in the essential increase of pick rates. The CP16 theory and control of stochastic systems. This speaker will explain why this happened. Fluids I progress in the theory has led to the develop­ Leonid A. Bunimovich ment ofreliable numerical algorithms for com­ 4:00 Integrability and Non-Integrability in Southeast Applied Analysis Center puting characteristic quantities in stochastic Bubble Dynamics and School of Mathematics dynamical systems. The speakers in this Alexander R. Galper, Te/-Aviv University, Israel Georgia Institute of Technology minisymposium will address the theoretical 4:20 Asymptotic Focusing of Particles in a developments in the area of stochastic bifur­ Wake Flow cations, stochastic functional equations, and Burns, R. W. Davis and F. Moore, T. J. E. random attractors; and the applications of National Institute of Standards and 10:00 AM-12:00 PM random dynamical systems to engineering Technology Ballroom I - Level B systems. 4:40 Perturbation Growth in Non-Autono­ mous Geophysical Flows MS25 Organizer: N. Sri Namachchivaya Petros J Ioannou, Harvard University Dynamics of Curves and Filaments University of Illinois, Urbana 5:00 Non-Normal Dynamics of Eddy Variance The motion and dynamical behavior of curves 10:00 The Role of Attractors in Stochastic Bifurcation Theory and Transport Properties in Geophysical and fIlaments provide excellent models forphysi­ Ludwig Arnold, University of Bremen, Flows cal problems ranging from DNA conformation Brian F. Farrell, Harvard University Germany (10.8 meters) to motion of solar flux tubes and 5:20 Experimental Evidence for Chaotic sunspot formation (108 meters), with many more 10:30 Bifurcation Theory for Stochastic Scattering in a Fluid Wake Differential Equations relevant examples in between. Although a clas­ Peter Baxendale, University of Southern John C. Sommerer, Hwar-Ching Ku, and sical subject, recent advances in experimental, Harold E. Gilreath, Johns Hopkins University California computational, and analytical techniques have 11 :00 Random Dynamical Systems in 5:40 Mixing and Transport Rates for a brought increased attention to the study of the Engineering Marangoni-driven Chaotic Flow dynamics of curve and fIlament systems. In this Gerard Alba-Soler and August Palanques­ S. T. Ariaratnam, University of Waterloo, rninisymposium a thin cross-section of these Mestre, University of Barcelona, Spain Canada new methods and results will be presented with 11 :30 Self Stabilization of Stochastic Systems an emphasis on applications to physical sys­ 6:00 PM-7:00 PM Volker Wihstutz, University of North tems. The purpose of this mini symposium is to Carolina, Charlotte acquaint the general applied math community Golden Cliff - Level B 12:00 Attractors for Random Dynamical with some of the extensive recent work on curve Systems Business Meeting dynamics, particularly curves with twist, and its Bjorn Schmalfusk, University of Bremen, SIAM Activity Group on Dynamical relation to physical systems. Germany Systems Organizer: Isaac Klapper Montana State University Conference Program DYNAMICAL SYSTEMS " 19

CONFERENCE PROGRAM Tuesday, May 20 10:00 AM-12:00 PM 1989 allowed for a fresh look from a "gauge­ regarded as admissible if they correspond to theoretic" perspective. Recent advances ob­ traveling wave solutions when viscosity is Wasatch A & B - Level C tained using this and other perspectives will added. This criterion may conflict with the be presented. The minisymposium will also classical Lax criterion. The speakers will dis­ MS27 provide a forum for mathematicians and bi­ cuss the use of vector field bifurcation theory Slow Evolution ologists to interact and present what they see to elucidate the wave structure of Riemann in Conservative Systems as the important problems in this area. The solutions. current mathematical methods employed are A common paradigm for the description of Organizer: Stephen Schecter chiefly analytic in nature and bog down for physical systems is that of a slow evolution North Carolina State University animals with complicated shapes. On the other through families of relatively simple states, 10:00 An Application of Vectorfield Bifurca­ hand the important biological questions are like equilibrium states or states of periodic tion to Conservation Laws that Change often of a rough qualitative nature. One of the motion. The mathematical foundation for ap­ Type main problems in this area is to know what the proximating this evolution is fairly well estab­ Barbara Lee Keyfitz, University of Houston problems are. This is only possible through lished if the underlying systems are dissipa­ 10:30 On the Influence of Viscosity on sustained communication between biologists tive. There are numerous examples, however, Riemann Solutions of Nonlinear and mathematicians or physicists. for which the underlying systems are conser­ Conservation Laws vative. The mathematical foundations for ap­ Organizers: Richard W. Montgomery, Suncica Canic. Iowa State University proximating the slow evolution are not well University of California, Santa Cruz; and II :00 The Role of Polycycles in Jair Koiller, Laboratorio Nacional de established in this case, except for one-de­ Nonuniqueness of Riemann Solutions Computacao Cientifica, Brazil gree-of-freedom systems, where the method Arthur Azevedo, Universidade de Brasilia, of averaging is effective. This minisymposium 10:00 Self-Propulsion of Spherical Swimmers Brazil; Dan Marchesin, Instituto de Matematica Pura e Aplicada, Brazil; Bradley is intended to describe whatis presently known Using Surface Deformations Aravi Samuel, Howard Stone; and Howard Plohr, State University of New York, Stony and to present physically important examples, Berg, Harvard University Brook; and Kevin Zumbrun, Indiana including those in which the simple states University undergo a transition from stable to unstable, 10:30 Discovering the Mysteries of Swimming Microorganisms on the Beaches of Rio II :30 Riemann Problem Solutions of illustrating the mathematical complexities that Kurt Ehlers, Laboratorio Nacional de Codimensions 0 and I can arise in these contexts. Computacao Cientifica, Brazil Stephen Schecter. Organizer; Dan Organizer: Norman R. Lebovitz II :00 Geometry of Microswimming Marchesin, Instituto de Matematica Pura e University of Chicago Richard W. Montgomery, Organizer Aplicada, Brazil; and Bradley Plohr, State University of New York, Stony Brook 10:00 Slow Evolution Near Families of Stable II :30 A Spectral Method for Stokes Flows and Equilibria of Hamiltonian Systems Applications to Microswimming Norman R. Lebovitz, Organizer Jair Koiller. Organizer; Joaquin Delgado, 10:00 AM-12:00 PM 10:30 Weakly Unstable Systems in the UNAM Mexico; and Marco Raupp, Chair: James A. Walsh, Oberlin College Presence of Neutrally Stable Modes Laboratorio National de Computacao Superior A - Level C John D. CralNford, University of Pittsburgh Cientifica, Brazil II :00 Some Experiments on Nonstationary CP17 Vibrations in a Single Degree-of­ 10:00 AM-12:00 PM Freedom Nonlinear Systems Teaching Dynamics Anil K. Baja). Mark Hood, and Patricia Maybird - Level C 10:00 Circle Homeomorphisms and Advanced Davies, Purdue University MS29 Calculus II :30 Accurate Phase After Slow Passage James A Walsh, Oberlin College Through Subharmonic Resonance Bifurcation Theory and Systems 10:20 Development of a Laboratory Based Jerry D. Brothers, Raytheon E-Systems; and of Nonlinear Conservation Laws Course in Nonlinear Dynamical Systems Richard Haberman, Southern Methodist N. Fleishon, J. McDill, K. Morrison, R. Systems of two conservation laws in one University Schoonover, J. P. Sharpe and N. Sungar, space dimension are PDEs without viscosity California Polytechnic State University that arise in modeling many physical systems, 10:00 AM-12:00 PM such as gas dynamics, three-phase flow in a 10:40 UG Course at lIT Delhi on Self­ Organising Dynamical Systems porous medium, elastic strings, and phase Ballroom III - Level B D. Subbarao, Indian Institute of Technology, transitions. The fundamental initial-value India MS28 problem for these systems is the Riemann problem, in which the initial data are piece­ II :00 Controlling the Chaotic logistic Map, Biological Microswimming An Undergraduate Project wise constant with a single jump. Riemann There are certain mystery swimmers - mi­ Erik M. Bollt United States Military Academy solutions can include shock waves, which are cro-organisms for which the means of propul­ sion are not understood. The underlying dy­ namics for swimming are defined by the Stokes equations with time-varying boundary condi­ tions. A geometric reformulation of the swim­ ming problem due to Shapere and Wilczek in 20 " DYNAMICAl SYSTEMS Conference Program

CONFERENCE PROGRAM ues ay, ay 0 10:00 AM-12:00 PM 11 :00 Exponentially Amplified Sampling and ity, Thus, while an understanding of low­ Reconstruction of Weak Signals Using frequency variability in oceanic and atmo­ Chair: Tomas Gedeon, Montana State Controlled Chaotic Orbits spheric flows is of crucial importance to un­ University Chance M Glenn, Sr., Johns Hopkins University derstanding the climate of our planet, it is Superior B - Level C 11 :20 Suppressed and Induced Chaos by Near possible to address this issue in the context of Resonant Perturbation of Bifurcations a simple but prototypical oceanic flow-the (PIS Larry Fabiny and Sandeep T. Vohra, Naval problem of adiabatic double-gyre circulation. Research Laboratory Neural and Excitable Systems I The short and noisy nature of the numerical simulations of even such highly simplified 10:00 Phase Synchronization in an Array of Coupled Integrate-and-Fire Neurons 1:30 PM-2:30 PM models necessitates the use of modern dy­ with Dendritic Structure namical system tools and sophisticated spec­ Paul C. Bressloff and P. N. Roper, tral estimation procedures to enhance our un­ Loughborough University, United Kingdom Chair: Charles R. Doering, University derstanding of these systems_ Further, this is a of Michigan, Ann Arbor 10:20 Desynchronization of Neural Assemblies necessary step to put forward phenomeno­ Stephen Coombes, Loughborough Ballroom I, II, III - Level B logical explanations of the variability. Good University, United Kingdom; and Gabriel J. estimates oflow-frequency variability in geo­ Lord, University of Bristol, United Kingdom IP7 physical systems using the available short and 10:40 A Bonhoeffer van der Pol Model for Biomolecular Motors noisy time series is a difficult problem_ Tradi­ tional methods are mostly based on ideas of Action Potential in Excitability Biological cells contain microscopic ro­ Fourier Transforms, etc_, and their usage tends Changing Media botic machinery that is used for cell A. Rabinovich, Ben-Gurian University, Israel; to be very restrictive for short and noisy time motility, for transport of vesicles and I. Aviram, Beer-Sheva, Israel; N. Gulko, Ben­ series, Therefore this very important aspect, organelles within cells, to move protein Gurian University, Israel; and E. Ovsyscher, with direct implications for long-time predic­ molecules across internal membranes, to Soroka Medical Center, Israel tion, has not received the attention it deserves, partition chromosomes at cell division, 11 :00 Post-Fertilization Traveling Waves on Eggs The more recent usage of data-adaptive bases and to manufacture the entire Gilberta Flores and Antonmaria Minzoni and filters, maximum-entropy methods, sin­ biomoIecular machinery of the cell. Un­ National University of Mexico, Mexico; , gular-spectrum analysis, etc_, may be more like the macroscopic machinery of ev­ Konstantin Mischaikow, Georgia Institute of suitable in this context. Technology; and Victor Moll, Tulane University eryday experience, these molecular mo­ Organizers: Balo T. Nadiga and Darryl 11 :20 Additive Neural Networks and Periodic tors function in a regime in which Brown­ ian motion plays an important role, D.Holm Patterns Los Alamos National laboratory Tomas Gedeon, Montana State University Chemical energy is used to rectify the Brownian motion and hence to drive a 3:00 Low Frequency Variability in Ocean Circula­ 11 :40 Computational and Experimental tion Models Driven by Constant Forcing Exploration of a Cortical Neuronal molecular motor in a particular direction_ The speaker will discuss two examples, Richard J. Greatbatch, Memorial University Network as a Dynamical System of Newfoundland, Canada David J Pinto, Joshua C. Brumberg, Daniel J. both associated with microtubules: Simons, and G. Bard Ermentrout, University kinesin, which is a motor protein that 3:30 Low-Frequency Variability in the of Pittsburgh, Pittsburgh "walks" along microtubules; and chro­ Reduced-Gravity Shallow Water Model mosome transport, which is driven by the Balu T Nadiga, Organizer; Len Margolin, depolymerization of the microtubule itself, Los Alamos National Laboratory; and Darryl 10:00 AM-12:00 PM D. Holm, Organizer Charles S. Peskin Chair_- Elbert E N. Macau, University 4:00 Low-Frequency Variability of Wind­ Courant Institute of Mathematical of Maryland, College Park Driven Ocean Gyres Sciences Magpie A & B - Level B Pavel S. Berloff and James C. McWilliams New York University University of California, Los Angeles ' (P19 4:30 Mathematical and Computational Issues Controlling Chaos in the Double Gyre Model 3:00 PM-5:00 PM John D. McCalpin, Silicon Graphics, Inc. 10:00 Nonlinear Control of Remote Unstable States in a liquid Bridge Convection Magpie A & B - Level B Experiment MS30 3:00 PM-5:00 PM Valery Petrov, Michael F. Schatz, Kurt A. Ballroom I - Level B Muehlner, Stephen 1. VanHook, W. D. low-Frequency Variability McCormick, J. B. Swift, and Harry L. in the Double-Gyre Circulation MS31 Swinney, University of Texas, Austin Model Reduction, Analysis, 10:20 Targeting from a Chaotic Scattering The high degree oflow-frequency variability Elbert E. N. Macau and Edward Ott characteristic of most geophysical systems is and Control of Spatia-Temporal University of Maryland, College Park very important for long-time predictions_ Such Dynamics Using Kl Methods naturally occurring systems are usually ex­ 10:40 Sustaining Chaos using Basin Boundary Understanding the dynamics of spatio-tempo­ ceedingly complex. However, highly simpli­ Saddles ral (ST) phenomena is important in many fied models of such systems also show a Ira B. Schwartz, Naval Research Laboratory; areas of research, such as turbulent flow, remarkable degree oflow-frequency variabil- and loana Triandaf, SAle, McLean, VA reaction-diffusion systems, and coupled os- Conference Program DYNAMICAL SYSTEMS " 21

CONFERENCE PROGRAM Tuesday, May 20 cillators. Because ST dynamics exhibits both involve the study of systems of ODE with 3:30 Nonlinear Optical Phenomena in low and and high dimensional behavior, it is 'evolution' in arc-length, while dynamic mod­ Periodic Structures important to characterize the behavior in a els involve systems of PDE in 1+1 dimen­ Herbert G. Winful. University of Michigan, Ann Arbor practical way, since many methods from low sions, but in both statics and dynamics math­ dimensional dynamics do not carryover to ST ematically unusual features, for example non­ 4:00 Modelocked Fiber laser Simulation and systems. One recent methodology for under­ local terms, can arise in modelling phenom­ Modeling standing ST dynamics is that of Karhunen­ ena such as contact. J. W Haus and G. Shaulov, Rensselaer Polytechnic Institute; and J Theimer, Rome Loeve (KL) decomposition. One of the main Organizer: John H. Maddocks Laboratory RUOCPA advantages in using KL methods is that an University of Maryland, College Park orthogonal basis can be constructed from 4:30 The Whitham Equations for Optical 3:00 Modeling Stable Configurations of measured data, and then used to project out Communications in NRZ Format Protein-bound DNA Rings Yuji Kodama, Osaka University, Japan low dimensional dynamic models. In this Jennifer A. Martino and Wilma K. Olson, minisympoium, the speakers will address ar­ Rutgers University eas of application of KL methods to both 3:00 PM-5:00 PM 3:30 A Combined Wormlike-Chain and Bead continuous and discrete applied areas in dy­ Model for Dynamic Simulations of long Ballroom III - Level B namics. Projected low dimensional behavior DNA of fluid mechanics, analysis and control of Hongmei Jian, Alexander Vologodskii; and MS34 bursting in reaction-diffusion, singularly per­ Tamar Schlick, New York University Topology in Dynamics turbed structural mechanics, and large arrays 4:00 Stability in Continuum Models of DNA The speakers in this minisymposium will of coupled oscillators will be discussed using Minicircles a KL point of view. Kathleen A. Rogers and Robert S. Manning, present ideas in dynamics that are primarily topological in nature especially to experimen­ Organizer: Ira B. Schwartz University of Maryland, College Park; and talists and theorists interested in dynamical U.S. Naval Research Laboratory John H. Maddocks, Organizer systems. 3:00 Analysis of Extensive Chaos by the 4:30 Stability of the Stationary States of the Karhunen-loeve Decomposition and by Elastic Rod Model Under an Applied Organizer: James A. Yorke a Hierarchy of Unstable Periodic Orbits External Force that Represent University of Maryland, College Park Henry Greenside, Duke University Configurations of DNA in living Cells 3:00 Crises: The Dynamics of Invariant Sets Thomas Connor Bishop and John E. Hearst, as a Parameter is Varied 3:30 Analysis and Control of Spatio-Temporal University of California, Berkeley Chaos in Reaction-Diffusion Processes JamesA. Yorke, Organizer; and Kathleen loana Triandaf, Science Applications Alligood, George Mason University International Corporation 3:00 PM-5:00 PM 3:30 Horseshoes and the Conley Index 4:00 New Measures of Coherence in Ballroom II - Level B Spectrum Disordered and Noisy Arrays of Konstantin Mischaikow, Georgia Institute of Coupled Phase Oscillators MS33 Technology Thomas W Carr. Southern Methodist 4:00 A Case Study of Topology Preserved in University; and Ira B. Schwartz, Organizer Periodically and Randomly an Environment with Noise: Fluid 4:30 Modeling the Dynamics of Wall Driven Dynamics in Nonlinear Flowing Past an Array of Cylinders Bounded Turbulence Optical Fibers Judy A. Kennedy, University of Delaware Larry Sirovich, Brown University and Mount The equation governing pulse propagation in 4:30 Stable Ergodicity and Stable Accessibil­ Sinai School of Medicine nonlinear optical fibers and waveguides under ity Charles C. Pugh and Mike Shub, University ideal conditions is the completely integrable of California, Berkeley 3:00 PM-5:00 PM nonlinear SchrOdinger (NLS) equation. In more Wasatch A & B - Level C realistic situations, however, this equation can be perturbed by effects such as continuous 3:00 PM-5:00 PM MS32 or discrete loss and periodic gain, amplified Chair: Amitabha Bose, New Jersey Continuum Models spontaneous emission noise, periodic varia­ Institute of Technology tions in the dispersion coefficient (also known Superior B - Level C of Biological Macromolecules as dispersion management), or by two-polar­ A number of biologically important molecules ization field envelope interactions. In this CP20 are extremely large and exhibit significant minisymposium (and the associated contrib­ Neural and Excitable Systems " behaviors on length scales much longer than uted paper session) the relevant physical and the atomic. Super-coiling of DNA and defor­ mathematical issues, both resolved and open, 3:00 Weakly Connected Weakly Forced mations of alpha-helical proteins are two prime will be discussed. Oscillatory Neural Networks examples. Recently there has been consider­ Eugene Izhikevich and Frank Hoppensteadt, Organizers: J. Nathan Kutz, Princeton Arizona State University able interest in simulating such molecules University; and William L. Kath, within the context of some form of continuum Northwestern University 3:20 Dynamics of Two Mutually Coupled Slow Inhibitory Neurons description. This minisymposium provides an 3:00 Propagation of Nonsoliton Pulses in opportunity for junior members from each of David Terman, Ohio State University; Nancy long Optically Amplified Transmission Kopel/, Boston University; and Amitabha four different research groups in this area to lines Bose, New Jersey Institute of Technology present their work. Static models typically Stephen Evangelides Jr, AT&T Research 22 " DYNAMICAL SYSTEMS Conference Prog ram

CONFERENCE PROGRAM Tuesday, May 20 3:40 Nonequilibrium Response Spectroscopy 3:20 The Melnikov Theory for Subharmonics The Construction of the Analytic Periodic of ION Channel Gating Kinetics and Their Bifurcations in Forced Solution in the Framework of the Mark M. Millonas, University of Chicago Oscillations Multiple Time Scale Method 4:00 Hodgkin-Huxley Neuronal Models Kazuyuki Yagasaki, Gifu University, Japan Sorinel Adrian Oprisan, "AI. L Cuza" Allan R. Willms, Cornell University 3:40 The Melnikov Vector as an Estimate for University, Romania 4:20 Preservation and Annihilation of Colliding Exponentially Small Splitting of Markov Random Fields and Dynamical Systems Pulses in a Model Excitable System Homoclinic Manifolds with Memory in Pattern Generation and M. Argentina and P. Coullet, Institut Non­ Alain Goriely, Craig Hyde, and Michael Recognition Processes Lineaire de Nice, France; and L Tabor, University of Arizona Sorinel Adrian Oprisan, "AI. L Cuza" Mahadevan, Massachusetts Institute of 4:00 Calculating Chaotic Strength University, Romania Technology David B. Levey and Paul D. Smith, University The Separatrix Map Approximation of Critical 4:40 Monitoring Changing Dynamics with of Dundee, United Kingdom Motion in the 311 Jovian Resonance Correlation Integrals: Case Study of an 4:20 Melnikov's Method for Stochastic Mult~ Ivan L Shevchenko, Russian Academy of Epileptic Seizure Degree of Freedom Dynamical Systems - Sciences, Russia David E. Lerner, University of Kansas Theory and Applications Application of Centre Manifold Theory to an Timothy Whalen, National Institute of Adaptive Control Problem Standards and Technology Graciela A. Gonzalez, Universidad de 3:00 PM-5:00 PM 4:40 Resonance Capture in Weakly Forced Buenos Aires, Argentina Chair: Peter Smereka, University of Mechanical Systems Extensive Chaos and Unstable Period Orbits Michigan, Ann Arbor D. Dane Quinn, University of Akron Scott M ZaIdi and Henry S. Greenside, Maybird - Level C Duke University 6:30 PM-7:30 PM Bifurcations in Systems of Coupled Bistable CP21 Units Synchronization 1/ Ballroom I, II, III - Level B Mark E. Johnston, University of Cambridge, United Kingdom 3:00 Spontaneous Synchronization and Poster Setup Critical Behavior in Oscillator Chains Chaotic Communications in the Presence of a Jeffrey L Rogers, Georgia Institute of Wireless Channel Technology 7:30 PM-9:30 PM Christopher Williams and Mark A. BeaCh, University of Bristol, United Kingdom 3:20 Synchronization and Relaxation in Ballroom I, II, III - Level B Globally Coupled Oscillators Forced Vibration of a Partially Delaminated Peter Smereka, University of Michigan, Ann Arbor Poster Session I Beam Roger P. Mendayand M. C. Harrison, 3:40 Coupling Topology and Global Dynamics Lagrangian Formalism of Hydrodynamic Forces University of loughborough, United Duncan J. Watts and Steven H. Strogatz, on Dust Particles Kingdom; and E. R. Green, leicester Cornell University Abdulmuhsen H. Ali, Kuwait University, University, United Kingdom 4:00 Synchronization of Spatio-Temporal Kuwait Dynamical Features Simulated by Recurrent Chaos Modelling and Phase Analysis of Immunolog~ Neural Networks Mikhail M Sushchik Jr., Nikolai F. cal Process at Primary Immune Fernanda Botelho, University of Memphis Rulkov, and Michael M. Rabinovich, Response University of California, San Diego Inertial Functions: Definition, Existence and Ivan Edissonov, Bulgarian Academy of Stability 4:20 Effect of Finite Inertia in Coupled Sciences, Bulgaria Kevin M Campbell, Michael E. Davies, and Oscillator Systems Global Bifurcations in Periodically Driven Zero­ Jaroslav Stark, University College London, Hisa-Aki Tanaka, Waseda University, Tokyo Dispersion Systems United Kingdom 4:40 Synchronized Chaos in Spatially S. M. Soskin, National Ukrainian Academy of Blownout Attractors Extended Systems and Interhemispheric Sciences, Ukraine Peter Ashwin and Philip Aston, University of Teleconnections Spectra of Frequency-Modulated Waves Surrey, United Kingdom; Matthew NiCOl, Gregory S. Duane and Peter J Webster, George D. Catalano, U.s. Military Academy University of Manchester Institute of Science University of Colorado, Boulder Symmetry Breaking Perturbations and Strange and Technology, United Kingdom Attractors Forced Symmetry-Breaking in Coupled 3:00 PM-5:00 PM Anna Utvak Hinenzon and Vered Rom­ Oscillator Networks Kedar, The Weizmann Institute of Science, Carlo Laing, University of Cambridge, Chair: Timothy Whalen, National Institute Israel United Kingdom of Standards and Technology Simulation of the Immune System on Parallel Superior A - Level C The Method of Melnikov for Perturbations of Computers Mult~Degree-of-Freedom Hamiltonian CP22 Massimo Bernaschi, IBM DSSC. Italy; Filippo Systems Castiglione, University of Catania, Italy; and Kazuyuki Yagasaki, Gifu University, Japan Melnikov Theory Sauro Succi, lAC CNR, Italy Reconstruction and Prediction of Stochastic 3:00 Asymptotic Expansions for Melnikov Invariant Measure for Dynamical Distributed Dynamical Systems Using Topology Functions for Conservative Systems Parameter Systems Preserving Networks Joseph Gruendler, North Carolina A& T State Zdzislaw W. Trzaska, Warsaw University of Markus Anderle and Sabino Gadaleta, University Technology, Poland Colorado State University Conference Program DYNAMICAL SYSTEMS ' 23

CONFERENCE PROGRAM Tuesday, May 20 Computing the Measure of Nonattracting Enhancement of Stochastic Resonance of Enhancing Stochastic Resonance in a Single Chaotic Sets FitzHugh-Nagumo Neuronal Model Unit Joeri Jacobs, Edward Ott, and Celso Driven by IIf Noise Carson C. Chow. Thomas T Imhoff, and Grebogi, University of Maryland, College Daichi Nozaki and Yoshiharu Yamamoto, James J. Collins, Boston University Park University of Tokyo, Japan Long-TIme Term Behaviour of Density Information Content of Cellular Automaton Chaotic Scattering in Micro Lasing Cavities Oscillator Traffic Simulations Tolga Yalccinkaya and Ying-Cheng Lai, Carios A. Vargas and Alejandro Ramirez­ Brian W. Bush, Los Alamos National University of Kansas Rojas, Universidad Autonoma Metropolitana Laboratory Intermingled Basins of Attraction: Azcapotzalco, Mexico Control of Differentially Flat Robotic Systems Uncomputability in a Simple Physical Using Spatial Disorder to Synchronize in the Presence of Uncertainty System Dynamical Systems Kristin Glass, Richard Colbaugh, and Tuna John C. Sommerer, Johns Hopkins Yuri Braiman, Emory University; and John Saracoglu, New Mexico State University, Las University; and Edward Ott, University of Lindner, The College of Wooster Cruces Maryland, College Park Continuum Limits of Convective-Diffusive Control of Uncertain Mechanical Systems Using A Model for Flows with Separation and the Lattice Boltzmann Methods Reduction and Adaptation Circular Hydraulic Jump Cecilia Fosser and C. David Levermore, Richard Colbaugh, Ernest Barany, Gil Shinya Watanabe, Vachtang Putkaradze, University of Arizona Gallegos, and Mauro Trabatti, New Mexico Tomas Bohr, Adam E. Hansen, Anders Numerical Analysis of Phase Synchronization State University Haaning, and Clive Ellegaard, Niels Bohr of Coupled Model Neurons Chemical Mixing Using a Chaotic Flow Insitute, Denmark Alexander K. Kozlov, Russian Academy of Xianzhu Tang, College of William and Mary; A Nonlinear Model to Classify Dynamic Time Sciences, Russia and Allen H. Boozer, Columbia University Series Characterization of Noise in Electrotelluric Symbolic TIme Series Analysis of Continuous Karin Schmidt. Friedrich Schiller University TIme Series Dynamical Systems Jena, Germany; and Hazel H. Szeto, Cornell Alejandro Ramirez-Rojas, Francisco Eugene R. Tracy. X.-l Tang, Reggie Brown, University Medical College Cervantes-de la Torre, and Carlos G. Pavia­ and S. Burton, College of William and Mary Synchronization of Structures by a Small Miller, Universidad Autonoma Metropolitana Signal Compression and Information Retrieval Number of Couplings Azcapotzalco, Mexico; Fernando Angulo­ via Symbolization Ya. I. Molkov, M. M. Sushchik, and A. V. Brown, Escuela Superior de Fisica y Xianzhu Tang and Eugene R. Tracy, College Yulin, Russian Academy of Science, Russia Matematicas, I. PN, Mexico of William and Mary Clusters in Globally Coupled Assemblies of Taking the Temperature of an Epileptic Seizure The Magnetic Pendulum Bistable Elements David E. Lerner, University of Kansas Francisco Cervantes de la Torre, J. L. A. V Yulin and M. M. Sushchik, Russian Dimension and KLD Correlation Lengths in a Fernandez Chapou, UAM-AZCAPOTZALCO, Academy of Science, Russia Two-Dimensional Excitable Medium CBI., Mexico Symmetry-Breaking Parametric Instabilities in Matthew C. Strain and Henry Greenside, Newton-Picard Methods for Robust Continua­ Circular Josephson Junction Arrays Duke University tion and Bifurcation Analysis of John J. Derwent and Mary Silber, Nonlinear Pattern Dynamics in Two-Dimen­ Periodic Solutions of Infinite-Dimen­ Northwestern University sional Josephson-Junction Networks sional Dynamical Systems with Low­ Numerical Evidence of Standing and Breathing Werner Guettinger and Joerg Dimensional Dynamics Solutions for the Oregonator with Oppenlaender, University of Tuebingen, Koen Engelborghs, Kurt Lust. and Dirk Equal Diffusivities Germany Roose, Katholieke Universite Leuven, Jack Dockery and Richard J. Field, Montana Dynamics in the Undergraduate Mathematics Belgium State University Program Scale-Sensitive Linear Analysis of Granivorous Using Linear Integro-Difference Equations to Richard F. Melka, University of Pittsburgh, Rodent Population Dynamics Emulate Solutions to a Temporally Stiff Bradford Nikkala A. Pack, North Carolina State System of PDE's Two Meaningful Alternative Representations of University; and Brian A. Maurer, Brigham Tyler K. McMillen, Utah State University Dynamic Behaviour Young University Ionic Diffusion Within the Confined Extracellu­ G. Arthur Mihram, Princeton, NJ; and Hamiltonian Moment Reduction for Describing lar Space of Glomeruli Danielle Mihram. University of Southern Vortices in Shear Flow Anita Rado, Alwyn Scott, and Leslie Tolbert, California S. Meacham, Florida State University; P. J. University of Arizona Trapping and Wiggling: Elastohydrodynamics Morrison, University of Texas, Austin; and G. Dynamical Behavior of a Model for Hormonal of Driven Microfilaments Flierl, Massachusetts Institute ofT echnology Regulation During the Menstrual Cycle Chris H. Wiggins, Princeton University, Controlling On-Off Intermittent Dynamics in Women Daniel X. Riveline and Albrecht Ott, Institut Yoshlhlko Naga( Xuan-Dong Hua, and Ying­ Paul M. Schlosser, Chemical Industry Curie, France; and Raymond E. Goldstein, Cheng Lai, University of Kansas Institute of Toxicology, Research Triangle University of Arizona Dynamics of Interacting Modes in EEG­ Park; and James F. Selgrade, North Carolina Bifurcations and Edge Oscillations in an Datasets State University Inhomogeneous Reaction-Diffusion Christian Uhf. Max-Planck-Institute of Scaling of Travelling Waves Produced from System Cognitive Neuroscience, Germany; Raul Electrostatic Instabilities Jonathan E. Rubin, Ohio State University, Kompass, University of Leipzig, Germany Anandhan Jayaraman and John David Columbus Crawford, University of Pittsburgh, Pittsburgh 24 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Tuesday, May 20 Wednesday, May 21 Breaking the Symmetry of an Attractor 8:30 AM-9:30 AM overview of the mathematical techniques cur­ Merging Crisis rently employed to model protein motors. Miguel A. F. Sanjuan, Universidad Some familiarity with cell biology would be Politecnica de Madrid, Spain Chair: Boris L. Altshuler, NEC helpful, but hopefully not necessary. Effects of Anisotropy on Nonlinear Evolution Research, Inc. Organizer: George F. Oster of Morphological Instability in Ballroom I, II, 1lI - Level B University of California, Berkeley Directional Solidification Alexander G%vin and Stephen Davis, IPS 10:00 Mathematical Models of Stochastic Northwestern University Ratchets Small Electronics and Quan­ Charles R. Doering, University of Michigan, Integra-Differential Model for Orientational Ann Arbor Distribution of F-Actin in Cells tum Chaos A/ex Mogi/ner, University of California, The transport properties of electronic 10:30 Cell Motions Driven by Actin Polymer­ Davis; Edith Geigant. University of Bonn, devices below the few-micrometer scale ization Germany; and Karina ladizhansky, show signatures of quantum interference George F. Oster, Organizer Weizmann Institute of Science, Israel and quantum confinement when the de­ 11 :00 The Bacterial Flagellar Motor: A New Canard Explosion and Bistability in the vices are cooled near absolute zero. This Mechanism for Energy Transduction Belousov-Zhabotinsky Reaction talk will summarize recent experiments Timothy C. Elston, Los Alamos National JOJ1-Paul Va roney, University of Guelph, measuring random (but perfectly repeat­ laboratory; and George Oster, Organizer Canada able) fluctuations in conductance of these 11 :30 Nana-Motors: Rotary Enzyme Com­ structures - known as quantum dots - and plexes the connection between the universal sta­ Ronald F. Fox, Georgia Institute of tistical properties of these fluctuations Technology and the universal spectral statistics of random matrix theory and quantum mani­ 10:00 AM-12:00 PM festations of chaos. Ballroom I - Level B There has been great progress in this field in the last few years, and the behav­ MS36 ior of quantum-coherent electrons is now Transport in Hamiltonian Systems well in hand, with good agreement be­ How does a blob of initial conditions evolve tween theory and experiment. This suc­ under a Hamiltonian flow? How do we char­ cess story applies when the unavoidable acterize such a motion, composed of regular coupling of electrons to the environment and chaotic components? These questions arise and each other is small. The great out­ in diverse fields such as celestial mechanics, standing problems in the field concern mechanical systems, Lagrangian advection in the the inclusion of interactions and fluids and geophysics, and particle confine­ decoherence. Some of these issues - some­ ment in plasma. Here, fundamental mecha­ what less of a success story - will also be nisms governing transport and their applica­ discussed. tions will be discussed; the role of self-simi­ Charles M. Marcus larities, of partial barriers and of degeneracies Department of Physics, Stanford in two, three, and four dimensional maps and University flows will be revealed. Finally, the implica­ tions of such theories on observable transport in fluids will be discussed. 10:00 AM-12:00 PM Organizer: Vered Rom-Kedar Magpie A & B - Level B Weizmann Institute of Science, Israel 10:00 Scaling Properties of Distributions of MS35 Exit Time and Poincare Recurrences Molecular Motors G. M. Zaslavsky, Courant Institute of Mathematical Sciences, New York University The advent of new technologies for measuring 10:30 Transport in Quadratic Volume forces and displacements at the molecular Preserving Maps scale has led to a resurgence in mathematical James D. Meiss, University of Colorado, modeling of molecular motors. These protein Boulder devices are thousands of times smaller than 11 :00 Lagrangian and Eulerian Transport: Are the smallest 'nanotechnology' has yet They Related? achieved, yet they power virtually all aspects George Haller, Brown University of the living cell. This mini symposium will present several case studies in how these mo­ 11 :30 Degenaracies, Instabilities and Transport tors are modeled and how the models compare Vered Rom-Kedar, Organizer to experimental data. The talks will give an Conference Program DYNAMICAL SYSTEMS " 25

CONFERENCE PROGRAM Wednesday, May 21 10:00 AM-12:00 PM 10:00 AM-12:00 PM variety of applications, including image pro­ cessing, material science, and biology. They Ballroom II - Level B Ballroom III - Level B moreover represent a new direction in math­ ematics, giving rise to challenging mathemati­ MS37 MS38 cal problems of pattern formation, spatial Fundamental Nonlinear Dynamics Applications of Synchronized chaos, traveling waves, and synchronization. of Diode lasers Chaos and Hyperchaos The talks in mathematical theory and applica­ tions on similar problems will clearly demon­ Diode lasers are widely used in applications, Recently interest has piqued in synchronized, strate the continuity between applications, such as consumer products as CD players and chaotic systems, primarily because they rep­ scientific computing, and mathematics. barcode scanners, and optical telecommuni­ resent a possible use of chaos in a communi­ cation. Many important aspects of their be­ cations setting. Advances in the area have led Organizers: John Mallet-Paret, Brown havior are due to nonlinear effects. We con­ to the ability to synchronize hyperchaotic University; and Shui-Nee Chow, Georgia Institute of Technology centrate on the two important mechanisms of systems, which allow the transmission of more optical injection and delayed optical feed­ complicated waveforms while retaining syn­ 10:00 Transition from Slow Motion to Pinning back, both of which cause effects that are chronization. We present here several of these in Lattice Equations fundamentally nonlinear in nature. Different recent advances including multiplexing of Christopher P. Grant, Brigham Young University models are studied, notably by using bifurca­ chaos, techniques for synchronizing tion theory and numerical simulations, in the hyperchaos and hyperchaotic circuits, and 10:30 Modeling Biological Systems by Means attempt to understand and control the dynam­ generalized versions of synchronization. The of Nonlinear Electronic Circuits ics. The models need to be checked, which purpose is to stimulate a wide audience inter­ IVberto P. Munuzuri and L. O. Chua, University of California, Berkeley; and V. leads to an intriguing interplay between ex­ ested in both the theoretical issues of chaos Perez-Munuzuri, University of Santiago de periments and theory. We want to communi­ synchronization and the practical uses for Compostela, Spain cate recent results on nonlinear dynamics of such behavior. Interested fields include dy­ diode lasers and how they have been obtained. namical systems analysts, electrical engineers, 11 :00 Traveling Waves in Lattice Dynamical Systems The intended audience includes researchers physicists, and communications. Wenxian Shen, Auburn University on lasers as well as applied mathematicians Organizer: Louis M. Pecora interested in concrete problems. As the meth­ 11 :30 Information Propagation, Pattern Naval Research Laboratory Formation and Reaction-Diffusion with ods are generally not unique to the field of 10:00 Synchronizing Hyperchaos for Cellular Neural Networks lasers, we also hope for interest from other Communications Patrick Thiran, Swiss Federal Institute of applied fields. The purpose is to exchange Mingzhou Ding, Florida Atlantic University Technology, Switzerland; and Gianluca ideas and hopefully stimulate further research. 10:30 Synchronizing Hyperchaotic Volume­ Setti, University of Bologna, Italy Organizer: Bernd Krauskopf Preserving Map Circuits 12:00 Traveling Waves, Equilibria, and the Vrije Universiteit, The Netherlands Thomas L. Carroll, Naval Research Computation of Spatial Entropy for 10:00 Bifurcations to Periodic Solutions in Laboratory Spatially Discrete-Reaction Diffusion Diode Lasers Subject to Injection 11 :00 Multiplexing Chaotic Signals Using Equations A. Gavrielldes and V. Kovanis, Phillips Synchronization John W. Cahn, National Institute of Laboratory; T. Erneux and G. Lythe, Lev S. Tsimring, N. F. Rulkov, M. M. Standards and Technology; Shui-Nee Chow, Universite Libre de Bruxelles, Belgium Sushchik, and H. D. I. Abarbanel, University Georgia Institute of Technology; John of California, San Diego Mallet-Paret, Organizer; and Erik S. 10:30 Bifurcation Analysis of a Phase­ Van Vleck, Colorado School of Mines Amplitude Model of an Injected Diode 11 :30 On Generalized Synchronization Laser Ljupco Kocarev, SI. Cyril and Methodius Bernd Krauskopf. Organizer; Daan Lenstra University, Republic of Macedonia 10:00 AM-12:00 PM and Wim van der Graaf, VrUe Universiteit, (This session will run until 1:00 PM) Chair: William R. Derrick, University of The Netherlands Montana 11 :00 Control of Optical Turbulence in High Superior A - Level C Brightness Semiconductor Lasers 10:00 AM-12:00 PM Jerome V. Moloney and David Hochheiser, Wasatch A & B - Level B CP23 University of Arizona II :30 Controlling Temporal and Spatio­ MS39 Bursting and Biochemical Oscilla­ Temporal Dynamics in Semiconductor Spatially Discrete Dynamical tions Lasers by Delayed Optical Feedback Markus Muenkel, University of Technology, Systems - Theory and Applications 10:00 Delaying the Transition to the Active Darmstadt, Germany; Christian Spatially discrete dynamical systems are ei­ Phase in Bursting Oscillators Thomas W Carr, Southern Methodist Simmendinger and Ortwin Hess, Institute of ther infinite-dimensional systems which pos­ Technical Physics, DLR, Germany University; and Victoria Booth, New Jersey sess a discrete spatial structure, generally Institute of Technology modeled on a regular lattice, or finite dimen­ sional systems which could be cellular neural 10:20 Slow Subsystem Bifurcations in Systems Exhibiting Bursting networks (CNN) or cellular automata. The Mark Pernarowski, Montana State University time variable can be either discrete or continu­ ous. Such systems arise as models in a wide 26 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Wednesda ,Ma 21 10:40 Effect of Weak Coupling on Oscillations II :40 Noise Sensitivity and Boundary Effects 1:30 PM-2:30 PM in Bursting Systems in Travelling Wave Instabilities Gerda de Vries and Arthur Sherman, Michael R. E Proctor, University of National Institutes of Health Cambridge, United Kingdom; Edgar Chair: Jerrold E. Marsden, Califor­ II :00 Interpretation of Chaos in the Belousov­ Knobloch, University of California, Berkeley; nia Institute of Technology Zhabotinsky Reaction and Steven M. Tobias, University of Ballroom I, /I, III - Level B Richard J Field, University of Montana Colorado, Boulder II :20 Bifurcations in One Model of the Bray­ 12:00 Defects and Domain Walls in the 2d IP9 liebhafsky Oscillations Complex Ginzburg-Landau Equation Nonlinear Control of William R. Derrick and Leonid V. Kalachev, Greg Huber, University of Chicago; Tomas University of Montana Bohr, Niels Bohr Institute, Denmark; and Lagrangian Systems Edward Ott, University of Maryland, College Recent advances in geometric mechanics, II :40 Phase-Locking and Chaos in Forced Park Excitable Systems .. motivated in large part by applications in Min Xie and Hans G. Othmer, University of control theory, have introduced new tools Utah 10:00 AM-12:00 PM for understanding and utilizing the struc­ ture present in mechanical systems. In par­ '2:00 Synchronous Bursting in Coupled Chair: Bruce J. Gluckman, Naval Surface ticular, the use of geometric methods for Neurons Warfare Center and The Children's analyzing Lagrangian systems with both M M Susheik, Va. L Molkov, and M. L Research Institute of George Washington symmetries and nonholonomic constraints Rabinovich, Russian Academy of Sciences, University Russia has led to a unified formulation of the Maybird - Level C dynamics that has important implications 10:00 AM-12:00 PM CP25 for a wide class of mechanical control systems. In this talk, the speaker will sur­ Chair: Rebecca Hoyle, University Stochastic Resonance vey recent results in this area, focusing on of Cambridge, United Kingdom 10:00 Coherence Resonance in a Noise-Driven thereiationships between geometric phases, Superior B - Level C Excitable System controllability, and curvature, and the role Arkady Pikovsky and Jurgen Kurths, of trajectory generation and tracking in CP24 Universitiit Potsdam, Germany nonlinear controller synthesis. An impor­ Waves and Ginzburg-Landau 10:20 Stochastic Resonance in a Neuronal tant class of examples are differentially flat Equations Network from Mammalian Brain systems, for which the tr,yectory genera­ Bruce J Gluckman, Naval Surface Warfare tion problem is conceptually simple and 10:00 Convective and Absolute Instability in Center and The Children's Research Institute computationally tractable. Examples will Finite Containers of George Washington University be drawn from robotic locomotion, flight Steven M Tobias, University of Colorado, 10:40 Spatio-Temporal Stochastic Resonance Boulder; Edgar Knobloch, University of control systems, and other areas, including in a System of Coupled Diode Resona­ videotape ofexperimental results performed California, Berkeley; and Michael R. E. tors Proctor, University of Cambridge, United at Caltech. Markus Locher, G. A. Johnson, and E. R. Kingdom Hunt, Ohio University; and F. Marchesoni, Richard M. Murray 10:20 Fronts Between Rolls of Different University of Illinois, Urbana California Institute of Technology Wavenumbers in the Presence of II :00 Spatio-Temporal Stochastic Resonance Broken Reflection Symmetry of a Kink Motion in an Inhomogeneous Rebecca Hoyle, University of Cambridge, phi 4 Model United Kingdom 3:00 PM-5:00 PM Igor Dikshtein, Russian Academy of. 10:40 Dynamics of a 2-Dimensional Complex Sciences, Russia; and Alexander Neiman, Ballroom I - Level B Ginzburg-Landau Equation with Chiral Saratov State University, Russia Symmetry Breaking MS40 II :20 Noise-Enhanced Information Transmis­ Keeyeol Nam. Michael Gabbay, Parvez N. sion in a Network of Globally Coupled Periodic Orbits in Chaotic Systems Guzdar, and Edward Ott, University of Oscillators Maryland, College Park The role of periodic orbits in chaotic systems is Paul Gailey Oak Ridge National Laboratory; well known to be of fundamental theoretical and II :00 Coupled Oscillators in Fluid Mechanics: Thomas Imhoff, Carson Chow, and James J practical importance. For example, chaotic be­ Wakes Behind Rows of Cylinders Collins, Boston University Patrice Le Gal, Jean-Francis Ravoux, and havior is often undesirable, and if the dynamics II :40 Spatio-Temporal Stochastic Resonance are low dimensional, it may be possible to re­ Isabelle Peschard, Universite d'Aix-Marseille I in Coupled Circuits & II-CNRS, France place chaos with stable periodic behavior. This Gregg A. Johnson, Naval Research minisymposium addresses recent theoretical re­ II :20 Vortex Dynamics in Dissipative Systems Laboratory; M. Locher and E. R. Hunt, Ohio sults that are of importance to practical situa­ Eisebeth Schroder, Niels Bohr Institute, University, Athens Denmark; and Ola Tornkvist, NAWFermilab tions. In particular, we address the role of nu­ Astrophysics Center merical simulations in identifying periodic or­ bits, the theoretical advantages to stabilizing low-periodic trajectories via feedback control, and the fundamental problem of the occurrence of stable periodicity in high dimensional chaotic systems. Conference Program DYNAMICAL SYSTEMS " 27

CONFERENCE PROGRAM Wednesday, May 21 Organizers: Ernest Barreto and Edward Ott 4:00 Vortex Dynamics and the Geometric 3:00 PM-5:00 PM University of Maryland, College Park Phase 3:00 Optimal Periodic Orbits of Chaotic Paul K. Newton, Organizer Magpie A & B - Level B Systems: Control and Bifurcations 4:30 Coriolis Forces as an SO(3) Gauge Field: Edward Ott, Organizer; and Brian R. Hunt, Applications to Molecular Dynamics MS43 University of Maryland, College Park Robert G. Littlejohn, University of California, Ouantum Chaos and Mesoscopic 3:30 From High Dimensional Chaos to Stable Berkeley Physics Periodic Orbits: the Structure of This minisymposium concerns the universal Parameter Space 3:00 PM-5:00 PM Ernest Barreto, Organizer; Brian R. Hunt, properties of quantum mechanical systems Celso Grebogi, and James A. Yorke, Ballroom III - Level B which are chaotic in their classical limit. The University of Maryland, College Park MS42 speakers will discuss both open (scattering) 4:00 Unstable Dimension Variability: A and isolated systems, with an emphasis on the Source of Nonhyperbolicity in Chaotic Dynamical and Statistical Model­ connection between quantum chaos and Systems ing of Biological Systems mesoscopic fluctuation phenomena seen in Eric J. Kostelich, Arizona State University; micron-scale electronic structures. Ittai Kan, George Mason University; Celso Due to its intrinsic nonequilibrium nature and complexity, biological systems have become Organizer: Charles M. Marcus Grebogi, University of Maryland, College Stanford University Park; Edward Ott, Organizer; and James A. an exciting field for applying dynamical sys­ Yorke, University of Maryland, College Park tem and statistical methods to discover its 3:00 Quantum Transport Through Micro­ structures: Signatures of Chaos 4:30 Periodic Shadowing underlying principles. The main theme of this minisymposium is to understand large-scale Harold U. Baranger, AT&T Bell Laboratories, Huseyin Kocak, University of Miami Lucent Technologies collective behavior from simple microscopic rules. The biological systems covered in this 3:30 Experiments in Quantum Chaos: 3:00 PM-5:00 PM session range from RNA mutation, DNA se­ Mesoscopic Fluctuations in Quantum Ballroom II - Level B quence matching, protein folding to flocking Dots Charles M. Marcus, Organizer dynamics. Simple microscopic rules based on MS41 experiment are used in modeling these bio­ 4:00 Chaotic Dynamics and Quantum level Applications of the Geometric logical systems. Continuum models can also Statistics be constructed for some cases in a coarse­ Jonathan P: Keating, University of Bristol, Phase United Kingdom grained level and analytical insight is often This minisymposium will focus on recent and gained by comparing the biological systems 4:30 Chaos, Quantum Mechanics, and ongoing developments concerning the 'geo­ to well studied physical systems. Universality metric', or 'Hannay-Berry' phase. The origi­ Boris L. Altshuler, NEC Research, Inc. Organizer: Yuhai Tu nal setting in the pioneering paper by Berry IBM T. J. Watson Research Center (1984) was the adiabatic theorem from quan­ 3:00 PM-5:00 PM tum mechanics dealing with a system coupled 3:00 RNA Virus Evolution: Fluctuation-Driven to a slowly changing environment. If the Motion on a Smooth landscape Chair: Jack Dockery, Montana State Hamiltonian varies adiabatically, then after a Herbert Levine, Lev Tsimring, and Douglas University Ridgway, University of California, San Diego; cyclic evolution of the environment param­ Wasatch A & B - Level C and David Kessler, Bar-llan University, Israel eters, the eigenstate returns to its original CP26 form, but with a geometric phase factor asso­ 3:30 Why Do Proteins look like Proteins? Hao U, Chao Tang, and Ned Wingreen, ciated with it. Since this paper appeared, there Reaction-Diffusion Equations NEC Research Institute; and Robert Helling, has been a flood of ongoing work on both University of Hamburg, Germany 3:00 Wave Propagation and its Failure in a classical and quantum analogues of this phase Discrete Reaction-Diffusion Equation factor and this minisymposium is meant to 4:00 DNA Sequence Matching and Nonequilibrium Dynamics with Paul C. Bressloff, Loughborough University, give a snapshot of current activity in this area. Multiplicative Noise United Kingdom Organizers: Jerrold E. Marsden, Terence Hwa and Miguel A. Munoz, 3:20 Spatial Patterns and Dynamics in California Institute of Technology; and University of California, San Diego; Dirk Filtration Combustion Paul K. Newton, University of Southern Drasdo and Michael Lassig, Max Planck Daniel A. Schult, Colgate University California Institute, Germany 3:40 The Evolution of Slow Dispersal Rates: 3:00 Geometric Phases and the Stabilization 4:30 Dynamics of Flocking: How Birds Fly A Reaction Diffusion System of Balance Systems Together Jack Dockery, Montana State University; Jerrold E. Marsden, Organizer; Anthony Yuhai Tu, Organizer; and John Toner, Vivian Hutson, Sheffield University, United Bloch, University of Michigan, Ann Arbor; University of Oregon Kingdom; Konstantin Mischaikow, Georgia Gloria Sanchez, University of Merida, Institue of Technology; and Mark Venezuela; and Naomi Leonard, Princeton Pernarowski, Montana State University University 4:00 Numerical Study of Bifurcation of 3:30 The Phase for the Spatial Three-Body Traveling Wave Solutions in a Problem Reactionn-Diffusion System Richard Montgomery, University of Mark Friedman, University of Alabama, California, Santa Cruz Huntsville 28 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Wednesday, May 21 4:20 Transition to an Effective Medium and 3:20 The Dynamics of Variable Time­ Traveling Waves in Heterogeneous Stepping ODE Solvers 6:30 PM-7:30 PM Reaction-Diffusion Systems Harbir Lamba and Andrew Stuart, Stanford Ballroom I, II, III - Level B M. Bar and M. Bode, Max-Planck-Institut fur University Physik Komplexer Systeme, Germany; A. K. 3:40 Multiple-Shooting Newton-Picard Poster Setup Bangia and Y Kevrekidis, Princeton University Methods for Computing Periodic 4:40 Rhombs, Supersquares, and Pattern Solutions of large-Scale Dynamical Competition in Reaction-Diffusion Systems with Low-Dimensional 7:30 PM-9:30 PM Systems Dynamics Ballroom I, II, III - Level B Stephen L Judd and Mary Silber, Kurt Lust, Koen Engelborghs, and Dirk Northwestern University Roose, Katholieke Universite Leuven, Poster Session II Belgium On the Solution to Nonlinear Thermal Ignition 3:00 PM-5:00 PM 4:00 Convergence of a Fourier-Spline Problem for Condensed Media with Representation of the Poincare Map Phase Transition with the Help of the Chair: David Peak, Utah State University Generator uGeometrical-Optical" Asymptotic Maybird - Level C Robert L Warnock, Stanford University; and Method James A. Ellison, University of New Mexico G. A. Nesenenko, Bauman State Technical CP27 4:20 Computation of Stability Basin for Large University, Russia Spatia-Temporal Chaos Close-to-Hamiltonian Nonlinear Systems Nonlinear Dynamics of Discrete Ecosystem Leonid Reznikov and Mark A. Pinsky. Models Subject to Periodic Forcing 3:00 Strange Attractors of the KdV-KS University of Nevada, Reno Equation: Spatio-Temporal Order and Danny Summers, Justin G. Cranford, and Chaos in 3D Film Flows Brian P Healey, Memorial University of Alexander L Frenkel and K. Indireshkumar, 3:00 PM-5:00 PM Newfoundland, Canada University of Alabama, Tuscaloosa The Geometric Phase in a Point Vortex Flow in Chair: Robert W. Ghrist, University of 3:20 Reduction of Complexity and Control of a Circle Texas, Austin B. N Shashikanth and P. K. Newton, Spatio-Temporal Chaos through Superior B - Level C Archetypes University of Southern California David Peak, Emily Stone, and Adele Cutler, CP29 Classification of Nonlinear Time Series Using Utah State University Cluster Analysis 3:40 Spatially Coherent States in Long Range Fluids" Thomas Schreiber, University of Wuppertal, Germany Coupled Maps 3:00 Granular Glasses and Fluctuational Sridhar Raghavachari and James A. Glazier, Hydrodynamics Application of the Theory of Optimal Control University of Notre Dame Sergei E. Esipov, James Franck Institute and of Distributed Media to Nonlinear 4:00 Detecting and Extracting Messages University of Chicago; Clara Saluena, Optical Fiber from Chaotic Communications using Universitat de Barcelona, Spain, James Vladimir Y. Khasilev, University of Rostov, Russia Nonlinear Dynamic Forecasting Franck Institute and University of Chicago, Dynamical Precursor of Black Hole Formation Kevin M Short, University of New and Thorsten Poschel, Humboldt-Universitat Numerical Simulation Hampshire zu Berlin, Germany Bruce N Miller and V. Paige Youngkins, 4:20 Local Dynamics and Spatio-Temporal 3:20 Continuous Avalanche Mixing of Texas Christian University Complexity Granular Solids in a Rotating Drum Finitely Representable Set-Valued Maps and Ralf W Wittenberg and Philip Holmes, Barry A. Peratt, Winona State University; and Computation of the Conley Index Princeton University James A. Yorke, University of Mary/and, Tomasz Kaczynski, Universite de Sherbrooke, College Park 4:40 Stability Analysis of Reaction-Diffusion Canada Waves Near Transitions to Spatio­ 3:40 Beltrami Flows and Contact Geometry On a Certain Class of Zero-Sum Discrete-Time Temporal Chaos Robert W Ghrist and John B. Etnyre, Stochastic Games Markus Bar. Max-Planck-Institut fur Physik University of Texas, Austin D. Leao Pinto, Jr., Universidade Estadual de Komplexer Systeme, Germany; Anil K. 4:00 Homogenization of the Navier-Stokes Campinas, UNICAMP, Brazil; Marcelo Dutra Bangia and Yannis Kevrekidis, Princeton Equation for Oscillatory Fluids Fragoso, National Laboratory for Scientific University Bjorn Birnir, University of California, Santa Computing, LNCC -CNPq, Brazil; and Joao Barbara Bosco Ribeiro do Val, Universidade Estadual de Campinas, UNICAMP, Brazil 3:00 PM-5:00 PM 4:20 The Temporal Dynamics of the Ocean Circulation Sudden Changes of Invariant Chaotic Sets in Chair: Robert L. Warnock, Stanford Steve Meacham, Florida State Planar Dynamical Systems University University; and Pavel Berloff, University of Carl Robert, University of Maryland, College Superior A - Level C California, Los Angeles Park; Kathleen T. Alligood, George Mason University; Edward Ott and James A. Yorke, 4:40 The Transition to Chaos for an Array of University of Maryland, College Park CP28 External Driven Vortices Numerical Methods Fred Feudel, Universitat Potsdam, Germany; Manifold Reduction and Flow Reconstruction and Parvez Guzdar, University of Mary/and, Using Karhunen-Loeve Transformations 3:00 Reliable Numerical Detection of Chaotic College Park and Neural Reconstruction Behaviour in Hamiltonian Systems Douglas R. Hundley, Colorado State Govindan Rangarajan, Indian Institute of University SCience, India Conference Program DYNAMICAL SYSTEMS " 29

CONFERENCE PROGRAM Wednesday, May 21 Influence of Measurement Desynchronization Chaos and Crises in More Than Two Dimen­ Using Manifolds to Generate Trajectories for on Dynamics of Ship Motion Control sions libration Point Missions System Silvina Ponce Dawson and Pablo Moresco, Kathleen C. Howell, Purdue University, West Vladimir G. Borisov, Institute of Control University of Buenos Aires, Argentina Lafayette Sciences, Russia; Fedor N. Grigor'yev and Steps Toward the long-Term Simulation of Symplectic Integrators Versus Ordinary­ Nikolay A. Kuznetsov, Institute for Turbulent Flows through Higher-Order Differential-Equation Solvers for Information Transmission Problems, Russia; Incremental Unknowns/Hierarchical Hamiltonian Systems and Pavel I. Kitsul, Mankato State University Basis Jon Lee, Wright Lab IFIB), Wright-Patterson When Can a Dynamical System Represent an Salvador Garcia, Universidad de Talca, Chile Air Force Base Irreversible Process? Stick-Slip Dynamics and Friction in an Array of Scaling in Forced laser Synchronization R. M. Kiehn, University of Houston Coupled Nonlinear Oscillators Carlos L. Pando, Universidad Autonoma de Mult~Parameter Bifurcation Analysis in H. G. E Hentschel, Yuri Braiman, and Puebla, Mexico Nonlinear Control Systems Fereydoon Family, Emory University Two-Dimensional Josephson-Junction Network Barry R. Haynes, Mark P. Davison, The A Hilbert Space Framework for the Analysis of Architecture for Maximum Microwave University of Leeds, United Kingdom Bifurcations in Control Radiation Emission Can Markoff Chain Theory Determine a Signals Aubrey B. Poore and Timothy J. Trenary, Joerg Oppenlaender and Werner Origin: Chaos or Random? Colorado State University Guettinger, University of Tuebingen, Charles R. Tolle, David Peak, and Robert Causal Bifurcation Sequences in Systems Germany Gunderson, Utah State University Containing Strong Periodic Forcing A Feedback Scheme for Controlling Dispersive Scaling Behavior of Transition to Chaos in E. A. D. Foster, Memorial University of Chaos in the Complex Ginzburg-landau Quasiperiodically Driven Dynamical Newfoundland, Canada Equation Systems Optimal Dynamic Nonlinear Pricing and Georg Flatgen, Yannis G. Kevrekidis, Ying-Cheng l.iJi, University of Kansas; Ulrike Capacity Planning Princeton University; Paul Kolodner, Bell Feudel, Max-Planck Arbeitsgruppe Susan Y. Chao and Samuel Chiu, University Laboratories, Lucent Technologies "Nichtlineare Dynamik", Universitat of California, Berkeley and Stanford Chaos Synchronization and Riddled Basins in Potsdam, Germany; and Celso Grebogi, University Two Coupled One-Dimensional Maps University of Maryland, College Park Stacked lagrange Tops: Analysis of Mult~ Tomasz Kapitaniak, Technical University of Noise-Induced Riddling in Chaotic Systems Component Systems Lodz, Poland; and Y. Maistrenko, Academy Ying-Cheng l.iJi, University of Kansas; and Debra Lewis, University of California, Santa of Sciences of Ukraine, Ukraine Celso Grebogi, University of Maryland, Cruz lattice Boltzmann Equation Model for College Park Recurrent Propagation of a Wave Front in an Potassium Dynamics in Brain Discontinuous Bifurcations and Crises in a DU Excitable Medium Longxiang Dai and Robert M. Miura, DC Buck Converter Yue-Xian U, University of California, Davis University of British Columbia, Canada Mario di Bernardo, University of Bristol, Non-Collision Singularities in a 6 Body Problem United Kingdom; Chris J. Budd, University of Instabilities of Hexagon Patterns in a Model for Rotating Convection Jeff Xia and Todd Young, Northwestern Bath, United Kingdom; and Alan R University Champneys, University of Bristol, United Filip Sain and Hermann Riecke, Kingdom Northwestern University Stationary Solutions of an Integro-Differential Equation Modeling Phase Transitions: The Invariant Manifolds and Solar System Use of Parametric Excitation for Unstable Fixed Point Detection and Capture Existence and local Stability Dyanmics Adam Chmaj, Brigham Young University Martin W Lo and Shane D. Ross, California Peter M. Van Wirt, United States Air Force, Institute of Technology Air Force Institute of Technology and Utah Control of Impact Dynamics State University; Robert Gunderson and Jan Awrejcewicz and Krzysztof Tomczak, lyapunov Transforms and Invariant Stability David Peak, Utah State University Technical University of Lodz, Poland Exponents William E. Wiesel, Air Force Institute of Dynamics of the Ginzburg-landau Equations Theoretical Analysis of Pattern Formation in Technology/ENY of Superconductivity Circular Domains Hans G. Kaper, Argonne National Gemunu H. Gunaratne, University of Dichotomies and Numerics of Stiff Problems Laboratory; Peter Takac, Universitat Rostock, Houston B. Katzengruber and P. Szmolyan, Germany Technische Universitat Wien, Austria Transport Properties in Disordered Ratchet Global Bifurcations in a Reaction-Difussion Potentials Computation of Invariant Tori Near II in the Model leading to Spatio-Temporal Chaos Fabio Marchesoni, University of Illinois, Earth-Sun System Martin Zimmermann and Sascha Firle, Urbana Gerard Gomez, Universitat de Barcelona, Uppsala University, Sweden; Mario Natiello, Numerical Simulation of Staanding Waves in a Spain; Angel Jorba and Josep Masdemont, Royal Institute of Technology, Sweden; Universitat Politecnica de Catalunya, Spain Vertically Oscillated Granular layer Michael Hildebrand and Markus Eiswirth, Chris Bizon, Mark Shattuck, Jack Swift, lyapunov Exponents from the Dynamics of Max-Planck Gesellschaft, Berlin, Germany; William McCormick, and Harry Swinney, Classical and Quantal Distributions Markus Bar, Max-Planck-Institut fur Physik University of Texas, Austin Aljendu K. Pattanayak and Paul Brumer, Komplexer Systeme, Germany; A. K. Bangia University of Toronto, Canada and Yannis G. Kevrekidis, Princeton Applications of the KAM Theory to Commensu­ University rate-Incomensurate Phase Transitions Deterministic Chaos and Analysis of Bambi Hu, Hong Kong Baptist University, Singularities Resonance Zones for Henon Maps Hong Kong and University of Houston Alexander Tovbis, University of Central Robert Easton, University of Colorado, Florida Boulder 30 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Wednesday, May 21 Thursday, May 22 The Modeling of Self-Oscillating Regimes in 8:30 AM-9:30 AM Since the last Snowbird meeting in 1995 where Catalytic Reaction Dynamics a similar mini symposium was organized and N. f. Ko/rsov, Chuvash State University, held, substantial advances have been made in Russia Chair: Rajarshi Roy, Georgia calculating the dynamical quantities of interest Image Encryption Based on Chaos Institute of Technology and relating them to transport coefficients, and in Jiri Fridrich, State University of New York, Ballroom I, II, III - Level B understanding the relationship between the dy­ Binghamton namical systems approach to transport and that Integrals of Motion and the Shape of the IP10 of irreversible thennodynamics. Of particular Attractor for the Lorenz Model Universal Spatia-Temporal interest in this connection is the role of entropy Sebastien Neukirch and Hector Giacomini, Chaos in Large Assemblies production in a fluid system, and how it can be Universite de Tours,France of Simple Dynamical Units related to various fractal structures in the phase Dynamics of the Two-Frequency Torus space dynamics. The speakers are all recognized Breakdown in the Driven Double Scroll Large assemblies of nonlinear dynamical leaders in this field and have made very impor­ Circuit units driven by a stochastic field with smooth tant contributions to it in the last few years. Murilo S. Baptista and Ibere L. Caldas, spatial variation generate strong turbulence Organizer: Robert Dorfman University of Sao Paulo, Brazil with various scaling properties. The driv­ University of Maryland, College Park Type-II Intermittency in the Driven Double ing field may actually be a self-generated Scroll Circuit internal field due to long-range mutual 10:00 Lyapunov Exponents and Irreversible Murilo S. Baptista and Ibere L. Caldas, coupling. Besides being persistent over a Entropy Production in Low Density University of Sao Paulo, Brazil Many-Particle Systems finite parameter range, this type of turbu­ Henk van Beijeren, University of Utrecht, Pulse Like Spatial Patterns Described by Higher lence is highly independent of what kind of The Netherlands; and Robert Dorfman, Order Model Equations dynamical units are involved, and whether Organizer A. I. Rotariu-Bruma and L.A. Peletier, or not each unit is coupled with its neigh­ 10:30 lyapunov Spectra of Various Model University of Leiden, The Netherlands; and borhood. Underneath the amplitude field W. C. Troy, University of Pittsburgh Systems in Nonequilibrium Steady States of our direct observation, a strongly inter­ Harald A. Posch and Christoph Del/ago, The Computer Analysis of Poincare Cross­ mittent field with a multifractal structure is University of Vienna, Austria Sections in Chemical Reactions discovered which is reminiscent of the Dynamics II :00 Entropy Production in Open Volume energy dissipation field in the fully-devel­ Preserving Systems Kozhevnikov IV and Koltsov N.J., Chuvash oped hydrodynamic turbulence. State University, Russia Pierre Gaspard, Universite Libre de Bruxel/es, Belgium Modular Networks for Animal Gaits Y oshiki Kuramoto Graduate School of Sciences and Martin Golubitsky, University of Houston; Ian II :30 Transport and Entropy Production in Stewart, University of Warwick, United Department of Physics low-Dimensional Chaos Kingdom; Pietrr:rLuciano Buono, University Kyoto University Tamas Tel, W. Breymann, and J. Vol/mer, of Houston; and J. J. Collins, Boston Eotvos University, Hungary University A Method for Solving the Linear Operator 10:00 AM-12:00 PM 10:00 AM-12:00 PM Problem Ky=hlxl Ballroom II - Level B L. K. Tolman, Brigham Young University Ballroom I - Level B Convergence Theorems for the Method MS44 MS45 Richard Wellman, Utah State University Dynamical Systems Theory and Dynamics of Laser Arrays: Coher­ On Some Random Generalized Functions Having a Mean Nonequilibrium Statistical Me­ ence, Chaos and Control Manuel L. Esquivel, Universidade Nova de chanics Advances in the design and fabrication of laser Lisboa, Portugal arrays have led to great interest in the power, The last few years have seen the growth of a new coherence, and stability properties of the light Flow-Reversal Systems with Fast Switching branch of statistical mechanics, the area of phys­ emitted by these arrays. This minisymposium Jan Rehacek, Los Alamos National ics that deals with the properties of systems of Laboratory; A10is Kiic, and Pavel Pokorny, will address important issues in the mathemati­ large numbers of particles. This new branch - the Institute of Chemical Technology, Czech cal modelling oflaser arrays including compari­ applications of dynamical systems theory to the Republic sons of predicted dynamics with experimental theory of transport in fluids and solids - is observations. The connections between coher­ founded on fundamental mathematical ideas of ence and dynamical behavior and the control of Sinai, Ruelle, and Bowen, but has a history that these properties will be an important aspect of reaches back to the 19th century development of the minisymposium presentations and discus­ statistical mechanics by Maxwell, Boltzmann, sions to be given by leading experts in the field. and Gibbs. Recent research has focused upon the Advances in the design and fabrication of laser relation between issues in transport theory such arrays have led to great interest in the power, as the calculation of transport coefficients like coherence and stability properties of the light viscosities, diffusion coefficients, etc., and the emitted by these arrays. Lyapunov exponents and KS entropies that char­ acterize the chaotic dynamics of the system. Organizer: Rajarshi Roy Georgia Institute of Technology Conference Program DYNAMICAL SYSTEMS " 31

CONFERENCE PROGRAM Thursday, May 22 10:00 Nonlinear Dynamics of Semiconductor 10:30 Invariant Manifolds for a Class of 10:00 AM-12:00 PM Laser Arrays Dispersive, Hamiltonian, Partial Herbert G. Winful, University of Michigan, Differential Equations Chair: Sharon R. Lubkin, University of Ann Arbor Claude-Alain Pillet. Universite de Geneve, Washington 10:30 Design of Laser Resonators for Switzerland; and Clarence Eugene Wayne, Superior B - Level C Enhanced Spatial Coherence and Mode Pennsylvania State University Control 11 :00 On Oscillations of Solutions of CP30 James R. Leger, University of Minnesota, Randomly Forced-Damped NLS Biology Minneapolis Equations 11 :00 Two Coupled Lasers Sergei B. Kuksin, Steklov Mathematical 10:00 A Two-Dimensional Model of Human Thomas Erneux, Universite Ubre de Institute, Russia Walking Bruxelles, Belgium; A. Hohl. A. Gavrielides, 11 :30 Resonances and Radiation Damping in Susan 1. Schenk, New Jersey Institute of and V. Kovanis, Phillips Laboratory Conservative Nonlinear Waves Technology 11 :30 Entrainment of Coupled Solid State Michael L Weinstein, Organizer 10:20 Bifurcation Analysis of Nephron Lasers Pressure and Flow Regulation T. A. Brian Kennedy Y. Braiman, A. Khibnik, 10:00 AM-12:00 PM Erik Mosekilde and Mikael Barfred, Technical and Kurt Wiesenfeld, Georgia Institute of University of Denmark, Denmark; and Niels­ Technology Ballroom III - Level B Henrik Holstein-Rathlou, University of Copenhagen, Denmark MS47 10:40 Pattern Formation In Vitro 10:00 AM-12:00 PM Control and Shadowing Sharon R. Lubkin, University of Washington Magpie A & B - Level B The talks in this session discuss applications 11 :00 Population Dynamics of Dungeness of control and shadowing. The applications Crab: Connecting Mechanistic Models MS46 to Data include laboratory experiments with a bounc­ Kevin Higgins and Alan Hastings, University Energy Transfer in Nonlinear ing ball and a double pendulum that use target­ of California, Davis; Jacob N. Sarvela, Partial Differential Equations ing to reach prespecified states; the control of University of Texas, Austin; and Louis W. In nonlinear and nonintegrable partial differ­ pendulation in a crane attached to a rocking Botsford, University of California, Davis ship; theoretical and numerical results on con­ ential equations (PDEs) modeling physical 11 :20 Global Asymptotic Behavior and systems, important phenomena are often un­ trol in a system whose phase space consists of Dispersion in Size-Structured, Discrete derstandable in terms of energy transfer among chaotic sets and KAM surfaces; and a discus­ Competitive Systems natural modes of the system. Such descrip­ sion of the limits of deterministic modeling. John E. Franke, North Carolina State tions can be, for example, in terms of (nonlin­ The notion of "control of chaos" has received University; and Abdul-Aziz Yakubu, Howard ear) bound state and radiation modes, or small a great deal of attention in recent years. The University scale and long scale structures. In this experiments to be discussed in this 11 :40 Dynamics of Mass Attack, Spatial mini symposium some conservative and dissi­ minisymposium are among the first to (1) deal Invasion of Pine Beetles into Lodgepole pative dynamical systems will be considered directly with the "targeting of chaos" problem Forests and analyzed from this perspective. Examples in a laboratory setting and (2) to discuss appli­ Peter White and James Powell, Utah State of phenomena to be examined are asymptotic cations with real industrial/military impor­ University; Jesse Logan and Barbara Bentz, USFS Intermountain Research Station stability, metastability, radiation damping, tance. singularity formation in Hamiltonian systems, Organizer: Eric J. Kostelich and turbulence in random and dissipatively Arizona State University 10:00 AM-12:00 PM perturbed Hamiltonian systems. The equa­ 10:00 Controllable Targets for Use with Chair: Thomas W. Carr, Southern tions considered are nonlinear wave and Chaotic Controllers Methodist University Schr6dinger equations and their perturbations. Thomas L. Vincent. University of Arizona Superior A - Level C The methods used are quite broad and involve, 10:30 A Chaotically Forced Pendulum: Ship for example, classical PDE and asymptotic Cranes at Sea CP31 methods, invariant manifolds, scattering theory Guohui Yuan, Brian Hunt, Celso Grebogi. Karhunen-Loeve Methods and stochastic analysis. The purpose of this Edward Ott, and James A. Yorke, University minisymposium is to present in a single forum of Maryland, College Park; and Eric 1. 10:00 A Study of Transition in Rayleigh­ recent evelopments on the question of energy Kostelich, Organizer Benard Convection using a Karhunen­ transfer in nonlinear and nonintegrable PDEs. 11 :00 Targeting in Soft Chaotic Hamiltonian Loeve Basis Systems I. Hakan Tarman, King Fahd University of Organizer: Michael I. Weinstein Petroleum and Minerals, Saudi Arabia University of Michigan, Ann Arbor Christian G. Schroer and Edward Ott, University of Maryland, College Park 10:20 On Dynamical Systems Obtained Via 10:00 Self-Focusing of the Nonlinear Galerkin Projection onto Bases of KL­ SChrodinger Equation and Its Behavior 11 :30 Limits to Deterministic Modeling James A. Yorke, University of Maryland, Eigenfunctions for Fluid Flows Under Small Perturbation Dietmar Rempfer, Cornell University George C Papanicolaou, Stanford University College Park 10:40 The Karhunen-Loeve Transformation: When Does it Work? Michael Kirby, Colorado State University 32 " DYNAMICAl SYSTEMS Conference Program

CONFERENCE PROGRAM Thursday, May 22 II :00 Coupled Mechanical-Structural Dynamical Systems: A Geometric 10:00 AM·12:00 PM 1:30 PM·2:30 PM Singular Perturbation-Proper Chair: Sue Ann Campbell, University Orthogonal Decomposition Approach of Waterloo, Canada Chair: Vered Rom-Kedar, Weizmann loannis r Georgiou and Ira B, Schwartz, Maybird - Level C Institute of Science, Israel Naval Research Laboratory Ballroom I, II, III - Level B II :20 Detecting Interacting Modes in Spatia­ CP33 Temporal Signals Bifurcation and Symmetry IPll Christian Uhl, Max-Planck-Institute of Complex Ginzburg-landau Cognitive Neuroscience, Germany 10:00 Stability of the Whirling Modes of a Hanging Chain (Kolodner's Modes) Equations as Perturbations of Warren Weckesser, Rensselaer Polytechnic Nonlinear SChrodinger Equations 10:00 AM·12:00 PM Institute General complex Ginzburg-Landau Chair: Patrick D. Miller, Brown University 10:20 Instabilities of Spatia-Temporally Wasatch A & B - Level C Symmetric Periodic Orbits (CGL) equations are considered as per­ A. M. Rucklidge, University of Cambridge, turbations of general nonlinear CP32 United Kingdom; and Mary Silber, SchrMinger (NLS) equations over a one Chaotic Advection, Turbulence, Northwestern University dimensional periodic domain. We iden­ and Transport I 10:40 On a Neutral Functional Differential tify some structures in the phase space of Equation Modelling a Simple Physical the NLS equations that persist under the 10:00 Chaotic Advection in an Array of Ouasi- System CGL perturbation, where "persists" 20 Vortices Sue Ann Campbell, University of Waterloo, means to be approximated in the C infty S. R. Maassen, H. J. H. Clercx, and G. J. F. Canada topology by a CGL structure as the per­ van HeUst Eindhoven University of 11 :00 Bifurcation leading to Hysteresis turbation tends to zero. Such structures Technology. The Netherlands Gregory Berkolaiko, University of necessarily play an important role in the 10:20 Enhancement of Stirring by Chaotic Strathclyde, United Kingdom dynamics of the CGL global attractor. Advection using Parameter Perturba­ 11 :20 Feedback-Assisted Instability Detection We find criteria for NLS rotating waves tion and Target Dynamics Jason S. Anderson, Georg Flatgen, and and traveling waves to persist and, in Yechiel Crispin, Embry-Riddle University Yannis G. Kevrekidis, Princeton University; some cases, determine their dynamical 10:40 Fractal Entrainment Sets of Tracers Katharina Krischer, Fritz-Haber-lnstitOt der stability. In the Lyapunov case this char­ Advected by Chaotic Temporally Max-Planck-Gesellschaft, Germany; and acterizes all omega-limit sets of the flow Irregular Fluid Flows Ramiro Rico-Martinez, Instituto Tecnologico and a complete bifurcation analysis can Joeri Jacobs, Edward Ott, Thomas de Celaya, Mexico be carried out. When the focusing NLS Antonsen, and James A. Yorke, University of II :40 Grazing Bifurcations in a Piezoelectric equation is perturbed, we show that none Maryland, College Park Impact Oscillator of the NLS orbits homo clinic to rotating II :00 Geometrical Dependence of Finite TIme Sandeep T. Vohra, Naval Research waves persist, but rather distinct lyapunov Exponents and Transport Laboratory homoclinic structures are produced by Xianzhu Tang, College of William and Mary 12:00 An Introduction to Groebner Bases and the CGL perturbation. and Columbia University; and Allen H. Invariant Theory C. David Levermore Boozer, Columbia University and Max-Planck r K Callahan and Edgar Knobloch, Institute, Germany University of California, Berkeley Department of Mathematics 11:20 Ouantifying Transport in Numerical University of Arizona Simulations of Oceanic Flows Patrick D. Miller and Christopher K. R, T. Jones, Brown University; Audrey M. 3:00 PM·5:00 PM Rogerson and Lawrence J. Pratt, Woods Hole Oceanographic Institution Ballroom I - Level B 11 :40 Anomalous Dispersion in 2-d Turbulence Antonello Provenza Ie, Instituto di Cosmo­ MS48 geofisica, Italy Noise and Heteroclinic Phenom­ ena Homoclinic and heteroclinic cycles are common in nonlinear dynamics, usually arising in global bifurcations but also occurring, in a structurally stable way, in some systems with symmetry. Introduction of noise to systems that exhibit homocliniclheteroclinic phenomena can result in qualitative changes in the dynamics, with the appearance of new characteristic timescales and the masking of chaotic behaviour being com­ mon. Because deterministic equations usually Conference Program DYNAMICAL SYSTEMS " 33

CONFERENCE PROGRAM Thursday, May 22 are not precise descriptions of physical systems, Organizers: Constance M. Schober, Old 4:00 Spatia-Temporal Pattern Formation in it is important in applications to know which Dominion University; and Gregor Combustion features of the deterministic dynamics persist Kovacic, Rensselaer Polytechnic Institute Alvin Bayliss and Bernard J. Matkowsky, under the addition of noise. This minisymposium 3:00 The Nonlinear Schrodinger Equation: Northwestern University looks at the effect of noise on a number of Asymmetric Perturbations, Traveling 4:30 Nonlinear Dynamics in Combustion systems that have homoclinic, heteroclinic, or Waves and Chaotic Structures Synthesis of Materials related phenomena. Various approaches will be Constance M. Schober, Organizer Vladimir A. Vol pert, Organizer discussed, including methods based on stochas­ 3:30 Homoclinic Orbits in the Maxwell-Bloch tic differential equations, Fokker-Planck equa­ Equations of Nonlinear Laser Optics 3:00 PM-5:00 PM tions, and numerical integration/simulation. Gregor Kovacic, Organizer; Jiyue J. [j and Victor Roytburd, Rensselaer Polytechnic Ballroom III - Level B Organizers: Vivien Kirk, University of Institute; and Thomas A. Wettergren, Naval Auckland, New Zealand; and Emily Undersea Warfare Center MSSI Stone, Utah State University 4:00 The Fate of Traveling Waves of the Riddling in Chaotic Systems 10:00 Probability Densities for Qualitatively Nonlinear Schrodinger Equation under Modified Dynamics Riddling and related bifurcations in dynami­ Ginzburg-Landau Type Perturbations cal systems with symmetric invariant sub­ Rachel Kuske, Tufts University; George C. David Levermore, Gustavo Cruz-Pacheco, spaces have been an area of recent study. Papanicolaou, Stanford University; and and Benjamin P. Luce, Los Alamos National Steve Baer, Arizona State University Laboratory Riddling refers to the situation where the 10:30 Noise-Controlled Dynamics in Normal basin of a chaotic attractor is riddled with 4:30 Smale Horseshoes in Perturbed holes that belong to the basin of another Forms Nonlinear Schrodinger Equations attractor. Current directions of research con­ Grant Lythe, Los Alamos National Charles Y. [j, Massachusetts Institute of Laboratory; and Steve Tobias, University of Technology sist of theoretical, numerical, and experimen­ Colorado, Boulder tal exploration of various physical conse­ 11 :00 Noisy Cycles, the Sequel quences associated with riddling and related Emily F. Stone, Organizer; and Dieter 3:00 PM-5:00 PM bifurcations. One such consequence is on-off Armbruster, Arizona State University Magpie A & B - Level B intermittency. The presentations in this 11 :30 A Heteroclinic Network with Noise minisymposium deal with several forefront Vivien Kirk, Organizer; and Mary Silber, MSSO problems on riddling. Topics to be covered Northwestern University Nonlinear Dynamics and Pattern include riddling bifurcation, blowout bifurca­ Formation in Combustion tion, intermingled basins, and on-off intermit­ 3:00 PM-5:00 PM tency in coupled map lattices, and unstable We consider both gaseous and solid fuel com­ periodic-orbit theory of blowout bifurcation. Ballroom II - Level B bustion, e.g., employed in combustion syn­ Organizer: Ying-Cheng Lai thesis of materials. Since the structure of the University of Kansas MS49 material produced, the propagation velocity, Melnikov Methods for Partial and maximum temperature are determined by 3:00 Riddling Bifurcation in Chaotic the mode of propagation of the combustion Dynamical Systems Differential Equations Celso Grebogi, University of Maryland, wave, assessment of the quality of the mate­ College Park In the past two years, several important break­ rial produced and the rate of flame spread throughs have been made in applying the Method must be considered because the maximum 3:30 Blowout-Type Bifurcations in Symmetric of Melnikov to demonstrating the existence of temperatures and velocities attained in dy­ Systems transverse homoclinic orbits and chaotic dy­ Peter Ashwin, University of Surrey, United namical modes can well exceed estimates Kingdom namics in near-integrable partial differential based on steady state analysis. Analytical, equations. This minisymposium will present numerical, and experimental approaches are 4:00 Synchronized Chaos, Intermingled Basins and On-Off Intermittency in some of them. The presented work deals with employed in a variety of problems, including conservative and dissipative perturbations of the Coupled Dynamical Systems the dynamics of cellular, target, spinning, and Mingzhou Ding, Florida Atlantic University nonlinear SchrOdinger equation, and the Max­ spiral flames. well-Bloch equations that describe the dynam­ 4:30 Characterization of Blowout Bifurcation ics oflight in laser cavities. The techniques used Organizer: Vladimir A. Volpert by Unstable Periodic Orbits Northwestern University in investigating these problems are the inverse Ying-Cheng Lai, Organizer spectral theory for integrable partial differential 3:00 Standing Waves, Heteroclinic Connec­ equations with periodic boundary conditions, tions, and Parity Breaking in Bifurca­ Backlund transformations and the dressing tions from Ordered States of Cellular Flames method, invariant manifold theory in infinite­ Michael Gorman, University of Houston; and dimensional settings, k normal forms, geometric Kay Robbins, University of Texas, San singular perturbation theory and the theory of Antonio orbits homoclinic to resonance bands, the en­ 3:30 Excitability in Premixed Gas Combus­ ergy-phase method for detecting multi-pulse tion homoclinic orbits, and numerical nonlinear spec­ Howard Pearlman, NAYI Lewis Research tral analysis. Careful comparisons with numeri­ Center cal simulations will also be presented. 34 " DYNAMICAL SYSTEMS Conference Program

CONFERENCE PROGRAM Thursday, May 22 3:40 Anomalous Transport in an Incompress- 4:40 Higher-Order Bistable Systems in One 3:00 PM-5:00 PM ible, Temporally Irregular Flow Space Dimension Chair: Michael E. Davies, University Shankar C. Venkataramani, James Franck William D. Kalies, J Kwapisz, R. C. A. M. College London, United Kingdom Institute; Thomas M. Antonsen Jr. and VanderVorst and T. Wanner, Georgia Maybird - Level C Edward Ott, University of Maryland, College Institute of Technology Park CP34 4:00 Space-Time Modeling and Pattern 3:00 PM-5:00 PM Time Series and Signal Processing Formation in Rotating Flows Erik A. Christensen, Levich Institute, City Chair: Hans True, Technical University of 3:00 No Noise is Good Noise? College of CUNY and New Jersey Institute of Denmark, Denmark Mark Muldoon, J P Huke, and D. S. Technology; Nadine Aubry, New Jersey Superior A - Level C Broomhead, University of Manchester Institute of Technology, and Levich Institute, Institute of Science and Technology, United City College of CUNY; Jens N. Sorensen and CP37 Kingdom Martin O. L Hansen, Technical University of Engineering and Control 3:20 Testing Nonlinear Markovian Hypoth- Denmark, Denmark eses in Dynamical Systems 4:20 Nonlinear Dynamics of Electrochemical 3:00 Stabilization of Uncertain Mechanical Christian Schittenkopf, Technische Oscillations, Surface, Morphology, and Systems Using Dynamic Feedback Universitat Munchen, Germany and Siemens Corrosion Ernest Barany and Richard Colbaugh, New AG, Germany; and Gustavo Deco, Siemens Elia V. Eschenazi and Ninj Balla, Xavier Mexico State University, Las Cruces AG, Germany University of Louisiana 3:20 On Time Optimal Control Trajectories of 3:40 Is There a Deterministic Relation /Phase 4:40 Chaotic Transport in a Chain of Biotechnological Processes Locking/ Between Sunspot Cycles and Vortices: Anomalous Diffusion and Levy Djamal Atroune, University of Georgia Interdecadal Variability of Atmospheric Statistics 3:40 Nonlinear Regenerative Machine Tool Temperature? Diego del Castillo-Negrete, University of Vibrations Milan Palus, Institute of Computer Science, Califomia, San Diego Gabor Stepan, Technical University of AS CR, Czech Republic; Dagmar Novotna, Budapest, Hungary Institute of Atmospheric Physics, AS CR, 3:00 PM-5:00 PM 4:00 Application of Symbolic Dynamics to Czech Republic; and Ivana Charvatova, Modeling and Control of an Internal Institute of Geophysics, AS CR, Czech Chair: Jinqiao Duan, Clemson University Combustion Engine Republic Superior B - Level C C. Stuart Daw and Matthew B. Kennel, Oak 4:00 Detecting Singularities of Non- Ridge National Laboratory; and Charles E. A. Deterministic Dynamics CP36 Finney, University of Tennessee, Knoxville Joseph P. Zbilut, Rush Medical College; Partial Differential Equations 4:20 Stabilization of Uncertain Mechanical David D. Dixon, University of California, Systems Using Bounded Controls Riverside; and Charles L Webber, Jr., Loyola 3:00 Analytical Properties of POE Jet Engine Richard Colbaugh, Ernest Barany, and Models University Chicago Kristin Glass, New Mexico State University, Hoskuldur Ari Hauksson, University of 4:20 Takens Embedding Theorems for Las Cruces Califomia, Santa Barbara; Andrzej Banaszuk, Forced and Stochastic Systems 4:40 Dynamical Response of a Spherical Jaroslav Stark, University College London, University of California, Davis; Bjorn Birnir Pendulum to Roll and Pitch Excitation United Kingdom and Igor Mezic, University of California, Santa Barbara Michael D. Todd and Sandeep T. Vohra, 4:40 Time-Series Analysis Tools for Nonlinear Naval Research Laboratory; and Frank A. System Identification 3:20 Analytic Aspects and Special Solutions Leban, Naval Surface Warfare Center of Nonlinear Partial Differential Joseph Iwanski and Elizabeth Bradley, 5:00 On a New Route to Chaos in Railway University of Colorado, Boulder Equations S. Roy Choudhury, University of Central Dynamic 5:00 Chaos and Detection Florida Hans True, Technical University of Denmark, Andrew M Fraser, Portland State University Denmark 3:40 Persistence of Invariant Manifolds for Nonlinear PDEs 3:00 PM-5:00 PM Don A. Jones, Arizona State University; and 5:20 PM Steve Shkoller, University of California, San Chair: Diego del Castillo-Negrete, Diego University of California, San Diego Conference Adjourns Wasatch A & B - Level C 4:00 Dynamics of a Nonlocal Kuramoto- Sivashinsky Equation CP35 Jinqiao Ouan and Vincent J Ervin, Clemson University Chaotic Advection, Turbulence, Fax & E-mail capabilities 4:20 Space Analyticity on the Attractor and Transport " Generated by the Set of All Stationary SIAM cannot offer e-mail access for Solutions for the Kuramoto-Sivashinsky individual attendees. We suggest you 3:00 Acoustic Turbulence Model bring your own laptop computer and use Alexander M. Balk, University of Utah Zoran Grujic, Indiana University, hotel facilities. If you do not have your 3:20 Interfacial Turbulence at Large Prandtl Bloomington own personal laptop computer, the hotel Number business office offers this service at a P. Colinet and J C. Legros, Universite Ubre nominal fee. de Bruxelles, Belgium Conference Program DYNAMICAL SYSTEMS " 3S SPEAKER INDEX

A Collins, James J.: MS3: 10:30, Sun ...... 8 G Constantin, Peter S.: IP4: 1:30, Mon ...... 16 Agur, Zvia: MSIO: 3:00, Sun ...... 11 Coombes, Stephen: CPI8: 10:20, Tue ...... 20 Gabbay, Michael: MS11: 8:00, Sun ...... 12 Alba-Soler, Gerard: CPI6: 5:40, Mon ...... 18 Cordo, Paul J.: MS3: 11:00, Sun ...... 9 Gadaleta, Sabino: Poster: 7:30, Tue ...... 22 Ali, Abdulmuhsen H.: Poster: 7:30, Tue ...... 22 Corron, Ned J.: CP9: 8:00, Sun ...... 13 Gailey, Paul: CP25: 11:20, Wed ...... 26 Ali, Mohamed A.: CP9: 7:20, Sun ...... 13 Crawford, John D.: MS27: 10:30, Tue ...... 19 Gal, Patrice Le: CP24: 11 :00, Wed ...... 26 Alstrom, Preben: MSI5: 11:00, Mon ...... 14 Crispin, Yechiel: CP32: 10:20, Thu ...... 32 Galper, Alexander R.: CPI6: 4:00, Mon ...... 18 Altshuler, Boris L.: MS43: 4:30, Wed ...... 27 Cross, Michael C.: MS6: 3:00, Sun ...... 10 Garcia, Salvador: Poster: 7:30, Wed ...... 29 Aranson, Igor: CPI3: 10:20, Mon ...... 15 Crowe, Kathleen: MS13: 7:30, Sun ...... 12 Gardner, Robert: MS 18: 11 :00, Mon ...... 15 Ariaratnam, S. T.: MS26: 11 :00, Tue ...... 18 Cvitanoviac, P.: CP5: 3:20, Sun ...... II Gaspard, Pierre: MS44: 11 :00, Thu ...... 30 Arnold, Ludwig: MS26: 10:00, Tue ...... 18 Gavrielides, A.: MS37: 10:00, Wed ...... 25 Ashwin, Peter: MS51: 3:30, Thu ...... 33 D Gedeon, Tomas: CPI8: 11:20, Tue ...... 20 Georgiou, Ioannis T.: CP3I: 11:00, Thu ...... 32 Aston, Philip J.: CP9: 7:00, Sun ...... 13 Dai, Longxiang: Poster: 7:30, Wed ...... 29 Atroune, Djamal: CP37: 3:20, Thu ...... 34 Gerstner, Wulfram: MSI7: 10:30, Mon ...... 14 Dangelmayr, Gerhard: CPl: 10:40, Sun ...... 9 Awrejcewicz, Jan: Poster: 7:30, Wed ...... 29 Ghrist, Robert CP29: 3:40, Wed ...... 28 Dankowicz, Harry: CP3: 11:00, Sun ...... 9 w.: Glass, Kristin: Poster: 7:30, Tue ...... 23 Davies, Huw G.: CP6: 3:20, Sun ...... 11 B Glass, Leon: MSIO: 3:30, Sun ...... II Davies, Michael E.: MS8: 3:30, Sun ...... 10 Glenn, Chance M., Sr.: CPI9: 11:00, Tue ...... 20 Bajaj, Anil K.: MS27: 11 :00, Tue ...... 19 Dawson, Donald: MSI2: 7:00, Sun ...... 12 Gluckman, Bruce J.: CP25: 10:20, Wed ...... 26 Balasuriya, Sanjeeva: MS2: 10:30, Sun ...... 8 Dawson, Silvina Ponce: Poster: 7:30, Wed ...... 29 Goldburg, Walter L: MSI5: 10:30, Mon ...... 14 Balk, Alexander M.: CP35: 3:00, Thu ...... 34 de la Torre, Francisco Cervantes: Poster: 7 :30, Tue Goldstein, Raymond E.: IP2: 1:30, Sun ...... 10 Bar, Markus: CP27: 4:40, Wed ...... 28 23 Goldstein, Raymond E.: MSl: 10:30, Sun ...... 8 Baranger, Harold U.: MS43: 3:00, Wed ...... 27 de Vries, Gerda: CP23: 10:40, Wed ...... 26 Golovin, Alexander: Poster: 7:30, Tue ...... 24 Barany, Ernest: CP37: 3:00, Thu ...... 34 del Castillo-Negrete, Diego: CP35: 4:40, Thu ...... 34 Gonzalez, Graciela A: Poster: 7:30, Tue ...... 22 Barreto, Ernest: MS40: 3:30, Wed ...... 27 Dennin, Michael: MSI: 11:00, Sun ...... 8 Goriely, Alain: CP22: 3:40, Tue ...... 22 Baxendale, Peter: MS26: 10:30, Tue ...... 18 Depassier, M. c.: CP8: 7:40, Sun ...... 13 Goriely, Alain: MS25: 10:30, Tue ...... 18 Belbruno, Edward: CP2: 11:40, Sun ...... 9 Derrick, William R.: CP23: II :20, Wed ...... 26 Gorman, Michael: MS50: 3:00, Thu ...... 33 Belmonte, Andrew: MSll: 7:00, Sun ...... 12 Derwent, John J.: Poster: 7:30, Tue ...... 23 Goz, Manfred E: CPll: 11:40, Mon ...... 15 Berkolaiko, Gregory: CP33: 11:00, Thu ...... 32 Devaney, Robert L.: IP3: 8:30, Mon ...... 14 Grant, Christopher P.: MS39: 10:00, Wed ...... 25 Birnir, Bjorn: CP29: 4:00, Wed ...... 28 di Bernardo, Mario: Poster: 7:30, Wed ...... 29 Gray, George R.: CPl: 11 :00, Sun ...... 9 Bishop, Thomas Connor: MS32: 4:30, Tue ...... 21 Ding, Jiu: MSI4: 7:30, Sun ...... 13 Greatbatch, Richard 1.: MS30: 3 :00, Tue ...... 20 Bizon, Chris: Poster: 7:30, Wed ...... 29 Ding, Mingzhou: Ms38: 10:00, Wed ...... 25 Grebogi, Celso: MS51: 3:00, Thu ...... 33 Blaom, Anthony: CP2: 10:00, Sun ...... 9 Ding, Mingzhou: MS51: 4:00, Thu ...... 33 Greenside, Henry: MS31: 3:00, Tue ...... 21 Bodenschatz, Eberhard: MS 1: 11 :30, Sun ...... 8 Dockery, Jack: CP26: 3:40, Wed ...... 27 Bokhove, Onno: CP8: 8:40, Sun ...... 13 Dockery, Jack: Poster: 7:30, Tue ...... 23 Grigor'yev, Fedor N.: Poster: 7:30, Wed ...... 29 Bokhove, Onno: MS7: 3:00, Sun ...... 10 Doering, Charles R.: MS35: 10:00, Wed ...... 24 Gruendler, Joseph: CP22: 3:00, Tue ...... 22 Grujic, Zoran: CP36: 4:20, Thu ...... 34 Bollt, Erik M.: CPI7: 11:00, Tue ...... 19 Duan, Jinqiao: CP36: 4:00, Thu ...... 34 Bollt, Erik M.: MS5: 11:30, Sun ...... 9 Duane, Gregory S.: CP21: 4:40, Tue ...... 22 Gunaratne, Gemunu H.: Poster: 7:30, Wed ...... 29 Borrelli, Robert L.: MS24: 4:00, Mon ...... 17 Dykman, Mark I.: MSI9: 11:30, Mon ...... 15 Bose, Amitabha: CP20: 3:20, Tue ...... 21 H Botelho, Fernanda: Poster: 7:30, Tue ...... 22 E Haberman, Richard: MS27: 11 :30, Tue ...... 19 Braiman, Yuri: CPI3: 11:40, Mon ...... 15 Haller, George: CPI5: 4:20, Mon ...... 17 Easton, Robert: Poster: 7:30, Wed ...... 29 Braiman, Yuri: Poster: 7:30, Tue ...... 23 Haller, George: MS36: 11:00, Wed ...... 24 Echebarria, BIas: CPI4: 5:20, Mon ...... 17 Braza, Peter A: CPI: 10:00, Sun ...... 9 Hansel, David: MS23: 5:30, Mon ...... 17 Edissonov, Ivan: Poster: 7:30, Tue ...... 22 Bressloff, Paul C.: CPI8: 10:00, Tue ...... 20 Egolf, David: MS6: 3:30, Sun ...... 10 Hauksson, Hoskuldur Ari: CP36: 3:00, Thu ...... 34 Bressloff, Paul CP26: 3:00, Wed ...... 27 c.: Ehlers, Kurt: MS28: 10:30, Tue ...... 19 Haus, J. W.: MS33: 4:00, Tue ...... 21 Brindley, John: MSI3: 7:00, Sun ...... 12 Elston, Timothy C.: MS35: 11:00, Wed ...... 24 Haydn, Nicolai: CPI2: 11:00, Mon ...... 15 Brown, Reggie: CP9: 7:40, Sun ...... 13 Engelborghs, Koen: Poster: 7:30, Tue ...... 23 Hayes, Scott: MS5: 10:30, Sun ...... 9 Bunimovich, Leonid A: IP6: 8:30, Tue ...... 18 Ercolani, Nicholas M.: MSI8: 10:00, Mon ...... 15 Haynes, Barry R.: Poster: 7:30, Wed ...... 29 Buono, Pietro-Luciano: Poster: 7:30, Wed ...... 30 Ermentrout, Bard: MS23: 4:00, Mon ...... 17 Hentschel, H. G. E: Poster: 7:30, Wed ...... 29 Bums, T. J.: CPI6: 4:20, Mon ...... 18 Erneux, Thomas: MS45: 11:00, Thu ...... 31 Higgins, Kevin: CP30: 11 :00, Thu ...... 31 Burtsev, S.: CP7: 8:00, Sun ...... 13 Eschenazi, Elia V.: CP35: 4:20, Thu ...... 34 Hinenzon, Anna Litvak: Poster: 7:30, Tue ...... 22 Bush, Brian W.: Poster: 7:30, Tue ...... 23 Esquivel, Manuel L.: Poster: 7:30, Wed ...... 30 Hochheiser, David: MS37: 11:00, Wed ...... 25 Evangelides, Stephen Jr.: MS33: 3:00, Tue ...... 21 Hogan, John: CP3: 11:40, Sun ...... 10 C Hoppensteadt, Frank c.: MSI7: 11:30, Mon ...... 14 Caldas, Ibere L.: Poster: 7:30, Wed ...... 30 F Hou, Thomas Y.: MS25: 11:30, Tue ...... 18 Calini, Annalisa: MS22: 5:30, Mon ...... 17 Howell, Kathleen C.: Poster: 7:30, Wed ...... 29 Fabiny, Larry: CPI9: 11:20, Tue ...... 20 Callahan, T. K.: CP33: 12:00, Thu ...... 32 Hoyle, Rebecca: CP24: 10:20, Wed ...... 26 Farrell, Brian E: CPI6: 5:00, Mon ...... 18 Campbell, Kevin M.: Poster: 7:30, Tue ...... 22 Hu, Bambi: Poster: 7:30, Wed ...... 29 Feudel, Fred: CP29: 4:40, Wed ...... 28 Campbell, Sue Ann: CP33: 10:40, Thu ...... 32 Hubbard, John H.: MS24: 4:30, Mon ...... 17 Feudel, Ulrike: CPI0: 7:20, Sun ...... 13 Canic, Suncica: MS29: 10:30, Tue ...... 19 Huber, Greg: CP24: 12:00, Wed ...... 26 Fibich, Gadi: MS25: 11:00, Tue ...... 18 Carr, Thomas CP23: 10:00, Wed ...... 25 Hundley, Douglas R.: Poster: 7:30, Wed ...... 28 w.: Field, Richard J.: CP23: 11:00, Wed ...... 26 Carr, Thomas MS31: 4:00, Tue ...... 21 Hunt, Brian R.: CPI0: 8:40, Sun ...... 13 w.: Finney, Charles E. A: CP37: 4:00, Thu ...... 34 Carroll, Thomas L.: MS38: 10:30, Wed ...... 25 Hunt, Brian R.: MS40: 3:00, Wed ...... 27 Flatgen, Georg: Poster: 7:30, Wed ...... 29 Catalano, George D.: Poster: 7:30, Tue ...... 22 Hunt, Fern Y.: MSI4: 8:00, Sun ...... 13 Flores, Gilberto: CPI8: 11:00, Tue ...... 20 Chao, Susan Y.: Poster: 7:30, Wed ...... 29 Hwa, Terence: MS42: 4:00, Wed ...... 27 Fosser, Cecilia: Poster: 7:30, Tue ...... 23 Chat", Hugues: MSI6: 11:00, Mon ...... 14 Hyde, Craig: CP2: 11 :00, Sun ...... 9 Foster, E. A. D.: Poster: 7:30, Wed ...... 29 Chiou-Tan, Faye Y.: MS3: 11:30, Sun ...... 9 Fox, Ronald E: MS35: 11:30, Wed ...... 24 I Chmaj, Adam: Poster: 7:30, Wed ...... 29 Franke, John E.: CP30: 11:20, Thu ...... 31 Choudhury, S. Roy: CP36: 3:20, Thu ...... 34 Fraser, Andrew M.: CP34: 5:00, Thu ...... 34 Ioannou, Petros J.: CPI6: 4:40, Mon ...... 18 Chow, Carson Poster: 7:30, Tue ...... 23 c.: Frenkel, Alexander L.: CP27: 3:00, Wed ...... 28 Ivey, Thomas: MS22: 5:00, Mon ...... 17 Christensen, Erik A.: CP35: 4:00, Thu ...... 34 Fridrich, Jiri: Poster: 7:30, Wed ...... 30 Iwanski, Joseph: CP34: 4:40, Thu ...... 34 Colbaugh, Richard: CP37: 4:20, Thu ...... 34 Izhikevich, Eugene: CP20: 3:00, Tue ...... 21 Friedman, Mark: CP26: 4:00, Wed ...... 27 Colbaugh, Richard: Poster: 7:30, Tue ...... 23 Froyland, Gary: MSI4: 7:00, Sun ...... 13 Colinet, P.: CP35: 3:20, Thu ...... 34 36 " DYNAMICAL SYSTEMS Conference Program SPEAKER INDEX

J Levine, Herbert: MS42: 3:00, Wed """ .. """"" ...... 27 Nesenenko, G. A.: Poster: 7:30, Wed "" .... """",, .. 28 Lewis, Debra: Poster: 7:30, Wed ...... " ...... 29 Neukirch, Sebastien: Poster: 7:30, Wed ...... """"" 30 Jacobs, Joeri: CP32: 10:40, TIlU "." .. """.""" .. "" .. 32 Li, Charles Y.: MS49: 4:30, Thu """"""""""" .. "" 33 Newton, Paul K.: MS41: 4:00, Wed " .... """" .. """ 27 Jacobs, Joeri: Poster: 7:30, Tue """""""""""""". 23 Li, Yue-Xian: Poster: 7:30, Wed .... " ...... " .... " .. " .. 29 Nicol, Matthew: Poster: 7:30, Tue .. " .. " ...... """ .. ".22 Jayaraman, Anandhan: Poster: 7:30, Tue """"""" 23 Littlejohn, Robert G.: MS41: 4;30, Wed " .. """ ..... 27 Niculae, Anne: CP4: 3:20, Sun ...... """""""" 11 Jian, Hongmei: MS32: 3:30, Tue """"""""""""".21 Lo, Martin W.: Poster: 7:30, Wed """" ...... " ...... 29 Nozaki, Daichi: Poster: 7:30, Tue " .... """ ...... ". 23 Johnson, Gregg A.: CP25: II :40, Wed """"""""" 26 Lobb, Chris J.: MS21: 5:00, Mon """"""""""""" 16 Nuz, Alexander: CPI4: 4:20, Mon "" .... """""""" 17 Johnson, Mark E.: MS20: 4:00, Mon """""""""" 16 Locher, Markus: CP25: 10:40, Wed ...... 26 Johnston, Mark E.: Poster: 7:30, Tue """""""""" 22 Lord, Gabriel J.: CP3: 10:20, Sun """""""""" .. "". 9 o Jones, Christopher K. R. T.: IPI: 8:30, Sun """""" 8 Lubkin, Sharon R.: CP30: 10:40, Thu """ .... "" ..... 31 Oliver, Marcel: MS2: 11 :00, Sun "" .. """" .. """ .. " ... 8 Jorba, Angel: Poster: 7:30, Wed """""""""""""" 29 Luce, Benjamin P.: MS49: 4:00, Thu " ...... " .... " .... 33 Oppenlaender, Joerg: Poster: 7:30, Tue .... ,,""""". 23 Judd, Stephen L.: CP26: 4:40, Wed """"""""""". 28 Lukens, James: MS21: 4:00, Mon """"""""""" ... 16 Oppenlaender, Joerg: 7:30, Wed ...... """" .. """ 29 K Lust, Kurt: CP28: 3:40, Wed .. """"" .. "" .. "" .... "". 28 Oprisan, Sorinel Adrian: Poster: 7:30, Tue .. """". 22 Luther, Gregory G.: CP4: 3:00, Sun " .... " ...... II Orlando, Terry P.: MS21: 4:30, Mon " .... "" .... " .. ". 16 Kaczynski, Tomasz: Poster: 7:30, Wed """""""". 28 Lythe, Grant: MSI2: 8:30, Sun"""""""" .... " ...... 12 Osborne, A. R.: MS2: 11:30, Sun " .. """""""""""" 8 Kalies, William D.: CP36: 4:40, Thu """""""""" 34 Lythe, Grant: MS48: 10:30, Thu """"""" .. """" ... 32 Osinga, Hinke: MS20: 4:30, Mon ...... " .... """""". 16 Kaloshin, Vadim Yu.: CPIO: 8:20, Sun """""""". 13 M Oster, George F.: MS35: 10:30, Wed .. " ...... "" .. ".24 Kantz, Holger: MS8: 4:00, Sun """""""""""""". 10 Othmer, Hans G.: CP23: II :40, Wed """" .... "" .. ". 26 Kaper, Hans G.: Poster: 7:30, Wed """""""""""" 29 Maassen, S. R.: CP32: 10:00, Thu " .. " .. " .. " .... " .. ". 32 Ouyang, Qi: MSI: 10:00, Sun """" .. """""" .... """. 8 Kapitaniak, Tomasz: Poster: 7:30, Wed """""""". 29 Macau, Elbert E. N.: CPI9: 10:20, Tue "" .. """ ..... 20 Kapitula, Todd: CPI5: 5:40, Mon """""""""""". 18 Mahadevan, L.: CP20: 4:20, Tue ...... " ...... 22 P Kapral, Raymond: MSI6: 11:30, Mon """"""""" 14 Mahadevan, L.: MS25: 10:00, Tue " .. " .. """"""" .. 18 Pack, Nikkala A.: Poster: 7:30, Tue """"""""""".23 Kath, William L.: CP4: 4:40, Sun""""""""""""" 11 Maier, Robert S.: MSI9: 11:00, Mon """" .. "".... " IS Palacian, J.: CP2: 11:20, Sun .. " .... " .. " ...... """""".9 Katzengruber, B.: Poster: 7:30, Wed """""""""". 29 Malchow, Horst: MS13: 8:30, Sun """""" .. " ...... 12 Palus, Milan: CP34: 3:40, Thu ...... " .... " .... """""" 33 Kaufman, Allan N.: CPI1: 10:00, Mon """""""". IS Marchesoni, Fabio: Poster: 7:30, Wed .... " .. "" ...... 29 Pando, Carlos L.: Poster: 7:30, Wed """"""" .. """ 29 Keating. Jonathan P.: MS43: 4:00, Wed """""""" 27 Marcus. Charles M.: IP8: 8:30, Wed .. """"""" ..... 24 Papanicolaou, George C.: MS46: 10:00, Thu """.31 Kennedy, Judy A.: MS34: 4:00, Tue """""""""". 21 Marcus, Charles M.: MS43: 3:30, Wed """""""". 27 Pattanayak, Arjendu K.: Poster: 7:30, Wed """"".29 Kennedy, T. A. Brian: MS45: 11:30, Thu """"""" 31 Marsden, Jerrold E.: MS41: 3:00, Wed " ...... " ...... 27 Peak, David: CP27: 3:20, Wed " .... """" .. """ ...... 28 Kevrekidis, y.: CP26: 4:20, Wed """"""""""""". 28 Martino, Jennifer A.: MS32: 3:00, Tue " ...... " .. ".21 Pearlman, Howard: MS50: 3:30, Thu " .. """""""" 33 Kevrekidis, Yannis G.: CP33: 11:20, Thu """""". 32 Matkowsky, Bernard J.: MS50: 4:00, Thu .. """ .... 33 Pearson, John: MSI8: 10:30, Mon ...... " ...... "" .. "" IS Keyfitz, Barbara Lee: MS29: 10:00, Tue """"""" 19 McCalpin, John D.: MS30: 4:30, Tue " .... """ ...... 20 Pecora, Louis M.: CPIO: 7:00, Sun .... " ...... " ..... 13 Khasilev, Vladimir Y.: Poster: 7:30, Wed """"""" 28 McClintock, Peter V. E.: MSI9: 10:30, Mon ...... 15 Pekarsky, Sergey: CP8: 7:00, Sun """""" .. """"". 13 Khibnik, A.I.: CPI: 11:20, Sun """"""""""""""" 9 McMillen, Tyler K.: Poster: 7:30, Tue " ...... 23 Peratt, Barry A.: CP29: 3:20, Wed """""" .. " ...... ". 28 Kiehn, R. M.: Poster: 7:30, Wed """""""""""""" 29 McWilliams, James C.: MS30: 4:00, Tue " .. " .... "" 20 Perline, Ron: MS22: 4:30, Mon ...... " .. " .... """""" 17 Kirby, Michael: CP31: 10:40, Thu """""""""""" 31 Meacham, Steve: CP29: 4:20, Wed .. """"" .. "" ..... 28 Pernarowski, Mark: CP23: 10:20, Wed """""""". 25 Kirk, Vivien: MS48: 11:30, Thu """""""""""""" 33 Meiss, James D.: MS36: 10:30, Wed .. " ...... " ...... 24 Peskin, Charles S.: IP7: 1:30, Tue .. """ .... " .... "" ... 20 Knobloch, Edgar: CPI1: 10:40, Mon """""""""" IS Melka, Richard F.: Poster: 7:30, Tue .. " ...... 23 Petrov, Valery: CPI9: 10:00, Tue .. """"" ...... " .... 20 Kocak, Huseyin: MS40: 4:30, Wed """"""""""". 27 Menday, Roger P.: Poster: 7:30, Tue " ...... " ...... 22 Pinsky, Mark A.: CP28: 4:20, Wed """"""" ...... 28 Kocarev, Ljupco: MS38: 11:30, Wed """""""""" 25 Mezic, Igor: MS13: 8:00, Sun .. " ...... 12 Pinto, D. Leao, Jr.: Poster: 7:30, Wed .. "" ...... "" ... 28 Kodama, Yuji: MS33: 4:30, Tue """""""""""""" 21 Mihram, Danielle: Poster: 7:30, Tue .... " ...... " ..... 23 Pinto, David J: CPI8: 11:40, Tue " ...... " .... "" ...... 20 Koiller, Jair: MS28: 11 :30, Tue """""""""""""'" 19 Miller, Bruce N.: Poster: 7:30, Wed " ...... " .. " ...... 28 Piro, Oreste: CP9: 8:40, Sun .... """ .... " .. "" ...... "" 13 Koltsov, N. I.: Poster: 7:30, Wed """"""""""""". 30 Miller, Jacob R.: CP6: 4:20, Sun """ .. " .. """ .... "". II Plohr, Bradley: MS29: II :00, Tue ...... " .. """" ... 19 Koon, Wang-Sang: CP8: 8:00, Sun """"""""""". 13 Miller, Patrick D.: CP32: 11:20, Thu .... " .. " .... " .... 32 Posch, Harald A.: MS44: 10:30, Thu """"" .. """ .. 30 Kostelich, Eric J.: MS40: 4:00, Wed """""""""". 27 Miller, Peter D.: CP7: 8:20, Sun ...... 13 Proctor, Michael R. E.: CP24: 11:40, Wed ...... "" .. 26 Kozhevnikov LV.: Poster: 7:30, Wed """""""""" 30 Miller, Walter: MSI4: 8:30, Sun "" ...... 13 Provenzale, Antonello: CP32: 11 :40, Thu ...... ". 32 Kozlov, Alexander K.: Poster: 7:30, Tue """"""" 23 Millonas, Mark M.: CP20: 3:40, Tue " .... " .. ""...... 22 Provenzale, Antonello: MSI5: 10:00, Man " ...... " 14 Krauskopf, Bernd: MS37: 10:30, Wed """"""""" 25 Mills, Michael J.: CP4: 4:20, Sun"""" .. """""""" 11 Pugh, Charles c.: MS34: 4:30, Tue .... " .. """"""".21 Kuksin, Sergei B.: MS46: 11:00, ThU """"""""". 31 Milton, John G.: MSIO: 4:00, Sun ...... " .... " .. """ .. 11 Kuramoto, Yoshiki: IPIO: 8:30, Thu """""""""". 30 Mischaikow, Konstantin: MS34: 3:30, Tue" ...... 21 Q Kurths, Jiirgen: CP25: 10:00, Wed """""""""""" 26 Mogilner, Alex: Poster: 7:30, Tue ...... " .... " ...... 24 Quinn, D. Dane: CP22: 4:40, Tue"" .... " ...... " .. """ 22 Kurths, Jiirgen: MSI6: 10:00, Mon """"""""""". 14 Molkov, Ya. I.: Poster: 7:30, Tue """"""" .. """ ..... 23 Kuske, Rachel: MS48: 10:00, Thu """""""""""" 32 Montaldi, James: CP6: 4:00, Sun .... " .... " ...... " .. " .. 11 R Kutz, J. Nathan: CP4: 3:40, Sun """""""""""""" II Montgomery, Richard: MS41: 3:30, Wed ...... 27 Rabinovich, A.: CPI8: 10:40, Tue ...... """""" 20 Montgomery, Richard w.: MS28: 11:00, Tue " ..... 19 L Rado, Anita: Poster: 7:30, Tue "" ...... """""""",, 23 Morrison, P. J.: Poster: 7:30, Tue """""" .. """ .. "" 23 Raghavachari, Sridhar: CP27: 3:40, Wed " .. " .... "" 28 Lai, Ying-Chen: MS5: 11:00, Sun """"""""""""". 9 Mosekilde, Erik: CP30: 10:20, Thu " .. """ .. " .... " ... 31 Ramirez-Rojas, Alejandro: Poster: 7:30, Tue " .. "" 23 Lai, Ying-Cheng: MS51: 4:30, Thu """""" .. " .... ".33 Moss, Frank: IP5: 2:30, Mon " .. " .... " ...... " ... 16 Rand, Richard H.: MS9: 4:30, Sun " ...... """"""" II Lai, Ying-Cheng: Poster: 7:30, Wed """ .. """ .... "" 29 Mueller, Carl: MSI2: 7:30, Sun " ...... """"""...... 12 Rangarajan, Govindan: CP28: 3:00, Wed .. """""" 28 Laing, Carlo: Poster: 7:30, Tue""""""" .. """ .. """ 22 Muenkel, Markus: MS37: 11:30, Wed" .. " .... "" ..... 25 Rehacek, Jan: Poster: 7:30, Wed "" ...... " ...... " .. 30 Lamba, Harbir: CP28: 3:20, Wed """"""""""" .... 28 Muldoon, Mark: CP34: 3:00, Thu ...... " ...... " ... 33 Reichl, Linda E.: MSI9: 10:00, Man .... " .. """""" IS Landsberg, Adam S.: CP13: 10:40, Mon """" ...... IS Muldoon, Mark: MS8: 3:00, Sun ...... "" .. """ .. " .. 10 Rempfer, Dietmar: CP31: 10:20, Thu "" .... """""" 31 Langer, Joel: MS22: 4:00, Mon """""""""""""'" 17 Munuzuri, Alberto P.: MS39: 10:30, Wed .... " ...... 25 Renardy, Yuriko Y.: CPI4: 4:00, Man " .. "" .. """". 17 Lathrop, Daniel P.: CPI1: 10:20, Mon .. "" .... """" IS Muraki, David: MSI8: 11:30, Mon """ .. """"...... IS Ricca, Renzo L.: CP13: 10:00, Man """"""" .... "" IS Latushkin, Yuri: CPI2: 10:00, Mon """" ...... """" IS Murray, Richard M.: IP9: 1:30, Wed """"""""" ... 26 Riecke, Hermann: MS6: 4:00, Sun """""" .. ""." ... 10 Layton, Harold E.: MSIO: 4:30, Sun """""""""". 11 Murray, Rua D. A.: MSI4: 9:00, Sun """ .. " .... "" .. 13 Lebovitz, Norman R.: MS27: 10:00, Tue """"""" 19 Rinzel, John: MSI7: 10:00, Man "" .. """"""""" .. 14 Lee, Jon: Poster: 7:30, Wed """"""""""""""""... 29 N Robert, Carl: Poster: 7:30, Wed ...... """"""". 28 Leger, James R.: MS45: 10:30, Thu """"" .. "" ...... 31 Roberts, A. J.: CPIO: 9:00, Sun .... " .... " ...... """". 13 Nadiga, Balu T.: MS30: 3:30, Tue .. " .. " .. " .. " ...... 20 Rogers, Jeffrey L.: CP21: 3:00, Tue "" .. " .... " ...... 22 Leruer, David E.: CP20: 4:40, Tue ...... """""" .... " 22 Nagai, Yoshihiko: CP5: 3:00, Sun """" .... " ...... 11 Lerner, David E.: Poster: 7:30, Tue .... " ...... """ .. ". 23 Rogers, Kathleen A.: MS32: 4:00, Tue "" .. " ...... ".21 Nagai, Yoshihiko: Poster: 7:30, Tue """ .. " .. "".... " 23 Rom-Kedar, Vered: CP8: 7:20, Sun .. """" .. " ...... ". 13 Levermore, C. David: IPI1: 1:30, Thu """""""" .. 32 Nam, Keeyeol: CP24: 10:40, Wed " .. """" .... "" ..... 26 Levey, David B.: CP22: 4:00, Tue .... " .... """"""... 22 Rom-Kedar, Vered: MS36: 11:30, Wed ...... ".24 Neiman, Alexander: CP25: 11:00, Wed .. " .. "" ...... 26 Romero, Luis F.: CPI: 11:40, Sun ...... " ...... 9 Levi, Mark: MS9: 3:30, Sun " .... " .. "" ...... """"""" II Nepomnyashchy, A. A.: CPI4: 4:40, Mon " .... "" .. 17 Levine, Herbert: MSI1: 7:30, Sun ...... """ .... ". 12 Rosa, Epaminondas, Jr.: MS5: 10:00, Sun .. " ...... 9 Conference Prog ram DYNAMICAL SYSTEMS " 37 SPEAKER INDEX

Rosenblum, Michael: MSI6: 10:30, Mon ...... 14 Timofeyev, Ilya: CPI: 10:20, Sun ...... 9 Y Rotariu-Bruma, A. I.: Poster: 7:30, Wed ...... 30 Tobias, Steven M.: CP24: 10:00, Wed ...... 26 Rottschafer, Y: CPIS: 4:40, Mon ...... 18 Todd, Michael D.: CP37: 4:40, Thu ...... 34 Yagasaki, Kazuyuki: CP22: 3:20, Tue ...... 22 Roulstone, Ian: MS7: 3:30, Sun ...... 10 Tolle, Charles R.: Poster: 7:30, Wed ...... 29 Yagasaki, Kazuyuki: Poster: 7:30, Tue ...... 22 Roytburd, Victor: MS49: 3:30, Thu ...... 33 Tolman, L. K.: Poster: 7:30, Wed ...... 30 Yalccinkaya, Tolga: CPIO: 8:00, Sun ...... 13 Rubin, Jonathan E.: Poster: 7:30, Tue ...... 23 Tovbis, Alexander: Poster: 7:30, Wed ...... 29 Ya1ccinkaya, Tolga: Poster: 7:30, Tue ...... 23 Rucklidge, A. M.: CP33: 10:20, Thu ...... 32 Tracy, Eugene R.: Poster: 7:30, Tue ...... 23 Yang, Jianke: CP7: 7:40, Sun ...... 13 Rulkov, Nikolai F: CP9: 8:20, Sun ...... 13 Trenary, Timothy J.: Poster: 7:30, Wed ...... 29 Yang, Tian-Shiang: CP4: 4:00, Sun ...... II Tresser, Charles: CP6: 3:00, Sun ...... 11 Yorke, James A.: MS34: 3:00, Tue ...... 21 S Triandaf, Ioana: MS31: 3:30, Tue ...... 21 Yorke, James A.: MS47: 11:30, Thu ...... 31 Young, Todd: Poster: 7:30, Wed ...... 29 Sain, Filip: Poster: 7:30, Wed ...... 29 Tribbia, Joe: MS7: 4:00, Sun ...... 10 Yuan, Guocheng: CP6: 3:40, Sun ...... 11 Saluena, Clara: CP29: 3:00, Wed ...... 28 True, Hans: CP37: 5:00, Thu ...... 34 Yuan, Guohui: MS47: 10:30, Thu ...... 31 Samelson, Roger M.: MS2: 10:00, Sun ...... 8 Trzaska, Zdzislaw W.: Poster: 7: 30, Tue ...... 22 Yulin, A. Y.: Poster: 7:30, Tue ...... 23 Samuel, Aravi: MS28: 10:00, Tue ...... 19 Tsimring, Lev S.: MS38: 11:00, Wed ...... 25 Sandstede, Bjorn: CPI5: 4:00, Mon ...... 17 Tu, Yuhai: MS42: 4:30, Wed ...... 27 z Sanjuan, Miguel A. F: Poster: 7:30, Tue ...... 24 Turitsyn, S. K.: CP7: 7:20, Sun ...... 13 Schecter, Stephen: MS29: 11 :30, Tue ...... 19 Zaslavsky, G. M.: MS36: 10:00, Wed ...... 24 Schenk, Susan J.: CP30: 10:00, Thu ...... 31 u Zbilut, Joseph P.: CP34: 4:00, Thu ...... 33 Schiff, Steven J.: CPS: 4:00, Sun ...... II Uhl, Christian: CP31: 11:20, Thu ...... 32 Zharnitsky, Vadim: CP8: 8:20, Sun ...... 13 Schittenkopf, Christian: CP34: 3:20, Thu ...... 33 Uhl, Christian: Poster: 7:30, Tue ...... 23 Zimmennann, Martin: Poster: 7:30, Wed ...... 29 Schmalfusk, Bjorn: MS26: 12:00, Tue ...... 18 Umeki, Makoto: CP7: 7:00, Sun ...... 13 Zoldi, Scott M.: Poster: 7:30, Tue ...... 22 Schmidt, Karin: Poster: 7:30, Tue ...... 23 Urbanskij, Marius: CPI2: 10:20, Mon ...... IS Schober, Constance M.: MS49: 3:00, Thu ...... 33 Schreiber, Thomas: MS8: 4:30, Sun ...... 10 v Schreiber, Thomas: Poster: 7:30, Wed ...... 28 Vakakis, Alexander E: CP3: 10:00, Sun ...... 9 Schroder, Elsebeth: CP24: 11 :20, Wed ...... 26 van Beijeren, Henk: MS44: 10:00, Thu ...... 30 Schroer, Christian G.: MS47: 11 :00, Thu ...... 31 van der Burgh, Adriaan H. P.: MS9: 4:00, Sun ..... 11 Schult, Daniel A.: CP26: 3:20, Wed ...... 27 van der Heijden, G. H. M.: CP3: 11:20, Sun ...... 9 Schwartz, Ira B.: CPI9: 10:40, Tue ...... 20 Van Vleck, Erik S.: MS39: 12:00, Wed ...... 25 Selgrade, James E: Poster: 7:30, Tue ...... 23 Van Wirt, Peter M.: Poster: 7:30, Wed ...... 29 Senn, Walter: MS23: 5:00, Mon ...... 17 VanderVorst, R.: CPI5: 5:20, Mon ...... 18 Sharpe, J. P.: CPI7: 10:20, Tue ...... 19 Vargas, Carlos A.: Poster: 7:30, Tue ...... 23 Shashikanth, B. N.: Poster: 7:30, Wed ...... 28 Vaynblat, Dimitri D.: CPI1: 11:20, Mon ...... IS Shefter, Michael G.: CP11: 11:00, Mon ...... 15 Venkataramani, Shankar C: CP35: 3:40, Thu ...... 34 Shen, Wenxian: MS39: 11:00, Wed ...... 25 Verhulst, Ferdinand: CP2: 10:20, Sun ...... 9 Shevchenko, Ivan I.: Poster: 7:30, Tue ...... 22 Vincent, Thomas L.: MS47: 10:00, Thu ...... 31 Shkoller, Steve: CP36: 3:40, Thu ...... 34 Vohra, SandeepT.: CP33: 11:40, Thu ...... 32 Short, Kevin M.: CP27: 4:00, Wed ...... 28 Volpert, Vladimir A.: MS50: 4:30, Thu ...... 33 Sirovich, Larry: MS31: 4:30, Tue\ ...... 21 Voroney, Jon-Paul: Poster: 7:30, Tue ...... 24 Skeldon, Anne C.: CPI4: 5:00, Mon ...... 17 Smereka, Peter: CP21: 3 :21, Tue ...... 22 w So, Paul T.: CPS: 3:40, Sun ...... 11 Wagon, Stan: MS24: 5:30, Mon ...... 17 Socolar, Joshua E. S.: MS6: 4:30, Sun ...... 10 Walsh, James A: CPI7: 10:00, Tue ...... 19 Sommerer, John C.: CPI6: 5:20, Mon ...... 18 Wang, Xiao-Jing: MS23: 4:30, Mon ...... 17 Sommerer, John C: Poster: 7:30, Tue ...... 23 Warnock, Robert L.: CP28: 4:00, Wed ...... 28 Soskin, S. M.: Poster: 7:30, Tue ...... 22 Watanabe, Shinya: Poster: 7:30, Tue ...... 23 Soto-Trevino, Cristina: CPI5: 5:00, Mon ...... 18 Wattenberg, Frank: MS24: 5:00, Mon ...... 17 Sparrow, Colin: CP6: 4:40, Sun ...... 12 Watts, Duncan J.: CP21: 3:40, Tue ...... 22 Stark, Jaroslav: CPIO: 7:40, Sun ...... 13 Wayne, Clarence Eugene: MS46: 10:30, Thu ...... 31 Stark, Jaroslav: CP34': 4:20, Thu ...... 34 Weckesser, Warren: CP33: 10:00, Thu ...... 32 Stepan, Gabor: CP37: 3:40, Thu ...... 34 Weinstein, Michael I.: MS46: 11:30, Thu ...... 31 Stone, Emily F: MS48: 11:00, Thu ...... 33 Wellman, Richard: Poster: 7:30, Wed ...... 30 Storti, Duane W: MS9: 3:00, Sun ...... II Whalen, Timothy: CP22: 4:20, Tue ...... 22 Strain, Matthew C.: Poster: 7:30, Tue ...... 23 White, Peter: CP30: II :40, Thu ...... 31 Subbarao, D.: CP17: 10:40, Tue ...... 19 Wicklin, Frederick J.: MS20: 5:00, Mon ...... 16 Succi, Sauro: Poster: 7:30, Tue ...... 22 Wiercigroch, Marian: CP3: 10:40, Sun ...... 9 Summers, Danny: Poster: 7:30, Wed ...... 28 Wiesel, William E.: Poster: 7:30, Wed ...... 29 Sushchik, Mikhail M. Jr.: CP21: 4:00, Tue ...... 22 Wiesenfeld, Kurt: MS3: 10:00, Sun ...... 8 Susheik, M. M.: CP23: 12:00, Wed ...... 26 Wiggins, Chris H.: Poster: 7:30, Tue ...... 23 Szmolyan, Peter: CP13: 11:20, Mon ...... IS Wihstutz, Volker: MS26: 11:30, Tue ...... 18 T Williams, Christopher: Poster: 7:30, Tue ...... 22 Willms, Allan R.: CP20: 4:00, Tue ...... 22 Tajdari, Mohammad: CP2: 10:40, Sun ...... 9 Winfree, ArthurT.: MSII: 8:30, Sun ...... 12 Tanaka, Hisa-Aki: CP21: 4:20, Tue ...... 22 Winful, Herbert G.: MS33: 3:30, Tue ...... 21 Tang, Chao: MS42: 3:30, Wed ...... 27 Winful, Herbert G.: MS45: 10:00, Thu ...... 31 Tang, Xianzhu: CP32: 11:00, Thu ...... 32 Wirosoetisno, Djoko: MS7: 4:30, Sun ...... 10 Tang, Xianzhu: Poster: 7:30, Tue ...... 23 Wittenberg, Ralf W: CP27: 4:20, Wed ...... 28 Tarman, I. Hakan: CP31: 10:00, Thu ...... 31 Worfolk, Patrick: MS20: 5:30, Mon ...... 16 Taylor, Thomas J.: CPI2: 10:40, Mon ...... IS Tel, Tamas: MS44: 11:30, Thu ...... 30 x Terman, David: MSI7: 11:00, Mon ...... 14 Xin, Jack: MSI2: 8:00, Sun ...... 12 Thiffeault, Jean-Luc: CPI4: 5:40, Mon ...... 17 Thiran, Patrick: MS39: 11 :30, Wed ...... 25 38 " DYNAMICAL SYSTEMS Conference Program ABSTRACTS

,~ Confers ~,r '-' 1/~c9 ~ {'-/ ~

~ (l ft DYNAMICAL e,'- ~ .~ _ V SYSTEMS U May 18-22, 1997 Snowbird Ski and Summer Resort • Snowbird, Utah Abstracts 39

CPO! Salt Lake City, UT Phase Jumps of 7r in a Laser with a Periodically Forced Injected Signal Amplitude and phase dynamics of a laser with injected CPO! signal are analyzed as the injection amplitude is varied by Bifurcation Diagram of Two Lasers with Injected a small periodic term. Two-timing analysis yields slow time Field evolution equations of a periodically forced Toda oscillator. We study the dynamics of two coupled solid state lasers The dynamics near resonance are shown to be of particular subject to an injected field. Using singular perturbation interest. Control of the output amplitude and jumps of and averaging techniques we arrive at a phase model for 7r in the phase by changing the injection amplitude and which we present a comprehensive bifurcation analysis. We frequency are discussed. particularly study routes to full entrainment as the injected Peter A. Braza field amplitude increases and observe that the entrain­ Department of Mathematics and Statistics ment does not monotonically improve with the injected University of North Florida field strength. Our numerical simulations suggest that the Jacksonville, FL 32224 phase model captures the relevant dynamics well beyond E-mail: [email protected] the asymptotic regime in which it is derived. A.I. Khibnik Theory Center, 657 Rhodes Hall CPO! Cornell University, Ithaca, NY 14853 Synchronized Patterns in Coupled Lasers Y. Braiman The Maxwell-Bloch equations for a laser with saturable ab­ Department of Physics sorber may undergo a Hopf bifurcation which, since there is Emory University, Atlanta, GA 30322 no distinguished polarization direction, occurs in an equiv­ ariant setting as a Hopf bifurcation with O(2)-symmetry. If T.A.B. Kennedy and K. Wiesenfeld several such lasers are coupled in a ring, the full system pos­ sesses a Dn x O(2)-symmetry. The Hopf bifurcation with School of Physics Georgia Institute of Technology, Atlanta, GA 30332 this symmetry predicts a number of different states which can be classified group-theoretically. These states and their symmetry properties are described and intrepreted as dis­ tinct synchronized radiation patterns for the coupled laser CPO! system. Moreover, the dynamics of the underlying normal Nonlinear Dynamics of Semiconductor Laser Ar­ form shows structurally stable heteroclinic cycles which in­ rays duce transitions between different patterns. The nonlinear dynamics of multi-stripe semiconductor Gerhard Dangelmayr lasers is investigated numerically as a function of the injec­ Colorado State University tion current, and number and width of and separation be­ Department of Mathematics tween stripes, and shown to exhibit a rich behaviour char­ Ft. Collins, CO 80523 acterized by regular or coherent phenomena, filamentation, E-mail: [email protected] incoherence, nonlinear interactions between the stripes, and chaos. For two-stripe lasers, it has been found that Michael Wegelin the average intensity may exhibit spatio-temporal oscilla­ University of Tiibingen tions, whereas, for five--stripe ones, bursting phenomena in Tiibingen, FRG the three central stripes was found. Luis F. Romero, Emilio L. Zapata, and J. 1. Ramos Departamento de Arquitectura de Computadores CPOl E. T. S. Ingenieros Industriales Spatio-Temporal Dynamics of Broad-Area Semi­ University of Malaga, Spain conductor Lasers with Optical Feedback E-mail: [email protected] The spatio-temporal behavior of broad-area semiconduc­ tor lasers with conventional optical feedback and grating feedback is studied experimentally. We investigate the ef­ CPO! fects of the collimating lens position, the external cavity Homoclinic Orbits for the Second Harmonic Gen­ length, the optical feedback level, and the injection current. eration of Light in an Optical Cavity In addition to interesting nonlinear dynamics, such prac­ We present two large families of Silnikov-type homo clinic tical effects as lateral-mode selectivity and beam quality orbits that exist in a model of an asymmetric crystal inter­ enhancement are exploited to build high power external­ acting with light in a passive optical cavity. This crystal cavity diode laser sources with high temporal and good generates the second harmonic frequency. The resulting spatial coherence. brightness of the light with the fundamental and second George R. Gray and Ming-Wei Pan harmonic frequencies is described by a two degree of free­ Department of Electrical Engineering dom near integrable system which is close to a 2:1 res­ University of Utah onance. These families of homo clinic orbits give rise to 3280 Merrill Engineering Building chaotic dynamics in the model. In order to obtain these Salt Lake City, UT 84112 orbits we use a combination of the Melnikov method and email: [email protected] geometric singular perturbation theory. Alejandro B. Aceves David H. DeTienne Department of Mathematics and Statistics Nichols Corporation University of New Mexico, Albuquerque, NM 87131 40 Abstracts

CP02 Darryl D. Holm Necessary and Sufficient Conditions for Finite­ Theoretical Division Time Blow-Up in ODE's Los Alamos National Laboratory, Los Alalmos, NM 87544 The problem of finding conditions on a nonlinear vector field so that the solutions have finite time singularities is Gregor Kovacic and I1ya Timofeyev discussed. A general theorem which provides necessary and Department of Mathematical Sciences sufficient conditions for the blow-up, in real finite time, of Rensselaer Polytechnic Institute, Troy, NY 12180 the general solution to an autonomous, n-dimensional non­ Email: [email protected] linear ODE is presented. This theorem involves only the asymptotic form of the local series solution in complex time and thus provides a simple algorithm for determining the CP02 existence of finite time blow-up. The application of this Fast Resonance Shifting as a Mechanism of Insta­ new result to fluid mechanics and Hamiltonian dynamics bility in Dynamics Illustrated by Comets is also considered. Fast resonance shifting is a newly confirmed mechanism Craig Hyde and Alain Goriely for instability of motion in conservative systems. It was Program in Applied Mathematics originally observed numerically in 1989 as a way a parti­ University of Arizona, TUcson, AZ 85721 cle can rapidly hop between resonant periodic orbits in the E-mail:[email protected]. three-body problem, traversing large regions of the phase [email protected] space. The motion of the zero mass particle was about the Earth under the gravitational perturbation of the Moon. Unlike the standard process of Arnold diffusion which pre­ CP02 dicts that this process should take on the order of a bil­ On Lie Transforms for Differential Systems lion years, this resonance shifting, termed the hop, took about 30 days. Because of the rapidity of the hop, it was We deal with Lie transforms for ordinary differential sys­ felt that perhaps there were numerical errors. However, in tems with a small perturbation. The theory, presented 1995, it was discovered by Brian Marsden of the Harvard­ firstly by Deprit for Hamiltonian systems and extended Smithsonian Center for Astrophysics, that perhaps some thereafter by Kamel and Henrard for differential equations, comets moving about the Sun were performing the hop. has been recently generalized by Meyer for one contravari­ This was confirmed, and a paper explaining this and its ant tensor fields. Here we give a systematic deduction of connection with the work of John Mather, will appear in all the algorithms, presenting also some variations for spe­ The Astronomical Journal. It turns out that the hop oc­ cific cases as well as some combinations of the different curs at a region about the secondary mass called the 'fuzzy approaches useful when the perturbation is composed by boundary' where it is conjectured a hyperbolic tangle ex­ terms of different type. ists. The fuzzy boundary of the Moon also has important J. Palacian applications to the motion of spacecraft, and using it, the Departamento de Matematica e Informatica author found a new low route to the Moon in 1990 which Universidad Publica de Navarra was demonstrated by a Japanese spacecraft in 1991 arriv­ 31006 Pamplona, Spain ing at the Moon in October of that year. Email [email protected] Edward Belbruno The Geometry Center R. Barrio 1300 South Second Street Departamento de Matematica Aplicada Minneapolis, MN 55454 Universidad de Zaragoza (612)626-1845 50 Zaragoza, Spain [email protected]

CP02 CP02 Evolution Towards Symmetry Hamiltonian G-space Normal Forms as a Geometric We trace the long-time effect of the asymmetries in a Framework for Perturbation Theory Hamiltonian which becomes slowly symmetric in time. We Hamiltonian G-space normal forms are appropriate as a ge­ establish the relation between evolution towards symmetry ometric framework for perturbation theory when a system's and dissipative mechanics while using the techniques of av­ integrability comes from a non-Abelian symmetry group. eraging and adiabatic invariants, supported by Mathemat­ A particular normal form appearing in Dazord and Delzant ica 2.2. For 2-dof we apply the Arnold-Neihstadt theory of (1987), which we call action-group coordinates, is a 'non­ passage through resonance. The analysis reveals significant Abelian' generalization of action-angle coordinates. We asymmetric dynamics even when the asymmetric contribu­ use these coordinates to generalize a theorem of Nekhoro­ tions to the potential have become negligibly small (joint shev (1977), deducing exponential estimates for momen­ work with R.A.J.G.Huveneers). tum maps in a class of Hamiltonian G-spaces with 'almost Ferdinand Verhulst G-invariant' Hamiltonian. Department of Mathematics Anthony Blaom University of Utrecht Mathematics P.O. Box 80.010 253-37 Caltech 3508 TA Utrecht, The Netherlands Pasadena, 91125 CA E-mail: [email protected] [email protected] Abstracts 41

CP03 linear String Using Nonsmooth Transformations Dynamical Friction Modeling We analyze forced oscillations of a string on a nonlinear We derive a two-degree-of-freedom dynamical model of fric­ discrete foundation. Applying nonsmooth transformations tion from a macroscopic description of the interactions be­ of the spatial variable (Pilipchuck, 1985), we eliminate sin­ tween nominally flat surfaces. In particular, coupling is gularities from the equations of motion and obtain two non­ introduced between the horizontal motion and the separa­ linear nonhomegeneous boundary value problems (BVPs)j tion of the surfaces. The nonlinear model is studied and these are solved with a perturbation method. The asso­ found to qualitatively agree with experimentally observed ciated boundary conditions are +smoothening relations+ properties of dynamical friction. For example, stick-slip related to the transformations. As an application, local­ behavior is found to be associated with a Hopf bifurcation ized oscillations of the string are studied, and discreteness from the steady-state equilibrium. effects are included in the analytical results. Harry Dankowicz Gary Salenger, and Alexander F. Vakakis Department of Mechanics Department of Mechanical and Industrial Engineering Royal Institute of Technology University of Illinois at Urbana - Champaign S-100 44 Stockholm, Sweden 1206 W. Green Street, Urbana, IL 61801 E-mail: [email protected] [email protected]

CP03 CP03 Graph Theoretic Implications for Piecewise Linear Stochastic Effect on Chaotic Dynamics of Cutting Systems of Arbitrary Dimension Process Ideas from graph theory are used to produce a basis for all A model of the cutting process is formulated based on a possible periodic orbits to the piecewise linear system of a random variation of the specific cutting resistance. The rocking block confined between two vertical walls. This ap­ cutting resistance is generated as a simple one-dimensional proach uncovers previously unknown solutions to the prob­ univariate Gaussian process using the spectral represen­ lem which are subsequently observed in numerical simula­ tation method. The random responses are discussed and tions. The method is then applied to a one dimensional compared with the deterministic ones within the ranges array of m impact oscillators used to model the dynamics of parameters where chaotic motion occurs. Contrary to of a heat exchanger. It is shown that the total number of a claim for continuous systems, in which the variance of distinct impacting configurations is given by the (m+4)th the noise dampens the chaotic vibration, such a behavior Fibonacci number. It is suggested that this method can is not observed in the discontinuous system. The chaotic be adapted to other piecewise linear systems of arbitrary response is still present and the stochasticity induces only dimension. huge impact forces during the transient period. John Hogan and Martin Homer Marian Wiercigroch University of Bristol University of Aberdeen, Kings College Department of Engineering Mathematics Department of Engineering Queens Building, University Walk Aberdeen AB9 2UE, United Kingdom Bristol BS8 1TR, United Kingdom E-mail: [email protected]. uk E-mail: [email protected] Alexander H-D. Cheng Department of Civil Engineering CP03 University of Delaware, Newark, DE 19716 Homoclinic Orbits and Localized Buckling of Cylin­ E-mail: [email protected] drical Shells Experimentally the buckling of a long elastic cylindrical shell under axial compression occurs locally along the ax­ CP04 ial length. However the standard analytic approach as­ Nonlinear Polarization-Mode Dispersion in Optical sumes the buckle pattern comprises of an interaction of pe­ Fibers with Randomly Varying Birefringence riodic eigenmodes axially and circumferentially. We intro­ Randomly varying birefringence leads to nonlinear duce the von Karman-Donnell equations, their discretiza­ polarization-mode dispersion (PMD) in addition to the tion, and seek homo clinic orbits for the resulting system of well-known linear PMD. Here we calculate the variance o.d.es. These yield axially localized solutions as observed of the field fluctuations produced by this nonlinear PMD. experimentally. Excellent agreement with experiments is We also derive the equilibrium probability distributions achieved with fewer modes than other methods. for these PMD terms, and track the evolution of the po­ Gabriel J. Lord and Alan R. Champneys larization state's probability distribution from its initial Department of Engineering Mathematics condition to its steady-state uniform distribution on the University of Bristol, BS8 1 TR, United Kingdom Poincare sphere. E-mail: [email protected] William L. Kath Engineering Sciences and Applied Mathematics Giles W. Hunt Northwestern University Department of Mechanical Engineering 2145 Sheridan Rd. University of Bath, BA2 7AY, United Kingdom Evanston, IL 60208-3125

P. K. A. Wai CP03 Department of Electronic Engineeering Forced Oscillations of a Discretely Supported Non- The Hong Kong Polytechnic University 42 Abstracts

Hung Hom, Kowloon, Hong Kong CP04 Switching-Induced Timing Jitter in Nonlinear Op­ Curtis R. Menyuk and J. W. Zhang tical Loop Mirrors Department of Electrical Engineering The interaction of two copropagating pulses of different University of Maryland frequencies in a nonlinear optical loop mirror switch is in­ Baltimore, MD 21228-5398 vestigated. An analytic solution in the limit of large veloc­ ity difference is found. This analysis, as well as numerical studies, demonstrates signal pulse acceleration due to con­ CP04 trol pulse evolution. Pulse Dynamics in Fiber Lasers Michael J. Mills and William L. Kath A comprehensive theoretical treatment of the modelock­ Engineering Sciences and Applied Mathematics ing pulse dynamics in a fiber laser cavity with a saturable Northwestern University Bragg reflector is given. The governing pulse dynamics is 2145 Sheridan Rd. shown to be given by a modified Ginzburg-Landau equa­ Evanston, IL 60208-3125 tion with time dependent gain and diffusion terms. In a particular asymptotic limit, chirped solitary wave solutions are shown to exist and we consider their stability. Analytic CP04 and numerical simulations show the derived model to be in Timing Jitter Reduction in a Fiber Laser Mode­ quantitative agreement with experimental results. Locked by an Input Bit Stream J. Nathan Kutz and Philip Holmes All-optical timing recovery devices offer an efficient way Program in Applied and Computational Dynamics of reducing timing jitter in long-distance communication Princeton University, Princeton, NJ 08544 systems. We perform analytical and numerical studies of Michael Weinstein timing jitter reduction in a clock-recovery device in which Department of Mathematics, University of Michigan a data stream is used to mode-lock a fiber laser through Ann Arbor, MI 48109 the effect of cross-phase modulation. This situation is an­ alyzed in the soliton limit where the laser pulses are close Keren Bergman to optical solitons and the effect of the data steam on the EECS Department, Princeton University laser pulses is predicted using soliton perturbation theory. Princeton, NJ 08544 Analytical and numerical results obtained here show a sig­ nificant reduction in the rms timing jitter in the output laser pulse stream from the level present in the input data stream. CP04 Phase Dynamics in Optical Communications Anne Niculae and William L. Kath Engineering Sciences and Applied Mathematics Inverse scattering theory and associated soliton perturba­ Northwestern University tion theories have been essential for the description of soli­ 2145 Sheridan Rd. ton communications systems modeled by (f)NLS and its Evanston, IL 60208-3125 near-integrable counterparts. In recent work, semiclassical modulation theory using low genus solutions of the (d)NLS was successfully applied to describe NRZ pulse propaga­ tion. In this talk, we take the point of view that the the­ CP04 ory of N-phase solutions of integrable evolution equations Optical Pulse Dynamics in Dispersion Managed unifies soliton and NRZ work. After showing how the /-L­ Fibers variable approach can be used to obtain several well known For both soliton and non-return-to-zero (NRZ) data for­ results from optical soliton theory, an analysis of phase mats transmission in optical fibers can be significantly shifts and phase dynamics is carried out. The concept of improved by periodically varying the group-velocity dis­ the geometric phase for solitons is also reviewed, and finite­ persion. Here, in the short period limit an averaged time phase shift formulae are generated. Phase dynamics pulse evolution equation is derived, resulting in a nonlin­ are often critical in optical communications devices, and ear Schrodinger (NLS) equation with phase-independent this approach provides a natural way to analyze them. perturbations that is relatively easy to deal with. In the Gregory G. Luther soliton case, analysis indicates that soliton-like pulses are University of Notre Dame, Notre Dame, IN and possible, and a theoretical expression for pulse energy in BRIMS, Hewlett-Packard Laboratories terms of the dispersion map parameters is obtained. Bristol, England Tian-Shiang Yang and William L. Kath Engineering Sciences and Applied Mathematics Mark S. Alber Northwestern University University of Notre Dame 2145 Sheridan Rd. Notre Dame, IN. Evanston, IL 60208-3125 Jerrold E. Marsden Caltech Pasadena, CA. CP05 Hopf's Last Hope: Spatiotemporal Chaos in Terms Jonathan Robbins of Unstable Recurrent Patterns BRIMS, Hewlett-Packard Laboratories, and Bristol University Spatiotemporally chaotic dynamics of a Kuramoto­ Bristol, England Sivashinsky system is described by means of an infinite hi­ erarchy of its unstable spatiotemporally periodic solutions. Abstracts 43

An intrinsic parametrization of the corresponding invariant methods and a recent transformational technique. The im­ set serves as accurate guide to the high-dimensional dy­ plications of these findings will be discussed. namics, and the periodic orbit theory yields several global Steven Schiff averages characterizing the chaotic dynamics. Children's National Medical Center F Christiansen, P Cvitanovic and V Putkaradze Washington, DC 20010 Center for Chaos and Thrbulence Studies E-mail: [email protected] Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark CP05 Detecting Unstable Periodic Orbit in Experimental CP05 Data Symplectic Integrators A general nonlinear method to extract unstable periodic Versus Ordinary-Different ial-Equation Solvers for orbits from chaotic time series is proposed. Utilizing the Hamiltonian Systems estimated local dynamics along a trajectory, we devise a In this talk, we compare the performance of symplectic in­ transformation of the time series data such that the trans­ tegrators and ordinary-differential-equation (ODE) solvers formed data is concentrated on the periodic orbits. Thus, for some Hamiltonian systems. Considered here are the one can extract unstable periodic orbits from the trans­ symplectic integrators of second-order (Feng, Ruth, Fried­ formed data by looking for peaks in a histogram approxi­ man & Auerbach), fourth-order (Forest & Ruth, Candy & mation of its distribution function. In the presence of noise, Rozmus) and sixth-order (Yoshida), and the ODE solvers the statistical significance of the results is assessed using of DEABM and DERKF (Shampine) of the SLATEK li­ surrogate data. brary and LSODE (Hindmarsh) of ODEPACK. These in­ Paul T. So tegrators and solvers are used for trajectory computation of Department of Neurosurgery the Holmes oscillator, two-body central force field motion, Children's National Medical Center Henon-Heiles system, and Fermi-Pasta-Ulam chain. The 8 Washington, D.C. main observation is that up to 10 iterations ODE solvers E-mail: [email protected] perform just as well as symplectic integrators in accuracy, but fall short in efficiency. Edward Ott Jon Lee Institute for Plasma Research Wright Lab (FIB) University of Maryland Wright-Patterson AFB, OH 45433 College Park, MD J [email protected] Steven J. Schiff Department of Neurosurgery Children's National Medical Center CP05 Washington, D.C. Characterization of Blowout Bifurcation by Unsta­ ble Periodic Orbits Tim Sauer Blowout bifurcation in chaotic dynamical systems occurs Department of Mathematics when a chaotic attractor, lying in some invariant subspace, George Mason University becomes transversely unstable. We establish quantitative Fairfax, VA characterization of the blowout bifurcation by unstable pe­ riodic orbits embedded in the chaotic attractor. We argue Celso Grebogi that the bifurcation is mediated by changes in the trans­ Institute for Plasma Research verse stability of an infinite number of unstable periodic University of Maryland orbits. There are two distinct groups of periodic orbits: College Park, MD one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted trans­ verse eigenvalues of these two groups are balanced. CP06 Yoshihiko Nagai and Ying-Cheng Lai Dynamic Period-Doubling Bifurcations of a Uni­ Department of Physics and Astronomy modal Map Kansas Institute for Theoretical and Computational Sci­ We consider period-doubling bifurcations of a general one­ ence dimensional unimodal map as a control parameter is swept University of Kansas, Lawrence, Kansas 66045 slowly through the autonomous bifurcation point. Matched E-mail: [email protected] asymptotic expansions are used to find invariant manifolds for the period-l and period-2 regions. A uniformly valid triple-deck MAE describes the transition of typical trajec­ CP05 tories from the neighbourhood of the repelling period-l From Billiards to Brains - Applying Periodic Orbit manifold to the period-2 manifold. The expansion shows Theory to Mammalian Neuronal Networks explicitly the effect on the transition of both sweep-rate Characterizing mammalian neuronal networks through un­ and noise. stable periodic orbits has much attraction. Despite re­ Huw G. Davies and Krishna Rangavajhula cent attempts to apply parametric control to such systems Department of Mechanical Engineering through unstable orbit information, rigorous proof for the University of New Brunswick existence of such orbits was lacking. Experimental data PO Box 4400,Fredericton, NB, from both rat neuronal networks and human epileptic foci E3B 5A3 Canada will here be presented, and analyzed both with recurrence 44 Abstracts

Email: [email protected] tions, including discrete event systems, queuing theory, and dynamic programming. I will discuss deterministic non­ expansive maps (mostly those that are also monotonic) CP06 from a dynamical point of view. At one level the dynam­ Finding All Periodic Orbits of Maps Using Newton ics of these maps is trivial; the most interesting behaviour Methods: Sizes of Basins is periodic. Nontheless there are natural dynamical ques­ tions that are important for the applications, and interest­ For a diffeomorphism F on R, it is possible to find period ing mathematically. These include, but are not limited to, k orbits of F by applying Newton methods to the function the existence of fixed points, the possible periods that may Fk - I, (I the identity function). For an initial point x, if occur, and questions about asymptotic behaviour. the process converges to a point p which is a period k point of F, we say x is in the Newton basin of p. We investigate Colin Sparrow the size of the Newton basin and how it depends on p and Department of Pure Mathematics and Mathematical k. Statistics, 16, Mill Lane, University of Cambridge, Jacob R. Miller Cambridge CB2 1SB, United Kingdom Department of Mathematics & Computer Science E-mail: [email protected] Shippensburg University, Shippensburg, PA 17257 E-mail: [email protected]

James A. Yorke CP06 Institute of Physical Science & Technology The Dynamics of Some Digital Filters University of Maryland A class of digital filters are described by the map f(x, y) = College Park, MD 20742-2431 (y, (-x + ay)mod.1) on the unit square, where a is an in­ E-mail: yorke )@ipst.umd.edu teger with -a-j2. These maps, whose dynamics was first considered in the engineering literature, relate to a classi­ cal result by Minkowski. Their dynamical properties are CP06 not understood for most non integer values of a. Using Nonanalytic Perturbation of Complex Analytic renormalization techniques, we describe the dynamics of Maps a few examples. (Joint work with Roy Adler and Bruce Kitchens) This study provides some connections between complex an­ alytic maps of the complex plane and more general real Charles Tresser 2 IBM T.J. Watson Research Center analytic maps (or Coo maps?) of R . We perform a nu­ merical case study on the families of maps of the plane Mathematics Department given by P.O. Box 218, Yorktown Heights, NY 10598 E-mail: [email protected] Z -> fA(Z' C) == Z + Z2 + C + Az where Z is a complex dynamic (phase) variable, and C and A are complex parameters. Note that for A = 0, the re­ CP06 sulting two-parameter family is a complex quadratic family. Collapsing of Chaos For A I- 0, the map fails to be complex analytic, but is still We discuss the computational collapsing effect of a family analytic (quadratic) when viewed as a map from 7(.2 to it­ of chaotic maps: flex) = 1 - 12x - Ill, I > 2. i. e., a large self. We treat A in this family as a perturbing parameter number of initial conditions are eventually mapped to an and ask how the two-parameter bifurcation diagrams in the unstable fixed point. We rigorously prove that this effect C parameter plane change as the perturbing parameter A is independent of the precision of a computer, and it does is varied. The most striking phenomenon that appears as not vanish as we increase the precision. In fact, this effect A is varied is that bifurcation points in the C plane for is closely related to correlation dimension of the attractor. the A 0 case evolve into fascinating bifurcation regions = Guocheng Yuan and James A. Yorke in the C plane for nonzero The points we concentrate A. Institute of Physical Science & Technology on for A = 0 are contact points between "bulbs" of the University of Maryland corresponding Mandlebrot set. College Park, MD 20742-2431 James Montaldi E-mail: (gcyuan, yorke)@ipst.umd.edu Institut NonLineaire de Nice 06560 Valbonne, France [email protected] CP07 The Application of the Direct Scattering Transform Bruce B. Peckham to the NRZ-to-Soliton Data Conversion Problem Department of Mathematics and Statistics University of Minnesota, Duluth We apply the direct scattering transform to analyze the Duluth, MN 55812 conversion of phase modulated non-return-to-zero signal [email protected] to the soliton format in the line with the sliding frequency guiding filters. Based on this we propose a recipe to opti­ mize this conversion. CP06 S. Burtsev and R. Camassa Dynamics of Non-Expanding Maps & Applications Los Alamos National Laboratory in Discrete Event Theory Los Alamos, NM Deterministic or stochastic maps which are non-expansive P. Mamyshev in the sup norm arise naturally in a number of applica- Abstracts 45

Bell Laboratories, Lucent Technologies and ni is the linear growth rate of the unstable modes. 101 Crawfords Corner Road For the periodic case, this property coincides with the es­ Holmdel, NJ 07733 timate of the mean passage time in the structually stable heteroclinic cycles in the dynamical systems with symme­ try and random perturbations shown by Stone and Holmes (1990), if constant terms are neglected in their expression. CP07 An application of the homoclinic solutions to the motion Mass Exchanges Among Korteweg-de Vries Soli­ of a closed vortex filament is also discussed. tons Makoto U meki We consider multisoliton solutions of an integrable vector Department of Physics version of the Korteweg-de Vries (KdV) equation having University of Tokyo the property that the sum of the vector components satis­ 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113 Japan fies the scalar KdV equation. The results are interpreted as Email: [email protected] a way of assigning internal degrees of freedom to the soli­ tons of the KdV equation. Each of the field components carries its own conserved mass, and in the process of colli­ sion solitons exchange mass of different types. The amount CP07 of mass shared during soliton collisions is calculated explic­ Vector Solitons and Their Internal Oscillations in itly, revealing that in each pairwise collision the smaller Birefringent Nonlinear Optical Fibers (and slower) of the two solitons always gives up more mass In this paper, single-hump vector solitons and their internal than it had initially. This leads to a mass deficit, as the oscillations in birefringent nonlinear optical fibers are stud­ smaller soliton borrows against the continuum. ied. It is found that the total number of single-hump vec­ Peter D. Miller tor solitons is continuously infinite and their polarizations Optical Sciences Centre can be arbitrary. The internal oscillations of these vec­ The Australian National University tor solitons are investigated by the linearization method. Canberra, ACT 0200, Australia All the eigenmodes of the linearized equations around the Email: [email protected] vector solitons are determined. The discrete eigenmodes cause to the vector solitons permanent internal oscillations Peter L. Christiansen which visually appear to be a combination of translational The Institute for Mathematical Modeling (IMM) and width oscillations in the A and B pulses. Lastly, the The Technical University of Denmark asymptotic states of the perturbed vector solitons are stud­ Lyngby 2800, Denmark ied within both the linear and nonlinear theory. It is found that the state of internal oscillations of a vector soliton is always unstable. It invariably emits energy radiation and CP07 eventually evolves into a single-hump vector soliton state. Soliton Stability in Optical Transmission Lines us­ Jianke Yang ing Semiconductor Amplifiers and Fast Saturable Department of Mathematics and Statistics Absorbers University of Vermont Soliton stability has been examined in a cascaded transmis­ 16 Colchester Avenue sion system based upon standard monomode fibers with Burlington, VT 05401 in-line seminconductor optical amplifiers (SOAs), sliding filters and saturable absorbers (SAs). Stabilization of pulse propagation in such a system can be achieved with a proper CPOS choice of the filter and SA parameters. Conditions of the A Leading-Order Singular Hamiltonian Perturba­ stable propagation, including a critical sliding rate are de­ tion Method in Fluid Dynamics termined. The impact of the saturable absorber on the We consider the question of constructing a singular Hamil­ soliton stability will also be discussed. tonian perturbation method for fluids with generalized S. K. Turitsyn Poisson brackets. In partial answer to this question we Institute fur Theoretische Physik I have found a leading-order singular Hamiltonian pertur­ Heinrich Heine Universitiit Dusseldorf bation method by combining asymptotic perturbation ex­ Universitiitsstrasse 1 pansions with preservation of the Hamiltonian formulation. 40225 Dusseldorf Germany These leading-order results have been exemplified by a sys­ tematic derivation of the Hamiltonian formulation for the incompressible, homogeneous fluid equations via a Mach­ CP07 number expansion and for the barotropic quasi-geostrophic Noisy Quasi-Periodicity in the Simulation of the fluid equations via a Rossby-number expansion. Discrete Nonlinear Schrodinger Equation O. Bokhove We study numerically the homoclinic structures in the non­ Woods Hole Oceanographic Institution linear Woods Hole MA 02543, U.S.A. Schrodinger (NLS) equation, which possesses exact solu­ tions found by Ablowitz & Herbst (1990). By using the integrable difference scheme (Ablowitz & Ladik 1976) for CPOS the NLS equation, the Runge-Kutta method for time in­ Variational Principle for Bifurcations in One Di­ tegration and changing numerical precisions, we find that mensional Hamiltonian Systems periodic and quasi-periodic motions are generated instead We consider the one dimensional bifurcation problem u" + of homo clinic orbits. The periods are proportional to AU = N(u) with two point boundary conditions and where -In t/ni , where 10 is the magnitude of numerical errors N(u) is a general nonlinear term which may also depend 46 Abstracts

on the eigenvalue A. We show that there exists a vari­ CPOS ational principle for the lowest eigenvalue of the linear On Smooth Hamiltonian Flows Limiting to Hyper­ problem which can be extended to the nonlinear prob­ bolic Billiards lem to obtain the complete bifurcating branch. We find Sufficient conditions are found so that families of smooth that A = max[(Jo"= N(u)du+(1/2)K(urn ))/ Jo"= ug(u)du], Hamiltonian flows limit to a billiard flow as a parameter where Urn denotes the amplitude and the maximum is taken 1 E -> o. This limit is proved to be C near non-singular over all positive functions 9 such that g(O) = 0, g' > O. The orbits and CO near singular (tangent to the billiard bound­ function K(um ) is obtained from the solution of the lin­ ary) orbits. These results are used to prove that scattering ear problem. Extensions to higher dimensional bifurcation (thUS ergodic) billiards with tangent periodic orbits or tan­ problems will also be presented. gent homo clinic orbits produce nearby Hamiltonian flows R. D. Benguria and M. C. Depassier with elliptic islands. This implies that ergodicity may be Departamento de Fisica lost for smooth potentials which are arbitrarily close to Universidad Cat6lica de Chile ergodic billiards. Casilla 306, Santiago 22, Chile Dmitry Turaev and Vered Rom-Kedar [email protected] The Weizmann Institute of Science, Israel

CPOS CPOS The Hamiltonian and Lagrangian Approaches to Approximation of KAM Tori and Aubry-Mather the Dynamics of Nonholonomic Systems Sets by Periodic Solutions of a Perturbed Burgers' There are many differences in the approaches and each has Equation its own advantages; the momentum equation and the re­ This investigation was stimulated by a series of papers construction equation was first found on the Lagrangian by J.K. Moser, in which he regularized Mather's varia­ side while the failure of the reduced two form to be closed tional principle to obtain smooth approximations of Aubry­ was first noticed on the Hamiltonian side. Clarifying the Mather sets. In this talk we provide an alternative way relation between these approaches is important for devel­ to construct approximations of invariant sets carrying oping the control, stability and bifurcation theory for such quasiperiodic solutions. Our approach is demonstrated on systems. a certain class of Hamiltonian systems with one and a half Wang-Sang Koon degrees of freedom by taking the Eulerian viewpoint and Department of Mathematics reducing the system to a perturbed Burger's equation with University of California, Berkeley, CA 94720-3840 zero viscosity. Next, a viscousity term is introduced to E-mail: [email protected] regularize the solutions. Vadim Zharnitsky Jerrold E. Marsden Center for Nonlinear Studies, MS-B258 Control and Dynamical Systems Los Alamos National Laboratory California Institute of Technology 116-81 Los Alamos, NM 87545 Pasadena, CA 91125 E-mail: [email protected]

CP09 CPOS Signal Masking Using Synchronized Chaos-Proof Instabilities and Degeneracies of the 4-Dimensional and Analysis by Perturbation Theory Stochastic Web Cuomo and Oppenheim (1993) adopted the concept of syn­ We discover a fundamental mechanism for the sharp in­ chronized chaos (Pecora and Carroll, 1990) to mask weak crease in the diffusion rate of the motion of a charged par­ messages. However, why and when the technique works ticle in a uniform magnetic field when a transversely prop­ was not clearly understood (Strogatz, 1995). Here, the agating time-periodic electric-field is slightly tilted. It is problem is analyzed by mathematically studying the weak associated with the degeneracies of the underlying motion; (integral) properties of Lorenz equations and by applying in fact, a two-degree of freedom mechanism of instability the regular perturbation theory. Finally, numerical imple­ induces this behavior. Unfolding of this degeneracy en­ mentation is used to verify the mathematical results. ables the analysis of the separatrix splitting in some limits, giving a power-law rather than the typical exponential es­ Mohamed A. Ali timate of the splitting. Department of Civil & Environmental Engineering Cornell University, Ithaca, NY 14853-3501 Sergey Pekarsky E-mail: [email protected] The Weizmann Institute of Science, Rehovot, 76100, Israel current address: California Institute of Technology CP09 Pasadena, CA, 91125 New Bifurcations from an Invariant Subspace E-mail: [email protected] Synchronisation problems give rise to chaotic dynamics in invariant subspaces. Many simple models which attempt Vered Rom-Kedar to understand these problems involve only one direction The Weizmann Institute of Science, which is normal to the invariant subspace. We consider Rehovot, 76100, Israel different phenomena which can occur in systems with par­ [email protected] ticular types of symmetry and when there are two normal Abstracts 47

directions. The theory is illustrated by some examples. tan eo us front emitting pacemakers. Philip J. Aston and Peter Ashwin Julyan H. E. Cartwright Department of Mathematical and Computing Sciences Centre de Citlcul i Informatitzaci6 and Departament de University of Surrey, Guildford GU2 5XH, United King­ Ffsica dom Universitat de les Illes Balears E-mail: [email protected] 07071 Palma de Mallorca, Spain [email protected] Matthew Nicol Department of Mathematics Emilio Hernandez-Garda and Oreste Piro UMIST, Manchester M60 1QD, United Kingdom Institut Mediterrani d'Estudis Avangats, IMEDEA (CSIC­ DIB) and Departament de Fisica, Universitat de les Illes Balears CP09 07071 Palma de Mallorca, Spain Determining Coupling that Guarantees Synchro­ [email protected], [email protected] nization between Identical Chaotic System Method We present a criteria for determining nontrivial coupling strengths that are guaranteed to produce synchronization CP09 between chaotic systems with drive response coupling. The On Generalized Synchronization of Chaos in Mu­ criteria is based on a linear analysis of perturbations. tually Coupled Systems When it is satisfied one is guaranteed that small perturba­ The term Generalized Synchronization of Chaos (GSC) tions away from synchronization exponentially decay with refers to the regime where chaos synchronization occurs be­ time. The criteria is simple to use, has a simple geomet­ tween systems with distinctly different individual chaotic ric interpretation, and most calculations can be performed behaviors. Although the methods for observation of GSC analytically. were originally developed only for drive-response systems, Reggie Brown we show that some of these methods can be extended and College of William and Mary applied to certain cases of GSC in systems with mutual Williamsburg, VA 23187 coupling. In this report we present a few examples of GSC E-mail: [email protected] in mutually coupled chaotic systems. Nikolai F. Rulkov, Mikhail M. Sushchik Jr., and Henry D.l. Abarbanel CP09 Institute for Nonlinear Science Automatic Gain Control using Synchronized Chaos University of California, San Diego La Jolla, CA 92093 The discovery of synchronization has suggested using chaotic waveforms for communications. Practically, chan­ nel imperfections may distort the transmitted waveform and degrade communications. Specifically, exact synchro­ CPIO nization requires unity gain in the channel when the driven Crisis in Quasiperiodically Forced Systems subsystem is nonlinear. In this presentation, an approach We study various types of crisis in quasiperiodically forced for automatic gain control using synchronized chaos is re­ systems connected with the appearance of strange non­ ported. This approach uses cascaded subsystems to sense chaotic attractors (SNA), i.e. nonchaotic attractors with synchronization quality, auxiliary states to track error sen­ a strange geometrical structure. The quasiperiodic force sitivity to channel gain, and adaptive gain to compensate leads to different bifurcations of SNA like band-merging for channel losses. and interior crisis which were up to now only known for Ned J. Corron chaotic attractors. These bifurcations can be identified by Dynetics, Inc. means of rational approximations of the quasiperiodic force P. O. Drawer B and the sensitivity with respect to the external phase. Huntsville, AL 35814-5050 Ulrike Feudel [email protected] Max-Planck-Arbeitsgruppe Nichtlineare Dynamik an der Universitiit Potsdam PF 601553, D - 14415 Potsdam, Germany CP09 Fax: +49-331-9771142 Stick-Slip Dynamics as an Elastic Excitable Media E-mail: [email protected] We demonstrate the equivalence of the Burridge-Knopoff model with asymptotically viscous friction to a van der Pol-FitzHugh-Nagumo model for excitable media with CPlO elastic couplings. The proposed lubricated creep-slip fric­ Fractal Dimensions of Chaotic Saddles of Dynami­ tion law is appropriate to a range of real materials, and cal Systems the observed dominance of low dimensional structures in Formulas for the information dimensions of the natural our model is consistent with the results of some laboratory measures of a nonattracting chaotic set and of its stable friction experiments. Other media such as active optical and unstable manifolds are derived in an idealized setting waveguides or electronic transmission lines may have sim­ and conjectured to hold for invertible dynamical systems ilar dynamics. Our studies also predict the existence of of arbitrary dimensionality. The formulas give the infor­ stable global oscillations bifurcating via period doubling mation dimensions in terms of the Lyapunov exponents into space-time configurations composed of a set of spon- and the decay time of the associated chaotic transient. As an example, the formulas are applied to the situation of 48 Abstracts

filtering of data from chaotic systems. University of Southern Queensland Brian R. Hunt, Edward Ott, and James A. Yorke Toowoomba, Qld, Australia 4350 University of Maryland Australia College Park, Maryland 20742 E-mail: [email protected] E-mail: [email protected]

CPIO CPlO Invariant Graphs And Strange Nonchaotic Attrac­ Which Dimension of Fractal Measures are Pre­ tors for Quasiperiodically Forced Systems served by Typival Projections? It is an open problem whether strange nonchaotic attrac­ Does the dimension of an attractor reconstructed in a low­ tors can be the graph of a non-smooth function. We show dimensional embedding space equal the dimension of the that this only occurs if the closure of the attractor con­ attractor in its true state space? We show for the dimen­ tains unstable orbits. More precisely we prove that if the sion spectrum Dq that under reasonable hyposeses the an­ attractor has an open neighbourhood in which all orbits swer is "yes" for 1 < q 2: 2 but we show by example that have negative Liapunov exponents, then the attractor is a smooth invariant circle. We discuss the relevance of this to answer may be "no" for q < 1 and q > 2. We present and use a new definition of Dq for q > 1 which is equivalent to the creation of strange nonchaotic attractors. previous definitions Jaroslav Stark Vadim Yu. Kaloshin Centre for Nonlinear Dynamics and its Applications Princeton University, Princeton, NJ University College London E-mail: [email protected] Gower Street, London, WC1E 6BT, United Kingdom Fax: +33-171-380-0986 Brian Hunt Email: [email protected] Maryland University at College Park College Park, MD E-mail: [email protected] CPIO Blowout Bifurcation Route to Strange Nonchaotic Attractors CPIO Strange nonchaotic attractors are attractors that are ge­ Nondifferentiable Attractors, Data Filtering, and ometrically strange, but have nonpositive Lyapunov ex­ Fractal Dimension Increase ponents. We show that for dynamical systems with an The long standing problem of why certain time-series filters invariant subspace in which there is a quasiperiodic attrac­ (e.g. LTI or low-pass) can increase the fractal dimension tor, the loss of the transverse stability of the attractor can of the reconstructed attractor has been solved. There is lead to the birth of a strange nonchaotic attractor. A phys­ a close relation to the situation of driving a stable system ical phenomenon accompanying this route to strange non­ with a chaotic signal as in studies of chaotic synchroniza­ chaotic attractors is a distinct type of on-off intermittent tion. In this talk we show that both filtering and chaotic behavior. driving can lead to an attractor in which the map from Tolga Yalqnkaya and Ying-Cheng Lai the chaotic driving system to the response system is non­ Department of Physics and Astronomy differentiable (leading to a fractal object). We show how The University of Kansas this can be detected in time-series using recently developed Lawrence, KS 66045 statistics which test for continuity and/or differentiability E-mail: [email protected] between reconstructed attractors. Louis M. Pecora and Thomas L. Carroll Code 6343, Naval Research Laboratory CPl1 Washington, DC, USA A Nonlinear Wave Equation Arising in Two-Phase Flow We present and analyze a two-dimensional nonlinear wave CPIO equation derived from the Navier-Stokes equations for two Determining General Fractal Dimensions from miscible fluids in the limit of small Froude number. It rep­ Small Data Sets resents a perturbation of the KdV equation and contains Fractal geometry is applied in many scientific fields. Even all low-order terms; the core part consists of the modi­ the fundamental task of estimating fractal dimensions from fied Kawahara equation. We look for plane solitary waves experimental samples suffers biases from the finite size of and investigate the existence and stability of one- and data sets. I discuss the application of a novel method of two-dimensional periodic solutions utilizing a Ginzburg­ unbiasedly determining generalised dimensions from data Landau approach. With regard to our main application, sets. Example data is generated from the Henon map and fluidized beds, the equation is able to mimick the bifurca­ obtained from plant and crab distributions. The method tion sequence of the original system but fails to capture generally appears unbiased, reliable and provides reason­ the higher-amplitude behavior. able error estimates, but currently takes O(1000N2) com­ Manfred F. G6z puter operations. This approach will be especially useful in Department of Chemical Engineering the discernment and estimation of fractal properties from Princeton University "small" data sets. Princeton, NJ 08544 A.J. Roberts [email protected] Department of Mathematics & Computing Abstracts 49

CP11 CPU Recent Progress in Multi-Dimensional Mode Con­ Two-period Quasiperiodic Attractors of Weakly version Nonlinear Gas Dynamics Equations We consider the short wavelength asymptotics of N­ New never-breaking solutions for the I-D weakly nonlinear component wave equations in weakly nonuniform media: equations of Gas Dynamics in a bounded domain with re­ 2:;=1 J dmx'Dij(x,x')'!f;j(X') = 0; with i,j = 1,2 .. . N. flecting boundary conditions are obtained numerically. The Euler equations in this case are reduced to a Burgers-like Here, x = (x, t) with x an m - I-dimensional set of spatial equation with a linear integral dispersive term. Balance variables. Two (or more) polarizations may become degen­ of nonlinearity and dispersion leads to the formation of erate for some values of x and k (the conjugate wavenum­ time periodic or quasiperiodic solutions. We explore the ber) leading to a breakdown of the WKB approximation long-time behavior of "arbitrary" initial conditions numer­ locally in the ray phase space. New local asymptotic ex­ ically. We found a family of attracting solutions. In the pansions must be developed, with important physical im­ phase space of Fourier coefficients these attractors are rep­ plications which will be discussed. resented by quasiperiodic orbits with two periods. Allan N. Kaufman, Alain J. Brizard & Jim Morehead Michael G. Shefter Massachusetts Institute of Technology Lawrence Berkeley National Laboratory 77 Massachusetts Avenue, Rm 2-486 Berkeley, CA 94720 Cambridge, MA 02139 E-mail: [email protected] E. R. Tracy Physics Department College of William & Mary Rodolfo R. Rosales Massachusetts Institute of Technology Williamsburg, VA 23185 77 Massachusetts A venue, Rm 2-337 Cambridge, MA 02139

CP11 A Robust Heteroclinic Cycle in an 0(2)XZ2 Steady-State Mode Interaction CP11 The Strongly Attracting Character of Large Ampli- A structurally stable heteroclinic cycle connecting circles tude Nonlinear Resonant Acoustic Waves Without of fixed points of opposite parity is identified in a steady­ Shocks state mode interaction with 0(2) XZ2 symmetry. These fixed points correspond to equilibria in the form of zigzag A new class of fully nonlinear solutions for the 1-D Eu­ (odd parity) and varicose (even parity) states. Because of ler equations of Gas Dynamics in a bounded domain the Z2 reflection symmetry this 1 : 1 mode interaction bears with reflecting boundaries is obtained numerically. These certain similarities to earlier work on 2 : 1 mode interac­ solutions never develop shocks (even though they carry tion with 0(2) symmetry. However, in the present case the large pressure variations) and can be characterized as cycle connects eight equilibria, four of which are selected large amplitude acoustic standing waves. In the infinite­ from the circle of zigzag equilibria and four from the cir­ dimensional space of the Fourier coefficients, the motion cle of varicose equilibria. Conditions for the existence and is quasiperiodic, confined to 2-D tori. Numerical experi­ asymptotic stability of the cycle are determined and com­ ments reveal the strongly attracting character of these so­ pared with numerical integration of the mode interaction lutions: Regardless of the initial conditions, these waves equations. always dominate the flow after a sufficient time. P. Hirschberg and E. Knobloch Dimitri D. Vaynblat Department of Physics Massachusetts Institute of Technology University of California 77 Massachusetts Avenue, Rm 2-486 Berkeley CA 94720 Cambridge, MA 02139 Rodolfo R. Rosales Massachusetts Institute of Technology 77 Massachusetts Avenue, Rm 2-337 CP11 Cambridge, MA 02139 Local Singularities in Breaking Waves We examine free surface wave singularities which show a violent focusing of energy. Periodic, modulated and fre­ CP12 quency locked wave states lead to a state with fluid spikes. Convergence of the Transfer Operator for Rational When the excitation of the free surface waves exceeds a well Maps defined threshold, the waves break leading to a aperiodic state with spikes, droplet ejection and air entrainment. Re­ We prove that the transfer operator for a general class of turn maps formed from wave height measurements lead to rational maps converges exponentially fast in the supre­ a low dimensional model of the dynamics. A local model mum norm and in Holder norms for small enough Holder of the fluid dynamics of the singularity is presented, along exponents to its principal eigendirection. with a comparison with observed experimental events. Nicolai Haydn Daniel P. Lathrop University of Southern California Department of Physics Los Angeles, CA 90089 Emory University E-mail: [email protected] Atlanta GA, 30322 E-mail: [email protected] CP12 An Exact Formula for the Essential Spectral Radius of the Matrix Ruelle Operator on Spaces of Hoelder 50 Abstracts and Differentiable Vector-Functions rational maps and the Perron-Frobenius operator of the corresponding jump transformation. There will be also The essential spectral radius of the matrix Ruelle transfer outlined the applications to the problems of geometry of operator £, (£(y) on CT,o. is equal the Julia sets. to exp sup {hv +.xv - (r + a)xv}. Marius Urbanskij vEErg University of North Texas, Denton, TX Here f is expanding, Erg is the set of all f-ergodic proba­ E-mail: [email protected] bility measures on a smooth manifold X, hv is the entropy of f with respect to 1/, CP13 .xv = lim ~ log lIAxiom A (i.e. uniformly hyperbolic diffeomophism induces 9700 South Cass A venue, Argonne,IL 60439 a unitary operator on the Hilbert space of functions square E-mail [email protected] integrable with respect to a Gibbs measure. Here there are natural Hilbert subspaces, associated with the stable and unstable manifolds of the dynamics, which comprise incoming and outgoing subspaces;these are the building CP13 blocks of the Lax-Phillips theory. This talk characterizes Synchronization by Disorder in Parallel Arrays of the scattering operator of this system in terms of the data Josephson Junctions of the dynamical system and Gibbs measure, in particular We present numerical evidence that spatial disorder can in terms of the Ruelle operator. The Fourier transform of induce phase and frequency synchronization in parallel ar­ the scattering operator, the scattering function, is shown rays of Josephson junctions driven by dc and/or ac current to be meromorphic in an annulus of the complex plane, and sources. Disorder - induced synchronization can occur even the poles and their residues are determined by the spectral if the junctions are chaotic when uncoupled. We will dis­ properties of the Ruelle operator. cuss the consequences of improved synchronization on the Thomas J. Taylor current - voltage characteristics of an array and provide a Department of Mathematics qualitative understanding of the effect. Arizona State University Yuri Braiman Tempe, AZ 85287-1804 Department of Physics Emory University, Atlanta, GA 30322 Duk-Hyung Lee E-mail: [email protected] Department of Mathematics Indiana Wesleyan University Marion, IN 46953 CP13 Flux Creep in 2-D Josephson Junction Arrays The phenomenology of flux creep in a 2-dimensional CP12 Josephson junction array subject to a slowly ramped mag­ Transfer Operators in Conformal Dynamics netic field is considered. We describe the dynamical ap­ Perron-Frobenius operators, equilibrium states, and invari­ proach of the system to a critical state (the so-called Bean ant measures for conformal endomorph isms f of the Rie­ state), and derive an associated automaton model to un­ mann sphere a: will be introduced and discussed. Given derstand subsequent avalanching behavior. potential ¢> : J(f) ...... 1R with P(f,1», where J(f) is the Adam S. Landsberg Julia set of f there exists a unique equilibrium state of Department of Physics ¢> and the associate Perron-Frobenius operator is almost Haverford College, Haverford, PA 19041 periodic. Its various stochastic and mixing properties will E-mail: [email protected] be discussed including the central limit theorem and the exponentia I decay of correlations. More special cases will Kurt Wiesenfeld be also explored, particularly the relationships between the School of Physics Perron-Frobenius operator of parabolic and non-recurrent Abstracts 51

Georgia Institute of Technology Atlanta, GA 30332 CP14 Ginzburg-Landau Equations for Hexagonal Pat­ terns CP13 Several interesting physico-chemical instabilities settle Relaxation of Magnetic Knots to Minimal Braids from a sub critical bifurcation. A center manifold reduc­ tion of balance equations leads to a new type of ampli­ By using new equations for the Lorentz force associated tude equations containing spatial gradient quadratic terms. with magnetic knots (Ricca 1996), we prove that inflex­ Their form is generic since is based only on symmetry con­ ional knots (i.e. knots with a finite number of points of siderations. For the specific case of surface-tension-driven inflexion in isolation) are unstable and are isotoped natu­ convection we will present the bifurcation diagrams (cor­ rally to inflexion-free knots, possibly in braid form. Further responding to amplitude and phase instabilities) and the relaxation to minimal braids has interesting consequences associated patterns ear the onset of convection. on the minimum possible number of crossings, with an in­ crease of lower bounds for magnetic energy. Blas Echebarria Depto. de Ffsica y Matematica Aplicada Renzo L. Ricca Universidad de Navarra, Pamplona, Spain Department of Mathematics Carlos Perez-Garda University College London Depto. de Fisica y Matematica Aplicada Gower Street, London WC1E 6BT Universidad de Navarra, Pamplona, Spain United Kingdom E-mail: [email protected]

CP14 Non-Potential Effects in Pattern Formation CP13 Geometrical Phase in Twisted Pairs used in Com­ It was shown recently that hexagonal convection pat­ munication Networks terns are described generally by non-potential amplitude equations which are not identical to standard Newell­ Twisted pairs are used in small communication networks to Whitehead-Segel equations. In frame of the new non­ remedy the problem of interference between parallel wires potential equations, we reconsider the problem of pattern carrying signals which tend to distort information by the selection, calculate the "Busse balloon" for different kinds electromagnetic fields they produce in their vicinity. The of periodic patterns (including nonequilateral hexagonal standard argument extended is that twisting helps in can­ patterns) and investigate the propagation of fronts and do­ cellation of fields in such a way that amounts to shield­ main walls between patterns. The nonlinear development ing the wires.We intend to explain how such cancellation of secondary instabilities is considered as well. takes place using the concept of Geometrical Phase; and therefore to place the classical point of view in this new Alexander Golovin perspective. Northwestern University, Evanston, IL 60208 Mayank Sharma and D.Subbarao' Asaf Hari, Alexander Nepomnyaschchy, Department of Electrical Engineering and Alexander Nuz, and Leonid Pismen Fusion Studies Program Technion - Israel Institute of Technology Plasma Science & Technolgy Group Haifa, Israel Centre for Energy Studies Indian Institute of Technology, Delhi New Delhi 110 016. India Email:[email protected] CP14 Multiplication of Defects in Hexagonal Patterms A typical object in hexagonal patterns is a penta-hepta CP13 defect, i.e. a bound state of two dislocations on different Transonic Solutions for a Quantum Hydrodynamic roll subsystems. For Marangoni convection, we numeri­ Semiconductor Model cally found a new type of nonlinear instability resulting in multiplying penta-hepta defects. Each act of multipli­ Quantum hydrodynamics is an emerging new field in semi­ cation includes creation of a new dislocation pair on the conductor modeling. Mathematically, these models are dis­ dislocation-free roll subsystem, and recombination of dis­ persively regularized hydro dynamical equations coupled to locations into two penta-hepta defects of the next genera­ a selfconsistently determined electric potential. The basic tion. The long-run pattern evolution displays wavelength question is to understand the dispersionless semiclassical selection mediated by moving penta-hepta defects. limit. In one space dimension this leads to a singularly perturbed system of ODEs with two slow manifolds cor­ A.A. Nepomnyashchy responding to supersonic and subsonic states. The main Technion - Israel Institute of Technology physical interest lies in solutions connecting these states. Department of Mathematics We present analytical and numerical results for this tran­ 32000 Haifa, Israel sition problem. E-mail: [email protected]

Peter Szmolyan P. Colinet and J.C. Legros Institut fuer Angewandte und Numerische Mathematik Universite Libre de Bruxelles, Technische Universitiit Wien Service de Chimie Physique E.P., C.P. 165, Wiedner Hauptstrasse 6-10 50 Av. F.D. Roosevelt, 1050 Brussels, Belgium A 1040 Wien, Austria E-mail: [email protected] E-mail: [email protected] 52 Abstracts

CP14 University of Texas, Austin, TX 78712-1060 Takens-Bogdanov Bifurcation on the Hexagonal [email protected] Lattice for Double-Layer Convection A bifurcation analysis is performed at a Takens-Bogdanov Neil J. Balmforth point, for solutions which are doubly periodic with respect Department of Theoretical Mechanics to a hexagonal lattice. Included are the analyses of rolls, University of Nottingham hexagons, triangles, rectangular patterns, traveling waves, NG7 2RD, Nottingham, United Kingdom twisted patchwork quilt, centrosymmetric solutions and heteroclinic orbits. The theory is applied to double-layer convection. A center manifold reduction scheme is used CP15 to convert the governing equations to six complex-valued Irregular Jumping in the Perturbed Nonlinear ordinary differential equations, and Landau coefficients are Schrodinger Equation computed for specific fluid parameters. Numerical results Numerical studies indicate that certain solutions of the based on the amplitude equations are given. damped-forced nonlinear Schrodinger equation (NLS) ex­ Yuriko Y. Renardy hibit irregular jumping in the time domain. The jumping Department of Mathematics and behavior appears to be centered around the set of plane Interdisciplinary Center for Mathematics waves. In this talk I discuss how multi-pulse homoclinic Virginia Polytechnic Institute and State University and heteroclinic orbits can be constructed in the infinite Blacksburg, VA 24061-0123 dimensional phase space of the NLS, which explains the E-mail: [email protected] irregular jumping behavior. For the case of pure forcing, I also describe an involved sequence of homoclinic bifurca­ Michael Renardy tions. Department of Mathematics and ICAM Gyorgy Haller Virginia Polytechnic Institute and State University Division of Applied Mathematics Blacksburg, VA 24061-0123. Brown University Box F, Providence, RI 02912 02912 Kaoru Fujimura E-mail: [email protected] Japan Atomic Energy Research Institute Tokai-mura, Ibaraki 319-11, Japan

CP15 Existence and stability of singular heteroclinic or­ CP14 bits for the Ginzburg-Landau equation Patterns in Long Wavelength Convection The existence and stability of travelling waves of the Transition from a spatially uniform state to a periodic Ginzburg-Landau equation is considered. The waves in pattern occurs in many applications. Using weakly non­ question exist on a slow manifold in phase space and con­ linear analysis the relative stability of hexagons and rolls nect two stable plane waves with different wave numbers. or squares and rolls can be found. We demonstrate for The existence of these waves is proven via the use of the an example p.d.e that, with little additional effort, the methods of geometric singular perturbation theory. Topo­ stability analysis can be extended. Our results include logical methods are used to prove the linear stability of the hexagon/rectangle transitions and the prediction that, on waves. The waves are shown to be nonlinearly stable in a domain with periodic boundary conditions, the observed polynomially weighted spaces. pattern at onset depends on the size of the domain. Todd Kapitula Anne C. Skeldon, University of New Mexico Department of Mathematics Albuquerque, NM 87131 City University London, EC1V OHB United Kingdom E-mail: [email protected] CP15 Mary Silber Singularly Perturbed and Non-Local Modulation Dept of Engineering Sciences & Applied Mathematics Equations Northwestern University In this talk, two systems of coupled modulation equations Chicago, II 60208 for systems which are subject to two distinct destabilizing mechanisms are compared. In both systems, stationary periodic solutions exist under the same conditions. How­ CP14 ever, a variety of heteroclinic and homo clinic connections The Takens-Bogdanov Bifurcation in Long-wave are found for the singularly perturbed system which do Theory not have a counterpart in the non-local system. Thus, un­ like the singularly perturbed system, the non-local system Considering anisotropic double-diffusion with fixed fluxes cannot describe 'localised structures'. as a model problem, we asymptotically derive planform equations that capture the bifurcation structure around V. Rottschiifer and A. Doelman the co dimension-two point at long wavelengths. The form University of Utrecht of the equations obtained is valid for a wide range of phys­ 3584CH Utrecht, The Netherlands ical systems. We investigate the simplified case where the system has Boussinesq symmetry. Jean-Luc Thiffeault CP15 Institute for Fusion Studies and An Accurate Description of Slow-Motion Dynam- Abstracts 53

ics in Singularly Perturbed Reaction-Diffusion Sys­ Center of Dynamical Systems and Nonlinear Systems tems Georgia Institute of Technology, Atlanta, GA 30332 and University of Leiden, Netherlands Metastable patterns and slow-motion dynamics are known to occur for singularly perturbed reaction-diffusion systems W.D. Kalies on finite intervals. [Carr, PegoJ provided the equations California Polytechnic Statc University of motion for such patterns in the case of a second-order San Luis Obispo, CA 93407 equation using maximum principles, energy estimates and Sturm-Liouville theory. Here, using a different approach, equations governing the motion of patterns are derived for J. Kwapisz Center of Dynamical Systems and Nonlinear Systems general reaction-diffusion systems with gradient terms un­ Georgia Institute of Technology, Atlanta, GA 30332 der various boundary conditions. Bjorn Sandstede WeierstraB Institute for Applied Analysis and Stochastics MohrenstraBe 39, 10117 Berlin, Germany CP16 Mixing and Transport Rates for a Marangoni­ [email protected] driven Chaotic Flow We treat the problem of a bubble moving in a viscous incompressible fluid due to the combined action of buoy­ CP15 Multiple-Pulse Homoclinic and Periodic Orbits in ancy and a temperature gradient. For some values of the parameters therein involved, perturbations of the flow lead Singularly Perturbed Systems to chaotic advection in the vicinity of the bubble. The This talk will focus on finite dimensional singularly per­ stated problem is formulated in suitable variables as a turbed systems. Criteria for the existence of multiple­ two-dimensional perturbed time-periodic Hamiltonian sys­ pulse homoclinic and periodic orbits are given. These or­ tem. We evaluate the phase space transport that takes bits consist of finitely many slow segments and multiple place near the bubble during a period of the velocity field fast excursions in between each pair of slow segments. The through the Melnikov function (or the adiabatic Melnikov techniques used include a higher-order adiabatic Melinkov function, for very large periods). To go beyond one period, function, and a modified version of the Exchange Lemma. we review the Topological Approximation Method devel­ Applications to a model of resonant sloshing in a tank and oped by V.Rom-Kedar to obtain a partial classification of to the FitzHugh-Nagumo equations will be given. the heteroclinic tangles created by the intersections of the Cristina Soto-Trevino invariant manifolds of fixed points. The results depend on Boston University several parameters which are found both using analytical 111 Cummington Street tools (Secondary Melnikov function) and numerical com­ Boston, MA 02215 putations. Gerard Alba-Soler and August Palanques-Mestre Department of Applied Mathematics and Analysis CP15 University of Barcelona Homotopy Classes for Stable Connections between Gran Via, 585, 08071-Barcelona, Spain Hamiltonian Saddle-Focus Equilibria E-mail: [email protected] We study homo clinic and heteroclinic connections between Saddle-Focus equilibria in Hamiltonian systems with two degrees of freedom. For fourth order ODE's or mechan­ CP16 ical Hamiltonian systems which have a non-negative La­ Asymptotic Focusing of Particles in a Wake Flow grangian density, and have two or more saddle-foci (four The understanding of a number of important natural phe­ complex eigenvalues) which are global minima, we find a nomena, as well as the development of many technological rich set of mulitbump homo clinic and heteroclinic connec­ applications, is based on an analysis of the dynamics of tions which occur as local minima of an appropriate action discrete particles, such as bubbles, drops, and solid parti­ functional. A subset of the solutions we find are also given cles, in fluid flows. In this talk, we discuss the dynamics of by Devaney's generic result on Hamiltonian systems in the small, rigid, dilute spherical particles in the wake of a bluff presence of saddle-foci, which requires a transverse inter­ body, under the assumption that the background flowfield section of stable and unstable manifolds. For our method is a spatially periodic array of Stuart vortices, which can no assumptions on the intersection of stable and unstable be considered to be a regularization of the Karman vortex manifolds are made. The method is based on minimiza­ street. We show that when inertia, measured by the di­ tion on open classes of curve of certain topological type. mensionless Stokes number, is small, there is an attracting These classes are defined as follows. In the case of two slow manifold in the phase space of the particle motion. saddle-foci, say Pl and P2, one considers curves starting at Using numerical methods to analyze the flow on the slow Pl or P2 and terminating at Pl or P2. In between curves manifold, we provide an explanation for the unexpected can wind around Pl and P2 which defines different topo­ "focusing effect" which has been observed in experiments. logical classes of curves. The fact that one can minimize In these experiments, over a range of small enough values and the fact that Pl and P2 are saddle-foci allows use to of the Stokes number, but not too small, glass beads in­ prove that for almost every topological type (most words jected into the wake of a bluff body tended to concentrate 2 in 7r~emi (R \ {PI, P2} )) there exists a minimizer in the as­ near the edges of the vortex structures shed by the body, sociated class, which is a local minimizer of action. The thus tending to "demix" rather than mix homogeneously. existence of these connections implies that the topological T. J. Burns entropy of the Hamiltonian system in question is strictly Computing & Applied Mathematics Laboratory positive and the systems are non-integrable. R. VanderVorst R. W. Davis and E. F. Moore 54 Abstracts

Chemical Science & Technology Laboratory the time dependence. An illustrative example drawn from NIST, Gaithersburg, MD 20899 the theory of baroclinic instability is analyzed. Petros J. Ioannou Harvard University, CP16 Pierce Hall, Non-Normal Dynamics of Eddy Variance and 29 Oxford Street, Transport Properties in Geophysical Flows Cambridge, MA 02138 E-mail: [email protected] Transient growth of synoptic scale disturbances in the at­ mosphere gives rise to the episodic occurrence of cyclones in midlatitude jets. This process of cyclogenesis is understood theoretically through analysis of the associated non-normal CP16 linear dynamical operator. By extension, the eddy veloc­ Experimental Evidence for Chaotic Scattering in a ity variance and statistical mean transports of heat and Fluid Wake momentum produced by cyclone waves can be understood We present experimental evidence of chaotic scattering in by solving the associated non-normal stochastic differential a fluid wake. Measurements of tracer particles and dye in equation. The variance and transports are found to be con­ the stratified wake of a moving cylinder are shown to be centrated in a small subspace of disturbances and the form consistent with four predictions based on simple models of the disturbances primarily responsible for producing the and direct numerical simulation: unstable periodic orbits variance and transports is found to be similarly restricted. were shadowed by tracer particles; streaklines marked by In addition, changes in the system and boundary condi­ wake-delayed dye are shown to be fractal; early time-delay tions producing the greatest change in variance and trans­ statistics of fluid elements interacting with the wake de­ ports can be identified using extension of the theory. This cayed exponentially; and finally, the fractal dimension of permits bounds to be placed on the sensitivity of climate the wake is consistent with the dynamics of the wake, as statistics to secular alterations in the atmospheric system measured by characteristic time delays and Lyapunov ex­ and its boundary conditions. Examples of applications of ponents. the theory to atmospheric mid-latitude jets will be shown. John C. Sommerer, Hwar-Ching Ku, & Harold E. Gilreath Brian F. Farrell Milton S. Eisenhower Research Center Harvard University The Johns Hopkins University Applied Physics Laboratory Pierce Hall, 29 Oxford Street Laurel, MD 20723-6099 Cambridge, MA 02138 . E-mail: [email protected] E-mail: [email protected]

CP17 CP16 Controlling the Chaotic Logistic Map, An Under­ Integrability and Non-Inegrability in Bubble Dy­ graduate Project namics Chaos control is completely presented, including targeting, It is well known that a slightly deformable spherical bubble stabilization, and the OGY technique in the context of the exhibits in a quiescent potential flow an erratic motion. To logistic map. By restricting the discussion to this one­ explain this phenomenon it is shown that the motion of a dimensional map, all the required tools and techniques are bubble is governing by a Hamiltonian system on a sphere. accessible to undergraduate students. The goal is both to The resulting equations of motion are similar to a time­ expose the students to recent and exciting developments in dependent non-linear pendulum with a certain chaotical chaos theory, as well as to strengthen skills and intuition behavior. Using the averaging over the quick deformations regarding the definition of chaos, finding periodic orbits, the Liapunov stability of a rectilinear translation is inves­ and determining stability. The presentation is in the form tigated. of the worksheet actually explored in small groups by West Alexander R. Galper Point students. School of Engineering Erik M. Bollt Tel-Aviv University Department of Mathematical Sciences Tel-Aviv, Israel 69978 United States Military Academy West Point, NY 10928-1786 E-mail: [email protected] CP16 Perturbation Growth in Non-Autonomous Geo­ physical Flows CP17 Stability theory of time independent geophysical flows has Development of a Laboratory Based Course in recently been reinvigorated with the introduction of meth­ Nonlinear Dynamical Systems ods for analyzing transient growth of perturbations arising We describe the development of a set of laboratory exper­ from the non-normality of the linearized dynamical oper­ iments designed as part of an interdisciplinary undergrad­ ator associated with these systems. The methods used in uate course in nonlinear dynamical systems. A number of the autonomous case are extended in this work to pro­ simple physical systems (mechanical and electrical), along vide a theory for perturbation growth in time dependent with computer experiments are investigated during the labs non-normal systems. Non-normality is found to lead to to demonstrate the ideas of fixed points, bifurcations,chaos unbounded growth of disturbances when the time depen­ etc. The coursebook we use is Nonlinear Dynamical Sys­ dence satisfies certain general conditions. The growth rate tems and Chaos by S. H. Strogatz. and functional form of the growing disturbance are found and related to the degree of non-normality and the form of N. Fleishon Abstracts 55

Physics Department Integrate-and-Fire Neurons with Dendritic Struc­ California Polytechnic State University ture San Luis Obispo, CA 93407 Synchronization in an array of integrate-and-fire neurons with dendritic structure is studied using a phase-reduction J. McDill and K. Morrison technique. It is shown that for long-range excitatory in­ Mathematics Department teractions the system can undergo a bifurcation from a California Polytechnic State University synchronous state to a state of travelling oscillatory waves. San Luis Obispo, CA 93407 Such a transition is due to the diffusive spread of activity along the dendritic cable. The stability of the synchronous R. Schoonover state is sensitive to the range of interactions and the natu­ Chemistry Department ral frequency of oscillations. This could provide an alterna­ California Polytechnic State University tive to axonal delays as an explanation for the tendency of San Luis Obispo, CA 93407 visual crt ex to synchronize in contrast to waves of activity observed in olfactory cortex. J. P. Sharpe and N. Sungar Physics Department Paul C. Bressloff California Polytechnic State University Dept. of Mathematical Sciences San Luis Obispo, CA 93407 Loughborough University Email: [email protected] Loughborough, Leics. LEll 3TU, United Kingdom Email: [email protected]

CP17 UG Course at lIT Delhi on Self-Organising Dy­ CP18 namical Systems Desynchronisation of Neural Assemblies The undergraduate students at lIT Delhi are selected We analyse a system of pulse-coupled integrate-and-fire through a very stringent national examination .At lITD neurons to identify mechanisms of desynchronisation. Bi­ we therefore have very outstanding students in their 4 year furcations of the firing phase are used to determine the im­ B.Tech program and the Institute being funded by the portance of pulse width, propagation delay, shunting cur­ fedaral government allows a genourous computer access­ rents, electrical synapses and dendritic structure for the ability. Students complete apart from their departmental case in which all neurons fire with the same frequency. In courses quite some external credits. As part of this external contrast to a model lacking anyone of these these simple credit program I have floated and have been successfully biological features, stable asynchronous behaviour is eas­ running now for four semisters an undergraduate interdis­ ily established for reciprocal excitatory coupling between ciplinary course on "SELFORGANISING DYNAMICAL neurons. SYSTEMS" for B.Tech. students mainly of 3rd year. Ex­ Stephen Coombes periences will be reported. Nonlinear and Complex Systems Group D.Subbarao Department of Mathematical Sciences Fusion Studies Program Loughborough University Plasma Science & Technology Group Loughborough, Leicestershire, Centre for Energy Studies LEl1 3TU, United Kingdom Indian Institute of Technology, Delhi E-mail: [email protected] New Delhi 110016, India Email: [email protected] Gabriel J. Lord Applied Nonlinear Mathematics Group Department of Engineering Mathematics University Of Bristol CP17 Queen's Building, University Walk, Circle Homeomorphisms and Advanced Calculus Bristol BS8 ITR, United Kingdom The study of circle homeomorphisms is an appropriate topic for an advanced calculus or introductory real anal­ ysis course. The ideas involved are substantial, yet often CP18 require proofs that are accessible to an advanced under­ Post-Fertilization Traveling Waves on Eggs graduate. In addition, the student is exposed to a true success story in dynamical systems, namely, the role of the We consider the one-dimensional model proposed by Lane, rotation number in determining the dynamics of a circle Murray and Manoranjan to describe the propagation of a homeomorphism. Along the way the appearance of Cantor wave of calcium, and a pulse of elastic deformation, ob­ functions and Arnold tongues only adds to the appeal of served on the surface of some vertebrate eggs shortly after fertilization. We study the equations with smooth non­ the subject, and are well within the grasp of a junior or linearities, and prove the existence of traveling fronts, for senior mathematics student. small values of a coupling parameter, by use of the Mel­ James A. Walsh nikov function. We also discuss a modified version of the Department of Mathematics equations for which we can prove uniform bounds on the Oberlin College, Oberlin, OR 44074 wave speeds, suggesting the existence of waves for all values E-mail: [email protected] of the coupling parameter. Gilberto Flores and Antonmaria Minzoni National University of Mexico CP18 Mexico DF 01000, Mexico Phase Synchronization in an Array of Coupled E-mail: [email protected] 56 Abstracts

used for perturbation treatment. Konstantin Mischaikow A. Rabinovich Georgia Institute Of Technology Physics Department School of Mathematics Ben-Gurion University Atlanta, GA 30332-0001 Beer-Sheva, 84105, Israel E-mail: [email protected] E-mail: [email protected]

Victor Moll 1. Aviram Department of Mathematics 35, Sderot Yeelim Tulane University Beer-Sheva, 84730, Israel New Orleans, LA 70118 N. Gulko Mathematics Department CP18 Ben-Gurion University Additive Neural Networks and Periodic Patterns Beer-Sheva, 84105, Israel We present an additive recurrent neural network with E-mail: [email protected] weight adjustment procedure, which is able to reproduce periodic patterns. The network has the form of a cascade E. Ovsyscher of nets. The presented periodic pattern is a discrete, peri­ Cardiology, Soroka Medical Center odic set of vectors with coordinates ±l. The pattern gives Beer-Sheva, 84105, Israel rise to the initial vector of weights for the weight adjust­ ment dynamics. The weights converge to a limiting set of weights and the activation dynamics with the limiting set CP19 of weights exhibits stable periodic orbit. This stable pe­ Suppressed and Induced Chaos by Near Resonant riodic orbit exhibits the presented periodic pattern in the Perturbation of Bifurcations following sense. Every orthant in can be described by nn We present experimental results which demonstrate for the its signature, which is an n-vector of ±1, where we put 1 first time that the onset of chaos in a nonlinear system can if the corresponding variable is positive in the orthant and be either suppressed or induced by the application of a per­ -1 if the variable is negative in the orthant. The stable pe­ turbation signal which is near-resonant to a sub harmonic riodic orbit visits consequtively orthants whose signatures of the fundamental system frequency. The technique rep­ form the presented periodic pattern. resents a feedback independent method for stabilizing or Tomas Gedeon destabilizing chaotic orbits. Shifts in the onset of chaos Department of Mathematics are demonstrated for perturbation signals which are near­ Montana State University resonant to the period-2 orbit and the period-4 orbit of the Bozeman, MT 59715 experimental system. E-mail: [email protected] Larry Fabiny and Sandeep T. Vohra Naval Research Laboratory Optical Sciences Division CP18 Washington, DC 20375 Computational and Experimental Exploration of a Cortical Neuronal Network as a Dynamical System A large scale model of a rat whisker barrel is reduced to CP19 a simple dynamical system while retaining the ability for Exponentially Amplified Sampling and Recon­ both qualitative and quantitative comparison with biologi­ struction of Weak Signals Using Controlled Chaotic cal data. The reduced model yields novel and quantitative Orbits physiological predictions concerning the sensitivity of bar­ We describe a method of sampling and reconstructing ex­ rel neurons to whisker deflection amplitude and velocity. tremely weak signals based upon fundamental properties This prediction is tested directly through in vivo extra­ of chaotic dynamics. We show that small, continuous-time cellular recording studies examing the responses of barrel disturbance signals coupled into a chaotic system can be re­ neurons in the rat. constructed from the discrete-time perturbations required David J Pinto, Joshua C. Brumberg, to maintain a controlled periodic orbit. We derive a scal­ Daniel J. Simons, and G. Bard Ermentrout ing function necessary to transform the control perturba­ University of Pittsburgh, Pittsburgh, PA 15261 tion sequence into the continuous-time, weak disturbance E-mail: [email protected] signal for a typical, dissipative chaotic system. We show experimental measurements supporting our assertions. Chance M. Glenn, Sr. CP18 ECE Department A Bonhoeffer van der Pol Model for Action Poten­ The Johns Hopkins University tial in Excitability Changing Media Baltimore, MD 21218 A simple Bonhoeffer-van der Pol model of a heart tissue is Email: [email protected] studied to find the influence of spatial changes of excitabil­ ity on action potential propagation. The velocity of propagation is shown to be a good measure CP19 of excitability for the model. Targeting from a Chaotic Scattering Numerical solutions are obtained for "normal" and "patho­ We consider a model of a Hamiltonian system with an or­ logical" situations. A piecewise linear analytical model is bit "parked" on an unstable periodic orbit embedded in Abstracts 57

an unstable ehaotic set. We then attempt by means of parameters that affect stability. In different parameter a small control to target a position outside the original regimes, only one combination or the other is relevant. chaotic invariant set. This work illustrates how this can be Thus a change of parameter that moves the system from accomplished using the example of the chaotic scattering one regime to the other changes which system parameters set resulting from billiard type motion in the presence of determine stability. Heuristic arguments for the existence three hard circular discs. of more complicated anti-phase, suppressed and n to m solutions are given. Elbert E. N. Macau & Edward Ott University of Maryland, College Park, MD 20742 David Terman Ohio State University, Columbus, OH 43210

Nancy Kopell CP19 Boston University, Boston, MA 02215 Nonlinear Control of Remote Unstable States in a Liquid Bridge Convection Experiment Amitabha Bose We demonstrate the stabilization of an unstable periodic New Jersey Institute of Technology orbit whose trajectory in phase space is distant from the Newark, NJ 07102 unperturbed dynamics (characterized by two frequencies) E-mail: [email protected] in a convective flow experiment (Research supported by NASA and ONR). The system is a liquid bridge composed of silicone oil with a Prandtl number of about 40. Tem­ perature is measured at a point near the free surface of the CP20 Weakly Connected Weakly Forced Oscillatory N eu­ liquid column, and control perturbations are applied with a thermoelectric element located on the opposite side of the ral Networks bridge from the temperature sensor. A model independent, A network of weakly connected oscillatory neurons is par­ nonlinear control algorithm uses a time series reconstruc­ titioned into pools of oscillators having similar frequencies. tion of a control surface to characterize the system dynam­ Neurons within a pool interact via phase deviations. Av­ ics (V. Petrov and K. Showalter, Phys. Rev. Lett. 76, eraging shows that interactions between pools are func­ 3312 (1996)). Large perturbations are required to capture tionally insignificant even when there are synaptic connec­ the targeted periodic orbit, but once captured, control can tions between them. A neuron can participate in various be maintained indefinitely with very small perturbations pools by changing its center frequency. Interactions be­ (V. Petrov et al. submitted to Phys. Rev. Lett.). tween pools with frequencies fh i= [22 can be revived if the weak forcing is resonant with [21 - [22. Possible applica­ Valery Petrov, Michael F. Schatz, Kurt A. Muehlner Stephen J. VanHook, W. D. McCormick, J. B. Swift, tions to thalamo-cortical interactions are discussed. and Harry L. Swinney Eugene Izhikevich and Frank Hoppensteadt University of Texas, Austin, TX 78712 Center for Systems Science & Engineering E-mail: [email protected] Arizona State University Box 87-7606, Tempe, AZ 85287-7606 E-mail: [email protected] CP19 Sustaining Chaos using Basin Boundary Saddles We present a general method for preserving chaos in non­ CP20 chaotic parameter regimes by using the natural dynamics Monitoring Changing Dynamics with Correlation Integrals: Case Study of an Epileptic Seizure of system, and apply it to a CO2 laser model. Chaos is preserved by redirecting the flow towards the chaotic region We describe a procedure, and the motivation behind it, along unstable manifolds of basin boundary saddles, with which rapidly and accurately tracks the onset and progress the use of small infrequent parameter perturbations. of an epileptic seizure. Roughly speaking one monitors Ira B. Schwartz changes in the relative dispersion of a re-embedded time US Naval Research Laboratory series. The results are robust with respect to variation of Special Project for Nonlinear Science adjustable parameters such as embedding dimension, lag Code 6700.3, Plasma Physics Division time, and critical distances. Moreover, the general method Washington, DC 20375 is virtually unaffected when the data is significantly cor­ E-mail: [email protected] rupted by noise. David E. Lerner Ioana Triandaf Department of Mathematics Science Applications International Corporation University of Kansas, Lawrence, KS 66045 Applied Physics Operation, McLean, VA 22102

CP20 CP20 Preservation and Annihilation of Colliding Pulses Dynamics of Two Mutually Coupled Slow In­ in a Model Excitable System hibitory Neurons We analyze the transition from annihilation to preserva­ Recent analytic and numerical works have counter­ tion of colliding waves in a one-dimensional model excitable intuitively suggested that a network of neurons connected medium. The analysis exploits the similarity between the by slow inhibitory synapses may synchronize. We present local and global phase portraits of the system. The tran­ rigorous analytic results on how and under what condi­ sition is an infinite-dimensional analog of the creation and tions two such neurons can display stable synchronous os­ annihilation of limit cycles in the plane via a homo clinic cillations. We show that there are two combinations of Andronov bifurcation, and has parallels to the nucleation 58 Abstracts theory of first-order phase transitions. demonstrated by considering the coupling of the northern and southern hemisphere atmospheric circulations, mod­ M. Argentina and P. Coullet eled through a spectral truncation of the barotropic vor­ Institut Non-Lineaire de Nice, Sophia-Antipolis, France ticity equation. Despite time delays in the coupling, the attractive properties of an invariant synchronization man­ L. Mahadevan ifold in phase space cause correlations between weather 1-310, Massuchusetts Institute of Technology regimes in the two hemispheres, though the regime transi­ Cambridge, MA 02139-4307 tions in each hemisphere are timed chaotically. The trun­ cated model results agree with atmospheric observations. Gregory S. Duane and Peter J. Webster CP20 Program in Atmospheric and Oceanic Sciences Nonequilibrium Response Spectroscopy of ION University of Colorado, Boulder, CO 80309 Channel Gating E-mail: [email protected] We describe a new electrophysiological technique called nonequilibrium response spectroscopy which involves ap­ plication of a rapidly fluctuation (up to 10 kHz) large am­ plitude voltage clamp to voltage dependent biomolecules. CP21 Spontaneous Synchronization and Critical Behav­ The response of the channel to a rapidly fluctuating field is ior in Oscillator Chains exquisitely sensitive to features of the gating kinetics which are difficult or impossible to adequately resolve by means of The ability of an ensemble of disparate coupled nonlinear traditional stepped potential protocols. Here we focus on oscillators to spontaneously synchronize is dependent on the application of dichotomous noise voltage fluctuations. the range and strength of interactions. This dependence Dichotomous noise is a broad band Markovian colored noise is examined by introducing coupling that decays with dis­ with a discrete state space. Since Markovian kinetic mod­ tance as r-<> to a one dimensional chain of phase oscillators. els of channel gating can be imbedded in a higher dimen­ Critical behavior is found to exist as the range of interac­ sional state space which takes into account the effects of tions is varied. Previously studied models and results are the voltage fluctuations, many features of the response of recovered in the appropriate limits. the channels can be calculated in closed form. This makes Jeffrey L. Rogers dichotomous noise and is generalization uniquely suitable Georgia Institute of Technology for model selection and kinetic analysis. While nonequi­ School of Physics librium response methods can be applied to gating and 837 State Street single channel current recording techniques, here we focus Atlanta, GA 30332-0430 on application of this method to whole cell ionic current E-mail: [email protected] measurements. We show how data from the rat brain, and human cardiac isoforms of the sodium channel expressed in mammalian cells can be acquired and analyzed, and how this data reveals new, previously hidden aspect of the ki­ CP21 netics of these channels. Synchronization and Relaxation in Globally Cou­ pled Oscillators Mark M. Millonas James Franck Institute and Laboratory We examine a class of coupled Hamiltonian systems in of Cardiac Electrophysiology which identical nonlinear oscillators are coupled through a University of Chicago, Chicago IL 60637 mean field. It is shown that the system has a steady desyn­ E-mail: [email protected],edu chronized solution which becomes linearly unstable as the strength of the coupling is increased. In the stable case we observe that the order parameter of the system decays to zero. Our system shares many similarities to the Vlasov­ CP20 Poisson equation and to Kuramoto's model. We apply this Hodgkin-Huxley Neuronal Models model to understand sound propagation in bubbly fluids. We analyze the applicability of Hodgkin-Huxley type Peter Smereka conductance-based models of neurons, addressing the ques­ Department of Mathematics tions of parameter estimation, system noise, and model University of Michigan, Ann Arbor, MI 48109 performance. Different experimental protocols and mathe­ E-mail: [email protected] matical procedures for obtaining good parameter estimates are discussed, and the quality of the parameter estimation is examined in view of the inherent system noise. Finally, performance criteria for the models are considered. CP21 Synchronization of Spatiotemporal Chaos Allan R. Willms Center for Applied Mathematics Frequently, when chaotic systems are coupled, the com­ 657 Rhodes Hall plexity and the dimension of the resulting attractor in­ Cornell University creases. If however chaotic oscillations in these systems are Ithaca, NY, 14853-3801 synchronized, the dimension can remain unchanged. The [email protected] simplest form of synchronized chaos is identical oscillations in coupled systems. We demonstrate that such behavior is possible even if each uncoupled system demonstrates spa­ tiotemporal chaos. We discuss some stability issues. As an CP21 example, we consider two complex Ginzburg-Landau sys­ Synchronized Chaos in Spatially Extended Systems tems coupled dissipatively. And Interhemispheric Teleconnections Mikhail M. Sushchik Jr., Nikolai F. Rulkov, Partial chaotic synchronization in extended systems is Abstracts 59

and Michael M. Rabinovich vector can be computed also for complex time. Institute for Nonlinear Science Alain Goriely, Craig Hyde, and Michael Tabor University of California, San Diego La Jolla, CA 92093 Program in Applied Mathematics E-mail: [email protected] University of Arizona, Tucson AZ 85721

CP21 CP22 Effect of Finite Inertia in Coupled Oscillator Sys­ Asymptotic Expansions for Melnikov Functions for tems Conservative Systems We analyze the collective behavior of a set of coupled We use Melnikov techniques to show chaos without an ex­ damped driven pendula. In the limit with no inertia, this plicit homo clinic solution. For one degree of freedom con­ model becomes the Kuramoto model which exhibits syn­ servative equations we infer the existence of a homo clinic chronization onset by a second order phase transition with­ solution from the potential. We add perturbations: lin­ out hysteresis. For the case with finite (non-small) inertia, ear damping and periodic forcing, yielding two Melnikov we show that the synchronization of the oscillators exhibits integrals obtained without explicit knowledge of the ho­ a first order phase transition synchronization onset (a dis­ moclinic solution, one integral as a formula asymptotic in continuous jump). There is also hysteresis between two terms of the forcing frequency. As an example we consider macroscopic states, a weakly and a strongly coherent syn­ a sliding mass restrained by a transverse spring. chronized state, depending on the coupling and the initial state of the oscillators. A self-consistent theory is shown Joseph Gruendler to determine these cooperative phenomena and to predict Department of Mathematics the observed numerical data in specific examples. North Carolina A&T State University Greensboro, NC 27401 Hisa-Aki Tanaka [email protected] Waseda University, Tokyo, Japan [email protected]

Allan J. Lichtenberg CP22 University of California, Berkeley, CA 94720 Calculating Chaotic Strength The dynamics of the phase-locked loop (PLL) are both rich and complex. Using Melnikov techniques, we present CP21 a new method of driving the PLL chaotically, superior in ef­ Coupling Topology and Global Dynamics ficiency to previously reported results. Moreover, we show how the same techniques can be used to identify the opti­ One of assumptions underpinning much existing work on mum excitation required to disrupt any weakly perturbed large, coupled, dynamical systems is that the coupling Hamiltonian system containing a homoclinic connection. topology takes a simple form: usually either mean-field or Our analytic work is confirmed both experimentally and nearest-neighbour coupling. Using graph-theoretic tech­ numerically. We close the talk with the discovery of a fas­ niques, we investigate the effects of a range of coupling cinating period-adding phenomenum in the driven PLL. topologies, exhibiting varying degrees of randomness, on the global dynamics of some commonly studied oscillator David B. Levey and Paul D. Smith systems. Our results indicate that the global dynamical Department of Mathematics and Computer Science properties of some systems are highly sensitive to the un­ University of Dundee derlying topology and others, not at all. Perth Road, Dundee DD1 4HN Scotland, United Kingdom Duncan J. Watts and Steven H. Strogatz E-mail: [email protected] Cornell University Ithaca, NY 14853 E-mail: [email protected] CP22 Resonance Capture In Weakly Forced Mechanical Systems CP22 The Melnikov Vector as an Estimate for Exponen­ We study capture into resonance in the spin-up of an unbal­ tially Small Splitting of Homoclinic Manifolds anced rotor, with two orthogonal, linearly elastic supports. As the angular velocity of the rotor approaches the natural The Melnikov vector gives an estimate of the splitting dis­ frequencies of the frame, resonance capture may occur. Us­ tance between invariant manifolds for an oscillatory per­ ing the method of averaging, we obtain a simplified system turbation of an integrable system in n-dimensions. When valid in the neighborhood of the resonance. The result­ the forcing is fast (i.e. the period of the forcing is pro­ ing dynamical equations, which characterize capture into portional to the perturbation istelf), the splitting distance and escape from resonance, are studied using a generalized becomes exponentially small and the use of the Melnikov Melnikov integral. vector as an estimate of the splitting distance cannot be justified. However, in many examples, conditions on the D. Dane Quinn perturbations parameters can be derived so that the Mel­ Department of Mechanical Engineering nikov vector is a correct estimate. In this talk, general The University of Akron necessary conditions on the parameters so that the Mel­ Akron, OH 44325-3903 nikov vector provides a valid estimate of the splitting dis­ E-mail: [email protected] tance are derived. To do so, the relationship between the Melnikov vector and the singularities of the solutions in complex time is analyzed in detail so that the Melnikov 60 Abstracts

CP22 Oscillations Melnikov's Method for Stochastic Multi-Degree of We consider a system of two ordinary differential equations Freedom Dynamical Systems - Theory and Appli­ describing oscillations in the, so-called, Bray-Liebhafsky cations reaction that is much less studied than the well-known Melnikov's method for finding necessary conditions for ho­ Belousov-Zhabotinsky reaction. It is shown how oscilla­ moclinic chaos is extended to stochastically forced multi­ tions appear through a Hopf bifurcation and disappear degree of freedom dynamical systems. The stochastic Mel­ through a homoclinic bifurcation. This work comple­ nikov process is derived formally by generalizing results ments the results previously published by Noyes, Field and from single degree of freedom stochastic systems, and new Kalachev (1995). concepts inherent in this generalization are addressed. Ap­ William R. Derrick plications to the dynamics of a stochastically forced buck­ Department of Mathematical Sciences led column are presented. The use of the Melnikov process University of Montana, as an estimator of mean exit rates from preferred regions Missoula, MT 59812 of phase space is investigated. Timothy Whalen Leonid V. Kalachev Building and Fire Research Laboratory Department of Mathematical Sciences National Institute of Standards and Technology University of Montana, Gaithersburg, MD 20899 Missoula, MT 59812 E-mail: [email protected]

CP23 CP22 Effect of Weak Coupling on Oscillations in Bursting The Melnikov Theory for Subharmonics and Their Systems Bifurcations in Forced Oscillations We perform a bifurcation study of the Ginzburg-Landau The sub harmonic Melnikov theory for periodic perturba­ equations to examine the interaction of a pair of coupled tions of planar Hamiltonian systems is improved. An ap­ biological oscillators. Coupling is weak, diffusive and non­ proximation to the associated Poincare map in action-angle scalar, and non-identical oscillators are permitted. Our coordinates is explicitly constructed, and existence, sta­ motivation is to understand the synchronization of bursting bility and bifurcation theorems for subharmonics are ob­ electrical activity in pancreatic ,a-cells. For models of the tained. In particular, simple formulas for determining the fast oscillations in bursting activity, we demonstrate the stability of subharmonics and invariant circles bifurcating ubiquity of stable, asymmetrically phase-locked solutions, from them at Hopf bifurcations are obtained. Several ex­ in accordance with recent experimental results. amples are given to illustrate our theory. Gerda de Vries and Arthur Sherman Kazuyuki Yagasaki Mathematical Research Branch Department of Mechanical Engineering National Institutes of Health Gifu University, Gifu 501-11, Japan BSA Building, Suite 350 E-mail: [email protected] 9190 WISCONSIN AVE, MSC 2690 Bethesda, MD 20892-2690 E-mail: [email protected] CP23 Delaying the Transition to the Active Phase in Bursting Oscillators CP23 In a general model for bursting oscillators, we examine the Interpretation of Chaos in the transition between the quiescent and active phases. Burst­ Belousov-Zhabotinsky Reaction ing oscillations are usually thought of as sharply turning on The Belousov-Zhabotinsky reaction is a chemical system or off as a parameter is modified; in ,a-pancreas cells this that exhibits essentially all behaviors associated with non­ describes the change in the insulin release rate as a func­ linear dynamic laws, e.g., oscillations, spatial patterns, ex­ tion of the glucose concentration. Our analysis shows that citability and chaos, as well as bifurcations among these there is a smooth, albeit quick, transition. Specifically, for behaviors. The chemistry of this reaction is well-enough a critical tuning of the parameters, the system trajectory understood that an excellent approximation to its dynamic remains along an unstable manifold delaying the onset of law can be formulated. We will discuss the experimentally bursting. observed chaos in this system as well as its dynamical in­ Thomas W. Carr terpretation. Department of Mathematics Richard J. Field Southern Methodist University Department of Chemistry Dallas, TX 75275-0156 University of Montana [email protected] Missoula, MT 59812

Victoria Booth Department of Mathematics CP23 New Jersey Institute of Technology Phase-locking and Chaos in Forced Excitable Sys­ Newark, NJ 07102 tems Periodically-forced excitable systems are mathematical models for a variety of important systems in physiology, CP23 including cardiac tissue, sensory systems, calcium dynam- Bifurcations in One Model of the Bray-Liebhafsky Abstracts 61

ics, and secretory systems. Sufficiently slow forcing leads and space-dependent cubic terms is investigated. It is to set-valued circle maps, but rotation numbers can be de­ found that small terms breaking the space-reflection sym­ fined as in the smooth invertible case. However a different metry permit the existence of stationary fronts between approach is needed for rapidly-varying (e.g piecewise con­ roll solutions of different wavenumbers, and allow unstable stant) forcing. In this talk we describe recent results for a roll states to invade stable ones, in contrast to both the piecewise-linear Fitzhugh-Nagumo type model. In partic­ symmetric and variational cases. ular, we show the existence of subharmonic solutions and Rebecca Hoyle that stable periodic solutions with different rotation num­ Department of Applied Mathematics and Theoretical bers coexist on open sets in parameter space (Xie, et al., Physics 1996). The latter result shows how an experimental obser­ Silver Street, Cambridge CB3 9EW, United Kingdom vation first due to Mines (1913) can be understood in the E-mail: [email protected] context of a flow. We also show that chaotic dynamics can occur in some regions of parameter space.

Min Xie and Hans G. Othmer CP24 Department of Mathematics Defects and Domain Walls in the 2D Complex University of Utah Ginzburg-Landau Equation Salt Lake City, Utah 84112 We look at the spiral-wave domains which appear sponta­ neously in the complex Ginzburg-Landau equation. The domains are separated by thin walls, forming cellular pat­ CP23 terns on length scales much larger than the basic wave­ Slow Subsystem Bifurcations in Systems Exhibit­ length. To good approximation, the walls are segments ing Bursting of hyperbolae, with a transverse structure which depends Many models of excitable cells exhibit bistability between on the angle they make with the phase contours. This fast subsystem equilibria and fast subsystem periodic so­ structure is analyzed by treating the walls as heteroclinic lutions. Depending on parameter values, such systems can connections of a system of ODEs. exhibit bursting oscillations where fast variables (mem­ Greg Huber brane potential) alternate between oscillatory and non­ James Franck Institute and Ryerson Laboratory oscillatory states on a slow time. It will be shown, that as University of Chicago (slow) parameters are varied, the solutions of a polynomial 5640 S. Ellis Avenue model exhibiting bursting undergoes a series of bifurca­ Chicago, IL 60637 tions. Singular perturbation techniques and Melnikov the­ ory will be used to split a two parameter space into regions Tomas Bohr where the model exhibits bursting and steady behavior, Center for Chaos and Turbulence continual spiking and other more complicated oscillations. Niels Bohr Institute Mark Pernarowski 2100 Copenhagen, Denmark Department of Mathematical Sciences Montana State University Edward Ott Bozeman, MT 59715 Department of Physics E-mail: [email protected] University of Maryland College Park, MD 20742

CP23 Synchronous Bursting in Coupled Neurons CP24 Coupled Oscillators in Fluid Mechanics: Wakes be­ The effects related to the influence of couplings on the oscillatory frequency of interacting neurons are investi­ hind Rows of Cylinders gated in the context of general properties inherent in dif­ The well-known Benard-Von Karman cylinder wake is one ferent models. The effects similar to those observed in of the simplest oscillators of fluid mechanics. We present the Hindmarsh-Rose (HR) model (B.D.!. Abarbanel et al., here the results of experimental and theoretical works de­ Neural Computation 8, 1996, 234-244) are found for a voted to coupled wakes of several cylinders placed side by broad class of models. It is shown, in particular, that the side in a row perpendicular to an incoming flow. First, the stepwise dependence offrequency on the value of inhibitory coupled wakes of a pair of cylinders are investigated and coupling occurs both at mutual synchronization of two neu­ in particular the occurence of the different locked and un­ rons each of which is described by the same model (HM, locked regimes, depending on the distance separating the Sherman-Rinzel, Chay) and at interaction of the neurons cylinders. This study is then extended to a large num­ described by different models (Sherman-Rinzel and Chay ber of wakes which can be modelled by a discrete com­ models). plex Ginzburg-Landau equation. As predicted by a sta­ M.M. Sushchik, Ya.I. Molkov, M.l. Rabinovich bility analysis of this model, an "optical mode" of vor­ Institute of Applied Physics, Russian Academy of Science tex shedding, with neighbours in antiphase, is observed for 46 Uljanov Str., 603600 Nizhny Novgorod, Russia weakly coupled wakes. At medium coupling, some space­ E-mail: [email protected] time chaos is visualized. Space-time dislocations associ­ ated with wakes extinctions are randomly generated. Fi­ nally, when the distance between the cylinders is small, the wakes merge into different cells and are locked in phase in CP24 an "acoustic mode" . Fronts Between Rolls of Different Wavenumbers in the Presence of Broken Reflection Symmetry Patrice Le Gal, Jean-Francis Ravoux and Isabelle Peschard Institut de Recherche sur les Phenomenes Hors Equilibre A nonvariational Ginzburg-Landau equation with quintic 62 Abstracts

UM 138, Universite d'Aix-Marseille I & II-CNRS CP24 12 Avenue General Leclerc, 13003 Marseilles, France Convective and Absolute Instability in Finite Con­ E-mail: [email protected] tainers We consider the effect of including realistic boundary con­ ditions on the solutions to nonlinear equations that ad­ CP24 mit unidirectional waves with a definite (non-zero) group Dynamics of a 2-Dimensional Complex Ginzburg velocity. As examples we present solutions to the Com­ Landau Equation with Chiral Symmetry Breaking plex Ginzburg-Landau equation and some simplified dy­ namo equations (which are used to describe the generation We examine numerically and analytically the effect of of magnetic fields in many astrophysical bodies). Linear adding a chiral symmetry breaking term to the 2-D Com­ theory yields solutions that resemble wall-modes. How­ plex Ginzburg Landau equation. We find the shift in the ever as the control parameter is increased solutions begin spiral wave frequency due to the addition of this term. The to fill the domain, until the nearly uniform wave-train be­ sign of this shift depends on the topological charge of the comes unstable in a secondary bifurcation. Both the linear and nonlinear instabilities are related to solutions becom­ spiral wave. This results in the domination of one type of charge over the other - a feature seen in a recent experiment ing absolutely unstable. Ideas for further applications of the theory will also be presented. on Rayleigh-Benard convection. Steven M. Tobias Keeyeol Nam, Michael Gabbay, JILA, University of Colorado, Boulder, CO 80309 Parvez N. Guzdar, and Edward Ott E-mail [email protected] University of Maryland College Park, MD 20742 Edgar Knobloch University of California, Berkeley, CA 94720

CP24 Michael R.E. Proctor Noise Sensitivity and Boundary Effects in Travel­ DAMTP, University of Cambridge ling Wave Instabilities Cambridge, CB3 9EW, England, United Kingdom We consider the dynamics of nonlinear waves in driven dis­ sipative systems in finite domains, when the waves travel preferentially in one direction. Only when there is absolute CP25 instability can the waves sustain themselves without forc­ Noise-Enhanced Information Transmission in a ing against dissipation at the boundary; however, even in Network of Globally Coupled Oscillators the convective regime large amplitude solutions can occur due to very small stochastic perturbations. We describe We demonstrate a novel form of information transmission these effects for the CLG equation and a related system produced by modulating the coupling strength between in­ modelling dynamo waves in the Sun. dividual elements in a system of globally coupled nonlin­ ear oscillators. The modulation varies the fraction of os­ Michael R.E.Proctor cillators that are synchronized and transmits information DAMTP, University of Cambridge, Cambridge CB3 9EW, through the temporal behavior of the center of mass of the UK system. Optimum information transmission occurs at the level of intrinsic oscillator noise that poises the system at E.Knobloch the boundary between synchronous and non-synchronous Physics Dept., University of California, Berkeley states. Berkeley, CA 94720 Paul Gailey Oak Ridge National Laboratory, P.O. Box 2008, S.M.Tobias MS 6070, Oak Ridge, TN 37831-6070 JILA, University of Colorado E-mail: [email protected] Boulder, CO 80309 Thomas Imhoff Neuro Muscular Research Center, Boston CP24 University, Boston, MA 02215 Vortex Dynamics in Dissipative Systems We derive the exact equation of motion for a vortex in two­ Carson Chow and three-dimensional non-relativistic systems governed by Department of Mathematics the Ginzburg-Landau equation with complex coefficients. Boston University, Boston, MA 02215 The velocity is given in terms of local gradients of the mag­ nitude and phase of the complex field and is exact also for and Jim Collins Neuro Muscular Research Center arbitrarily small inter-vortex distances. and Department of Biomedical Engineering Boston University, Boston, MA 02215 Elsebeth Schroder [email protected] Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark E-mail: [email protected] CP25 Stochastic Resonance in a Neuronal Network from Ola Tornkvist Mammalian Brain NASAjFermilab Astrophysics Center, Stochastic Resonance (SR), a phenomenon whereby ran­ MS-209, P.O. Box 500, Batavia, IL 60510 dom noise optimizes a system's response to an otherwise 63 Abstracts subthreshold signal, has been postulated to provide a role experimental evidence that the signal-to-noise ratio (SNR) for noise in information processing in the brain. We used a of the output signal of a single diode resonator can be sig­ time varying electric field to deliver both signal and noise nificantly improved by coupling it diffusively into an ar­ directly to a neuronal network from mammalian brain. As ray of up to 32 non-identical resonators. It is shown that the noise amplitude was increased, we observed resonance the maximum SNR coincides with optimal spatiotemporal between the network's behavior and a subthreshold signal, synchronization of the chain. We investigate the effects of a clear demonstration of SR. different boundary conditions, and briefly address the con­ nection between spatiotemporal stochastic resonance and Bruce J. Gluckman the presence of kink-antikink pairs in the array. Naval Surface Warfare Center Carderock Division, Code 684, Bldg. 19-A252 Markus Locher, G. A. Johnson and E.R. Hunt Bethesda, MD 20084-5000 Dept. of Physics and Astronomy E-mail: [email protected] Condensed Matter and Surface Sciences Program and Ohio University, Athens, OH The Children's Research Institute of George Washington University F. Marchesoni Washington, DC 20010 Dept. of Physics, University of Illinois at Urbana- Champaign

CP25 Spatiotemporal Stochastic Resonance in Coupled CP25 Circuits Spatio-Temporal Stochastic Resonance of a Kink Motion in an Inhomogeneous

Alexander Neiman CP25 Department of Physics, Saratov State University Coherence Resonance in a Noise-Driven Excitable Astrakhanskaya str. 83, 410071 Saratov, Russia System [email protected] We study the dynamics of the excitable Fitz Hugh - Nagumo system under external noisy driving. Noise ac­ tivates the system producing a sequence of pulses. The CP26 coherence of these noise-induced oscillations is shown to Wave Propagation and its Failure in a Discrete be maximal for a certain noise amplitude. This new ef­ Reaction-Diffusion Equation fect of coherence resonance is explained by different noise­ dependencies of the activation and the excursion times. A We construct exact travelling wave solutions of a discrete simple one-dimensional model based on the Langevin dy­ reaction-diffusion equation describing a system of coupled namics is proposed for the quantitative description of this bistable elements. The form of the bistable potential is cho­ phenomenon. sen such that in a moving frame the waveform satisfies an integrable discrete equation that can be solved explicitly. Arkady Pikovsky and Jiirgen Kurths Arbeitsgruppe We analyse the linear stability of the wave and study wave "Nichtlineare Dynamik" propagation failure in the case of weak coupling using the Universitaet Potsdam anti-integrable limit. We also discuss some other applica­ Am Neuen Palais 10, Postfach 601553 tions of the anti-integrable limit in studying the behaviour D-14415 Potsdam, Germany of nonlinear discrete systems including the formation of lo­ E-mail: [email protected] calized states in a self-organizing neural network and local­ ized breathers in the nonlinear discrete Schrodinger equa­ tion. CP25 Paul C. Bressloff Spatiotemporal Stochastic Resonance in a System Loughborough University of Coupled Diode Resonators Loughborough, United Kingdom Recently, the phenomenon of array enhanced stochastic E-mail: [email protected] resonance was demonstrated via numerical simulations by Lindner et al. [Phys. Rev. Lett. 75, 3 (1995)]. We present 64 Abstracts

CP26 Northwestern University The Evolution of Slow Dispersal Rates: A Reaction Evanston, lL 60208 Diffusion System [email protected] We consider n phenotypes of a species in continuous but heterogeneous environment. It is assumed that the pheno­ types differ only in their diffusion rates. Assuming haploid CP26 genetics and a small rate of mutation it is shown that the Transition to an Effective Medium and Traveling only nontrivial equilibrium is a population dominated by Waves in Heterogeneous Reaction-Diffusion Sys­ the slowest diffusing phenotype. We, also, prove that if tems there are only two possible phenotypes then this equilib­ Pattern formation and wave propagation is studied in com­ rium is a global attract or and conjecture that this is true posite reaction-diffusion systems with spatially varying ki­ in general. Numerical simulations supporting this conjec­ netic properties in one and two spatial dimensions. Pertur­ ture and suggesting this is a robust phenomena are also bation theory is used to derive front equations of motion discussed. in a two-component bistable medium, capturing effects like Jack Dockery stationary or oscillatory front pinning with broken reflec­ Department of Mathematical Sciences tional symmetry. Direct numerical simulations for strongly Montana State University varying media show a transition to "effective medium"­ Bozeman, MT 59717 behavior, with "effective" traveling waves (pulses as well Email: [email protected] as spirals), that reflect the geometric arrangement of the heterogeneities. These "effective" waves lose stability as Vivian Hutson the length scale of the heterogeneity is increased. Further School of Mathematics and Statistics insight into the spatiotemporal dynamics is obtained from Sheffield University stability analysis of stationary as well as periodic solutions Sheffield, S3 7RH, United Kingdom of the PDE and by application of the Karhunen-Loeve de­ composition. Konstantin Mischaikow M. Bar and M. Bode School of Mathematics Max-Planck-lnstitut fiir Physik komplexer Systeme Georgia lnstitue of Technology Bayreuther Strasse 40, Haus 16, Atlanta, GA 30332 01187 Dresden, Germany

Mark Pernarowski A. K. Bangia and Y. Kevrekidis Department of Mathematical Sciences Department of Chemical Engineering Montana State University Princeton University Bozeman, MT 59715 Princeton, NJ 08544

CP26 Numerical Study of Bifurcations of Traveling Wave CP26 Solutions in a Reaction-Diffusion System Spatial Patterns and Dynamics in Filtration Com­ We study continuation and bifurcations of connection or­ bustion bits which are traveling wave solutions to the generalized Filtration combustion refers to highly exothermic heteroge­ FitzHugh-Nagumo equations. Our analysis includes de­ neous reactions in porous media involving both gaseous and tection of the heteroclinic loop and its continuation in 3 solid reactants. The reactions are characterized by high ac­ parameters, and searching for twisting points. tivation energies resulting in thin reaction zones. Using an­ Mark J. Friedman alytical and numerical methods, we describe instabilities, Mathematical Sciences Department dynamic behavior and pattern formation of propagating University of Alabama in Huntsville combustion wave solutions. The heterogeneous nature of Huntsville, AL 35899 the process allows interaction of instabilities and phenom­ E-mail: [email protected] ena characteristic of solid combustion with those of gaseous combustion. Daniel A. Schult CP26 Colgate University, Hamilton, NY Rhombs, Supersquares, and Pattern Competition in Reaction-Diffusion Systems We use equivariant bifurcation theory to investigate steady, CP27 spatially-periodic solutions of a general two-variable sys­ Stability Analysis of Reaction-Diffusion Waves tem of reaction-diffusion equations which undergoes a Thr­ Near Transitions to Spatiotemporal Chaos ing instability. We then apply our results to specific mod­ The stability of solitary pulses, wavetrains and fronts near els, such as Lengyel and Epstein's CIMA reaction model the onset of spatiotemporal chaos in a two-species reaction­ or the Schnakenberg model, to investigate transitions be­ diffusion (RD) model in one spatial dimension is inves­ tween e.g. hexagonal and rhombic patterns. The general tigated within a discretized, finite box representation of results are also used to construct models which produce the original RD model in a comoving reference frame and "supersquare" patterns in numerical simulations. compared to results from direct numerical simulations. We Stephen L. Judd and Mary Silber have identified three different scenarios of pulse bifurcation: Department of Engineering Sciences a collision of pulse branches, a saddle-node bifurcation and and Applied Mathematics a Hopf bifurcation leading to modulated traveling wave so- Abstracts 65

lutions in the PDE. In addition, stability results in two Maps spatial dimension are presented for rotating waves in small We show that the dynamics of logistic maps with a scaling circles (strong influence of boundary) and for the breakup form of connectivity exhibit a transition from spatial disor­ of rotating spiral waves in large domains. der to a spatially uniform, temporally chaotic state as the Markus Bar scaling is varied. We show that this transition is associated Max-Planck-Institut fUr Physik komplexer Systeme with a gap in the eigenvalue spectrum of the connectivity Bayreuther Strasse 40, Haus 16, matrix. We also study the Lyapunov spectrum of the sys­ 01187 Dresden, Germany tem away from the transition. Sridhar Raghavachari and James A. Glazier Anil K. Bangia and Yannis Kevrekidis Department of Physics Dept. of Chemical Engineering, Princeton University University of Notre Dame Princeton, NJ 08544 Notre Dame IN 46556 E-mail: [email protected]

CP27 Strange Attractors of the KdV-KS Equation: Spa­ CP27 tiotemporal Order and Chaos in 3D Film Flows Detecting and Extracting Messages from Chaotic We report analytic and numerical studies of a (1+2)D Communications using Nonlinear Dynamic Fore­ evolution equation (EE) approximating the surface of a casting film flowing down an inclined plane. The EE contains This paper will consider the use of nonlinear dynamic a control parameter, the dispersivity-to-dissipativity ra­ forecasting to detect and extract message signals from tio >.. In (l+l)D case, the EE is the KdV one with chaotic communication systems, using additive-message, (important) additional (dissipative) KS terms; it becomes modulated-message, and chaotic control schemes. The role just the KS equation for >. ---> O. We emphasize time­ of one-step and multi-step predictions will be considered, asymptotic regimes. Although the attractors are strange and techniques will be discussed which can aid in rebuild­ (the largest Liapunov exponents are positive) for all cas~s, ing the hidden message signal from short multi-step pre­ it is only for small >. that the surface waves appear dIs­ dictions. Some suggestions will be made about increasing ordered: For large >., they become unusual spatially or­ the security in chaotic communication systems. dered patterns, travelling collections of coherent soliton-like structures. Movies of these asymptotic regimes (see also Kevin M. Short http://euler.math.ua.edu:1997) will be shown. Statistical Department of Mathematics correlation results will be presented for both disordered University of New Hampshire and ordered wavy surfaces. Durham, NH 03824 E-mail: [email protected] Alexander L. Frenkel and K. Indireshkumar University of Alabama Tuscaloosa, AL 35487 CP27 Local Dynamics and Spatiotemporal Complexity CP27 We study a degenerate local bifurcation in the Brusselator Reduction of Complexity and Control of Spatio­ reaction-diffusion equations, for which standard up-folding Temporal Chaos through Archetypes and analysis predicts complex behavior, including Sil'nikov chaos. However, we show that such local analysis provides The space-time description of complex, extended systems only limited understanding of spatiotemporal complexity, can sometimes be reduced to a few effective degrees of as the normal form has very restricted validity. Our ap­ freedom by archetype decomposition-a variant of princi­ proach involves careful comparison of the predictions of pal component analysis that searches for archetypal pat­ computed high-order (nonunique) normal forms with the terns in raw data. We show that it is possible to identify untransformed system on the center manifold, and approx­ low dimensional unstable periodicity in systems exhibit­ imations of the full PDE. ing spatio-temporal chaos via archetype analysis. We also demonstrate how archetype analysis can facilitate the im­ Ralf W. Wittenberg plementation of strategies for controlling unstable periodic Program in Applied and Computational Mathematics modes in such systems. Princeton University, Princeton, NJ 08544 David Peak Philip Holmes Physics Department Program in Applied and Computational Mathematics, and Utah State University Department of Mechanical and Aerospace Engineering Logan, UT 84322-4415 Princeton University, Princeton, NJ 08544 E-mail: peakd@cc. usu. edu

Emily Stone and Adele Cutler Mathematics and Statistics Department CP28 Utah State University The Dynamics of Variable Time-Stepping ODE Logan, UT 84322-3900 Solvers Most commercial IVP solvers for ODEs use variable time­ stepping algorithms to minimize the work done in an CP27 integration. These numerical algorithms define discrete Spatially Coherent States in Long Range Coupled dynamical systems which are discontinuous due to the timestep-selection mechanism; furthermore, they are in 66 Abstracts a phase space one dimension higher than that of the leads to correct results. ODE itself. We study Runge-Kutta methods controlled Govindan Rangara jan by commonly-used timestep selection strategies and prove Department of Mathematics convergence results for trajectories on finite time intervals Indian Institute of Science as the tolerance r -+ O. We also examine the existence and Bangalore 560 012, India convergence properties of invariant limit sets for a class E-mail: [email protected] of discontinuous mappings that includes these numerical schemes. Harbir Lamba and Andrew Stuart CP28 SCCM program, Division of Mechanics and Computation Convergence of a Fourier-Spline Representation of Stanford University the Poincare Map Generator California 94305-4040 E-mail [email protected] The generating function for a Poincare map of an Hamilto­ nian system may be approximated as a truncated Fourier series in angle variables, the Fourier coefficients being spline functions of action variables. The coefficients may be CP28 constructed numerically from data on the underlying flow Multiple-shooting Newton-Picard Methods for as given by a symplectic integrator. We investigate the Computing Periodic Solutions of Large-Scale Dy­ convergence of the representation as the number of Fourier namical Systems with Low-Dimensional Dynamics modes and the number of spline knots tend to infinity. We will present a method based on multiple shooting to Robert L. Warnock compute and determine the stability of both stable and Stanford Linear Accelerator Center unstable periodic orbits of certain classes of partial differ­ Stanford University, Stanford, CA 94309 ential equations. The method uses a clever combination E-mail: [email protected] of a direct linear system solver and a Picard iteration to solve the linearised systems and generates good estimates James A. Ellison for the dominant Floquet multipliers almost for free. The Department of Mathematics and Statistics method is particularly interesting when a fine discretisation University of New Mexico, Albuquerque, NM 87131 is needed. E-mail: [email protected] Kurt Lust, Koen Engelborghs, and Dirk Roose Department of Computer Science K.U.Leuven, Celestijnenlaan 200A, CP29 B-300l Heverlee, Belgium Homogenization of the Navier-Stokes Equation for E-mail: [email protected] Oscillatory Fluids The Navier-Stokes equation is homogenized for oscillatory fluids. This results justifies multi-scales expansions com­ CP28 mon in physics and engineering. We capture the contri­ Computation of Stability Basin for Large Close-to­ butions of small fluctuations to the mean flow and apply Hamiltonian Nonlinear Systems the method to prove the Kolmogorov scaling for oscillatory It is shown that stability boundaries for multidimensional fluids. close-to-Hamiltonian models of power systems include frac­ Bjorn Birnir tal layers consisting of irregular mosaic of instable islands Department of Mathematics imbedded in stability areas. Novel computational method­ University of California, Santa Barbara, CA 93106 ology revealing transient stability phenomena occurring on E-mail: [email protected] short time intervals is proposed. Developed method runs much faster than the numerical simulation and provide the averaging of results obtained by numerous individual sim­ ulations. CP29 The Transition to Chaos for an Array of External Leonid Reznikov and Mark A.Pinsky Driven Vortices Mathematics Department University of Nevada, Reno We will present results on the transition from laminar flow Reno, NV 89511 to chaos for an array of externally driven vortices. The [email protected] strength of the forcing plays the role of the bifurcation parameter. We investigate numerically the changes in the bifurcation sequence and the generation of large scale shear flows for different number, as well as different aspect ratios, CP28 of the driven vortices. The influence of stress-free and no­ Reliable Numerical Detection of Chaotic Behaviour slip boundary conditons, will also be presented. in Hamiltonian Systems Fred Feudel One is often interested in long-term stability of complicated Max-Planck-Arbeitsgruppe Nichtlineare Dynamik an der Hamiltonian systems and reliably determining whether Universitat chaotic behaviour exists. For this, we require a fast nu­ Potsdam, PF605315, D - 14415 Potsdam merical integration algorithm that is explicitly symplectic. Fax: 49-331-9771142 Otherwise, we can get spurious chaotic behaviour where E-mail: [email protected] none exists. We present an efficient symplectic integration algorithm using Lie perturbation theory. We present exam­ Parvez Guzdar ples where conventional numerical integration techniques Institute for Plasma Research, University of Maryland lead to spurious chaotic behaviour whereas our algorithm Abstracts 67

College Park, MD 20742 Department of Mathematics and Statistics Fax: (301 )-405-1678 Winona State University, Winona, MN 55987 E-mail: [email protected] E-mail: [email protected]

James A. Yorke Institute for Physical Science and Technology CP29 University of Maryland, College Park, MD 20742 Beltrami Flows and Contact Geometry E-mail: [email protected] We consider the class of nonsingular Eulerian flows which are parallel to their curl: Beltrami flows. We establish a surprising connection between these flows and recent re­ sults in contact geometry. We draw an equivalence between CP29 Granular Glasses and Fluctuating Hydrodynamics such Beltrami flows and abstract "Reeb fields", which al­ lows us to conclude the general existence of closed flowlines. The properties of dense granular systems are analyzed from This leads to results on hydrodynamic stability of Beltrami a hydro dynamical point of view, based on conservation flows. laws for the particle number density and linear momen­ tum. We discuss averaging problems and the peculiarities Robert W. Ghrist and John B. Etnyre ofthe sources of noise. We perform a quantitative study by University of Texas, Asutin combining analitical and available experimental results on Austin, TX 78712 creep during compaction and data from molecular dynamic E-mail: [email protected] simulations of convective flow. We study boundary condi­ tions and finally show that the numerical integration of the hydrodynamic equations gives the expected evolution for CP29 the time-dependent fields. The Temporal Dynamics of the Ocean Circulation Clara Salueiia Because of dynamical constraints, the large-scale move­ James Franck Institute and the Department of Physics, ment of water in the general circulation of the ocean is University of Chicago, 5640 S. Ellis Ave. ,Chicago, IL 60637 irregular but not fully turbulent. Such circulations are typ­ USA, and ically modelled numerically as high-dimensional, damped, Departament de Fisica Fonamental, Universitat de 4 6 dynamical systems (with dimension of 0(10 - 10 )). Us­ Barcelona, ing long-term integrations of such a model, we explore both Diagonal 647, Barcelona 08028, Spain the regime in which low-dimensional dynamics prevails and E-mail: [email protected] the "higher Reynolds number" regime beyond this. Simple low-dimensional models of the low-dimensional regime are Sergei E. Esipov presented and their limitations are discussed. The nature James Franck Institute and the Department of Physics, of the transition from low-dimensional behavior to progres­ University of Chicago, 5640 S. Ellis Ave.,Chicago, IL 60637 sively higher-dimensional behavior is described using a va­ USA riety of statistical techniques. It is shown that in this tran­ sition, long-term variability is introduced with time scales Thorsten Piischel that are centennial and longer. Understanding the nature Institut fUr Physik, Humboldt-Universitat zu Berlin, of this transition is therefore an open problem of direct rel­ InvalidenstraBe 110 Berlin D-10115, Germany evance to the study of fluctuations in the Earth's climate. Steve Meacham Florida State University, Tallahassee, FL 32306 CP30 E-mail: [email protected] Global Asymptotic Behavior and Dispersion in Size-Structured, Discrete Competitive Systems Pavel Berloff The model for this study is a 2 species discrete competi­ University of California at Los Angeles tive system with two size classes, small and large. During Los Angeles, CA 90024 each time period, some small ones and some large ones die, some small ones become large, and some small ones stay small. As the small ones become large, they disperse CP29 between two patches. Conditions for the global conver­ Continuous Avalanche Mixing of Granular Solids gence to the extinction states of each of the two competing in a Rotating Drum species are investigated. The relationship between the car­ We consider the avalanche mixing of a monodisperse collec­ rying capacities of the species and their persistence is also tion of granular solids in a slowly rotating drum. A rather explored. Through numerical studies, we show that this clever computer model of this process has been developed model has a riddled basin of attraction. for the case where the drum rotates slowly enough that John E. Franke each avalanche ceases completely before a new one begins Department of Mathematics [METCALFE, SHINBROT, MCCARTHY, AND OTTINO, Na­ North Carolina State University ture, 374(1995)]. We develop a mathematical model for the Raleigh, NC 26795-8205 mixing in both this discrete avalanche case and in the more useful case where the drum is rotated quickly enough to Abdul-Aziz Yakubu induce a continuous avalanche in the material but slowly Department of Mathematics enough to avoid significant inertial effects. Our discrete Howard University model yields results which are consistent with those ob­ Washington, DC 20059 tained by Metcalfe et al. Barry A. Peratt 68 Abstracts

CP30 is associated with the myogenic rcsponse of the afferent ar­ Population Dynamics of Dungeness Crab: Con­ terioles that cause these vessels to contract when the blood necting Mechanistic Models to Data pressure rises. Another mechanism is associated with the so-called tubuloglomerular feedback which involves a reg­ We show density dependent features of Dungeness crab bi­ ulation of the arteriolar flow resistence in response to the ology may cause large amplitude population fluctuations composition of the fluid leaving the loop of Henle. Each when subjected to small environmental perturbations. We response can become unstable and produce self-sustained do this by fitting a stochastic mechanistic model to time oscillations with characteristic periodicities. Based directly series data (42 years). The model incorporates cannibal­ on the involved physiological mechanisms we have devel­ ism, density-dependent egg production, age-structure, and oped a model of these feedback regulations. It is shown a realistic multivariate noise structure. The best fitting how a Hopf bifurcation leads the system to perform self­ parameter values for the underlying deterministic models sustained oscillations ifthe tubuloglumerular feedback gain produce equilibrium (6 ports) and limit cycles (2 ports) in becomes sufficiently strong, and how a further increase of the absence of environmental perturbations. this parameter produces a folded structure (crossroad area) Kevin Higgins of overlapping period-doubling cascades. The numerical Institute of Theoretical Dynamics analyses are supported by existing experimental results. University of California, Davis CA Erik Mosekilde and Mikael Barfred E-mail: [email protected] Center for Chaos and Thrbulence Studies Department of Physics Alan Hastings The Technical University of Denmark Division of Environmental Studies DK-2800 Lyngby, Denmark University of California E-mail: [email protected] Davis CA

Jacob N. Sarvela Niels-Henrik Holstein-Rathlou Department of Computer Sciences Department of Medical Physiology University of Texas University of Copenhagen Austin TX Copenhagen, Denmark

Louis W. Botsford Wildlife, Fish and Conservation Biology CP30 University of California A Two-Dimensional Model of HUInan Walking Davis, CA A two-dimensional, four-angle model of human walking will be presented. The model will examine both the swing phase as well as the double support phase of gait through CP30 the use of Lagrangian equations of motion. A relaxation Pattern Formation In Vitro method scheme is employed to solve the resulting nonlin­ Spectacular spatial patterns can be formed from the ag­ ear ordinary differential equations so that the model can gregation of microorganisms in an initially homogeneous predict such gait characteristics as toe-off angle and step environment, as they interact with each other and with length. To conclude, animations of both experimental data their environment. Well-known examples are the chemo­ and model predictions will be shown to demonstrate the tactic aggregation patterns of the slime mold Dictyostelium model's accuracy. discoideum and the bacterium E. coli. We study a chemo­ Susan J. Schenk tactic model of pattern formation in cultures of E. coli and New Jersey Institute of Technology Salmonella typhimurium, and a non-chemotactic aggregat­ & Essex County College, Newark, NJ 07102 ing eukaryotic system which appears to generate pattern in E-mail: [email protected] vitro using only traction forces. Our simplest assumptions lead to experimentally observed patterns in all of these systems under a wide range of conditions. The eukaryotic CP30 system is dependent on the cells dynamically changing the Dynamics of Mass Attack, Spatial Invasion of Pine material properties of their extracellular matrix. Beetles into Lodgepole Forests Sharon R. Lubkin The Mountain Pine Beetle (MPB) is a major cause of de­ Dept. of Applied Mathematics FS-20 struction of valuable wood resources in the western States. University of Washington A system of Partial Differential Equations (PDE) has been Seattle, WA 98195 developed to model the spatial redistribution of the MPB. E-mail: [email protected] This model is based on movement of the flying MPB in re­ sponse to the chemical environment and incorporates the trees natural defense mechanism of pitching out nesting CP30 MPB. Solving this system ofPDE's can be computationally Bifurcation Analysis of Nephron Pressure and Flow time consuming so a system of Integro-Difference Equa­ Regulation tions (IDE) has been developed to approximate the sys­ By regulating the secretion of salts, water and metabolic tem of PDE's. In this talk we will compare the numerical endproducts, the kidney plays an important role in con­ scheme for solving both the IDE and PDE systems by look­ trolling the blood pressure. To protect their own function ing at the dynamics they suggest for the MPB mass attack. and secure a relatively constant blood flow, the kidneys also dispose of mechanisms that can compensate for varia­ Peter W. White and James Powell tions in the arterial blood pressure. One such mechanism Department of Mathematics and Statistics Abstracts 69

Utah State University, Logan, UT 84322-3900 usc. Our discussion shows that for the projections con­ [email protected] sidered here, one has to take into account the question of http://xserver.math.usu.edu/staff/pwhite/main.html completeness of the set of basis functions. Jesse Logan and Barbara Bentz Dietmar Rempfer USFS Intermountain Research Station Cornell University, Ithaca, NY 14853 860 North 1200 East, Logan, Utah 84321

CP31 CP31 A Study of Transition in Rayleigh-Benard Convec­ Coupled Mechanical-Structural Dynamical Sys­ tion using a Karhunen-Loeve Basis tems: A Geometric Singular Perturbation-Proper Orthogonal Decomposition Approach A Karhunen-Loeve (K-L) basis is generated from a database obtained from a numerical simulation of the We combine the theories of GSP(geometric singular per­ Boussinesq equations governing Rayleigh-Benard ther­ turbation), and POD (proper orthogonal decomposition) ~o mal convection in a horizontally periodic convective box study coupled infinite-dimensional dynamical systems m bounded vertically by a free surface at some reference val­ mechanics. We present results for a mechanical-structural ues of the parameters; Rayleigh number, Prandtl number system consisting of a pendulum coupled to a linear vis­ and aspect ratio of the convective box. The K-L basis is coelastic rod. Whenever the coupling between the flexible then used to reduce the Boussinesq equations into a trun­ rod and the pendulum is sufficiently small, the GSP pre­ cated system of amplitude equations, in which Rayleigh dicts a global, nonlinear, two-dimensional invariant man­ number is kept as a parameter while Prandtl number and ifold. The long-time(slow) dynamics reside on this mani­ aspect ratio are fixed at the reference values, by a Galerkin fold. The POD method confirms this result since it predicts projection. This system is studied as a model of the tran­ a single coherent structure for sufficiently small coupling. sition as Rayleigh number is increased passed its critical Using the POD method, we identify the additional degrees­ value from the linear stability theory. of-freedom that are activated as the coupling increases. 1. Hakan Tarman Ioannis T. Georgiou and Ira B. Schwartz Department of Mathematical Sciences Naval Research Laboratory King Fahd University of Petroleum and Minerals Special Project for Nonlinear Science Dhahran 31261, Saudi Arabia Plasma Physics Division Fax: (966) (3)860-2340 Washington DC 20375 E-mail: [email protected] E-mail: [email protected]

CP31 CP31 Detecting Interacting Modes in Spatiotemporal The Karhunen-Loeve Transformation: When does Signals it work? A method for the analysis of spatiotemporal signals is pre­ The application of the Karhunen-Loeve (KL) procedure. to sented. The approach aims at a decomposition of the signal the analysis of spatio-temporal data sets and models begms q(x, t) = L. ~i(t)Ui by the simultaneous identification of with computing data dependent eigenvectors which form the spatial ~odes Ui and the underlying dynamical sys­ the basis of an orthogonal transformation. This transfor­ tem a~;jat = J[{~j}]. This is accomplished by minimizing mation produces what is often referred to as an "optimal a least-square-fit potential which yields a nonlinear eigen­ representation" of the data. This talk will focus on when value problem with higher order correlation tensors. Ap­ the KL procedure is actually optimal, and when it is not. plications of the algorithm to hydrodynamic instabilities We also will consider how dimensionality reducing map­ [Uhl, C., Friedrich, R. , Haken, H., Phys.Rev.E 51, 5, 3890] pings can be constructed to improve upon the KL approach are presented. Medical applications are shown in a second when it in fact fails to be optimal. contribution. Michael Kirby Christian Uhl Department of Mathematics Max-Planck-Institue of Cognitive Neuroscience Colorado State University Inselstrasse 22-26 Ft. Collins, CO 80523 D-04103 Leipzig, Germany Fax: 970-491-2161 E-mail: [email protected] E-mail: [email protected]

CP32 CP31 Enhancement of Stirring by Chaotic Advection us­ On Dynamical Systems Obtained Via Galerkin ing Parameter Perturbation and Target Dynamics Projection onto Bases of KL-Eigenfunctions for Fluid Flows The problem of maintaining chaotic stirring of particles ad­ vected in a fluid is treated using occasional perturbations In this talk we present an analysis of the application of of a system parameter and prescribed chaotic target dy­ Galerkin projection in conjunction with Karhunen-Loeve namics. The symmetry of the regular motion is broken by eigenfunctions obtained from complex (transitional or tur­ forcing it to follow a chaotic reference model. Results of bulent) unsteady fluid flows. We will show that for certain simulations are presented for a class of eggbeater models. cases it is possible to derive dynamical systems represen­ Increasing the allowed maximum level of the control effort tations for arbitrary subregions of a flow that are exact causes the regular motion to become chaotic in a shorter in a certain sense. It will turn out, however, that these exact representations may nevertheless be only of limited 70 Abstracts

time. Woods Hole Oceanographic Institution Woods Hole, MA 02543 Yechiel Crispin Department of Aerospace Engineering Embry-Riddle University Daytona Beach, FL 32114 CP32 [email protected] Anomalous Dispersion in 2-d Turbulence Here I discuss some recent results on single-particle disper­ sion in 2D turbulence and point vortex systems. The main CP32 result is the identification of different regimes of anomalous Fractal Entrainment Sets of Tracers Advected by dispersion at intermediate times, related to the dominance Chaotic Temporally irregular Fluid Flows of hyperbolic or elliptic regions and to the presence of co­ herent vortices. The chaoticity of Lagrangian trajectories We model a two-dimensional open fluid flow that has is studied as well. A comparison with the dispersion prop­ temporally irregular time dependence by a random map erties of ocean subsurface floats (SOFAR floats) is consid­ ~n+l Mn(~n)' where on each iterate n, the map Mn is = ered. chosen from an ensemble. We show that a tracer distri­ bution advected through a chaotic region can be entrained Antonello Provenzale on a set that becomes fractal as time increases. Theoret­ Instituto di Cosmo-geofisica ical and numerical results on the multifractal dimension Torino, Italy spectrum are presented. Joeri Jacobs, Edward Ott, Thomas Antonsen and James Yorke CP32 Institute for Plasma Research Geometrical Dependence of Finite Time Lyapunov University of Maryland Exponents and Transport College Park, MD 20742 We report a general relationship between finite time Lya­ E-mail: [email protected] punov exponents and the geometry of stable foliations of low dimensional dynamical systems (Physica D 95 283- 305). In short, the finite time Lyapunov exponent varies CP32 smoothly along the stable foliation and its spatial deriva­ Chaotic Advection in an Array of Quasi-2D Vor­ tive is determined by the divergence of the tangent (unit) tices vector of the stable foilation. Numerical illustrations will Chaotic advection and mixing of passive tracers in an array be presented for both conservative and dissipative systems. One immediate application is the prediction of a new class of quasi-2D vortices are studied by laboratory experiments in stratified fluids. The vortices result from the inverse en­ of barriers for diffusive transport in hamiltonian systems. ergy cascade of quasi-2D turbulence in a rectangular con­ Xianzhu Tang tainer. Both the Eularian flow field and the Lagrangian College of William and Mary, Williamsburg, VA motion of particles and blobs of dye are measured with and Columbia University, New York, NY 10027 digital image-analysing techniques. Numerical simulations E-mail: [email protected] and methods of dynamical systems theory are applied to analyse the (chaotic) transport properties of these tracers. Allen H. Boozer S.R. Maassen, H.J.H. Clercx and G.J.F. van Heijst, Department of Applied Physics Columbia University, New York, NY 10027 Eindhoven University of Technology, and Max-Planck Institute, Germany Fluid Dynamics Laboratory, P.O. Box 513,5600 MB Eindhoven, The Netherlands E-mail: [email protected] CP33 Bifurcation Leading to Hysteresis CP32 We study the bifurcation which leads to hysteresis in a Quantifying Transport in Numerical Simulations of bistable differential equation with periodic modulation of Oceanic Flows a parameter. Such equations are of interest in optical bista­ Geometric methods from dynamical systems are used to bility and ferromagnetism. As the modulation amplitude characterize Lagrangian transport in numerical simulations increases, two stable and one unstable cycles join to form of two-dimensional, oceanic flows. Numerical methods a hysteresis loop. We study the equation for approximating invariant manifolds of hyperbolic fixed points are applied to the fully aperiodic, time-dependent x'(t) = L(x(t)) + hJ-£(t), vector fields, identifying two-dimensional "stable" and "un­ stable" manifolds emanating from regions of strong hyper­ bolicity. The lobes resulting from the transverse intersec­ where L is piecewise linear and h is amplitude. We make use of symmetry properties and of explicit solutions of the tions of these invariant structures are used to quantify the equation to study the bifurcation process and critical value transport between different regions of the flow. of the amplitude. Patrick D. Miller and Christopher K.R.T. Jones Division of Applied Mathematics Gregory Berkolaiko Brown University, Providence, RI 02912 Department of Mathematics E-mail: [email protected] University of Strathclyde Glasgow, G11XH Scotland Audrey M. Rogerson and Lawrence J. Pratt United Kingdom E-mail: [email protected]. uk Abstracts 71

CP33 An Introduction to Groebner Bases and Invariant Ramiro Rico-Martinez Theory Instituto Tecnol6gico de Celaya Depto. de Ingenieria Quimica, Celaya, Gto. 38010 Mexico The problem of finding normal forms for systems with sym­ metry is closely related to the Nineteenth Century alge­ braic subject ofInvariant Theory and Hilbert's Fourteenth CP33 Problem. The recent development of Groebner bases facil­ Instabilities of Spatio-Temporally Symmetric Peri­ itates a systematic search for the invariants and the rela­ odic Orbits tionships among them. In addition, Groebner bases have a wide variety of applications to other problems. We develop tools to analyse the instabilities of periodic or­ bits with spatio-temporal symmetries. Convection prob­ T. K. Callahan and Edgar Knobloch lems with periodic side boundary conditions often have University of California temporally periodic solutions that are invariant under ad­ Berkeley, CA 94720 vancing a fraction of their temporal period combined with [email protected] a reflection or rotation. Translations generate a continuous family of these solutions, and bifurcations that break the spatio-temporal symmetry can lead to solutions that drift along this group orbit. We discuss examples from magne­ CP33 On a Neutral Functional Differential Equation to convection and other hydrodynamic systems. Modelling a Simple Physical System A.M. Rucklidge Department of Applied Mathematics and Theoretical One of the simplest differential equations used to model Physics physical systems is a damped harmonic oscillator. In cer­ University of Cambridge tain applications If there is a time delay associated with Cambridge CB3 9EW, United Kingdom this feedback, our "simple" model has become a neutral E-mail: [email protected] functional differential equation, which can be quite com­ plicated to analyse. We will consider the stability of the trivial solution to this equation, with particular regard to M. Silber the role of the time delay in thise stability. We will show Department of Engineering Sciences and Applied Mathe­ that bifurcations to periodic solutions can occur and for­ matics mulate conditions under which two such bifurcations may Northwestern University, Evanston, IL 60208 E-mail: [email protected] interact, leading to a co dimension two bifurcation. Finally, we will show that such interactions may be resonant, i.e. that the frequencies of the associated periodic solutions, w1,w2, may obey w1:w2=m:n, where m and n are integers. CP33 Grazing Bifurcations in a Piezoelectric Impact Os­ Sue Ann Campbell Department of Applied Mathematics cillator University of Waterloo Grazing bifurcations tend to occur in non-smooth dynam­ Waterloo, Ontario N2L 3G1, Canada ical systems and are considered important physical exam­ Email: [email protected] ples of a general type of bifurcations called border-collision bifurcations. Recently, there has been a flurry of activity in numerically analyzing the Nordmark map which predicts CP33 the generic behavior of grazing bifurcations in impacting Feedback-Assisted Instability Detection systems. In this work, an experimental investigation of grazing bifurcations in a piezoelectric impact oscillator is We present a new methodology for the experimental study carried out. Experimental verification of the predictions of instabilities in nonlinear dynamical systems. In contrast regarding observation of "maximal" periodic orbits close to the usual "settle and observe" protocol, we use feed­ to grazing impact is carried out. back algorithms motivated by numerical bifurcation theory to experimentally trace bifurcation diagrams. Bifurcation Sandeep T. Vohra points of the experiment become steady states of an aug­ Naval Research Laboratory mented dynamical system in which the control parameter Optical Sciences Division is considered as an additional state variable. We demon­ Washington, DC 20375 strate the applicability of the proposed method in several cases, including convergence on "hard" bifurcations such as a subcritical Hopf. We study the issue of single state CP33 measurement versus complete state measurement as well as Stability of the Whirling Modes ofa Hanging Chain the effect of noise on our identification/feedback scheme. (Kolodner's Modes) Jason S. Anderson, Georg Flatgen, Yannis G. Kevrekidis In 1955, Kolodner proved the existence of infinitely many Department of Chemical Engineering families of relative equilibria for the equations that model Princeton University, Princeton, NJ 08544 a hanging chain. The stability of these nonlinear whirling E-mail: [email protected] modes has remained an open problem. I prove the linear or­ [email protected] bital stability of the relative equilibria that are sufficiently close to the vertical equilibrium (i.e. that have sufficiently Katharina Krischer small angular momentum). The proof relies on the invari­ Fritz-Haber-Institut der Max-Planck-Gesellschaft ance of the stable equilibrium under the action of the sym­ Faradayweg 4-6, 14195 Berlin, Germany metry group 80(2). The stability of Kolodner's whirling E-mail: [email protected] modes is an application of a more general result. Consider 72 Abstracts

a simple mechanical system (Q, K, V, G) with symmetry such data, with the specific goal of elucidating differences group G = 50(2) whose action is not free, and let qo be between good and bad ODE models. In particular, we fo­ an equilibrium for which V"(qo) > O. I prove (under some cus our efforts on leveraging these tools to derive a tter fairly general conditions) the linear orbital stability of all model of the driven, damped pendulum. relative equilibria sufficiently close to qo' A finite dimen­ Joseph Iwanski sional system to which the theory applies is the double Department of Applied Math spherical pendulum. University of Colorado VVarren VVeckesser Campus Box 526 Department of Mathematical Sciences Boulder CO 80309-0430 Rensselaer Polytechnic Institute Troy, NY 12180 Elizabeth Bradley E-mail: [email protected] University of Colorado Campus Box 430 Boulder CO 80309-0430 CP34 Testing Nonlinear Markovian Hypotheses in Dy­ namical Systems CP34 VVe present a statistical approach for detecting the Marko­ No Noise is Good Noise? vian character of dynamical systems by analyzing their flow Much recent activity in nonlinear signal processing theory of information via higher order cumulants. As an extension has been directed towards the discovery and exploitation of Theiler's method of surrogate data this cumulant based of structure in noise processes. This talk will describe cer­ information flow is used as the discriminating statistic in atin exotic processes which in a very real sense may be testing the observed dynamics against a hierarchy of null regarded as "good noise" =97 the sense being that we have hypotheses corresponding to nonlinear Markov processes of developed techniques to make a separartion - in principle increasing order. Numerical results on nonlinear processes, an exact separation - of signals from noises in this class. autoregressive models and noisy chaos are discussed. The class includes deterministic, chaotic processes as well Christian Schittenkopf as nonlinear stochastic processes equivalent to random dy­ Technische Universitiit Miinchen namics on a fractal set. Our interest in these is twofold: ArcisstraBe 21 we hope, and there is some evidence to believe, that these 0290 Munich, Germany noises are relevant to genuine applications in signal process­ also: Siemens AG, Corporate Research and Development ing; more generally, we would like to know "VVhat makes Otto-Hahn-Ring 6 a good noise?" 81730 Munich, Germany Mark Muldoon, J.P. Huke and D.S. Broomhead E-mail: [email protected] Mathematicss Department UMIST Gustavo Deco P.O. Box 88, Manchester M60 1QD, Great Britain Siemens AG, Corporate Research and Development E-mail: [email protected] Otto-Hahn-Ring 6 81730 Munich, Germany CP34 Is There a Deterministic Relation (Phase Locking) CP34 between Sunspot Cycles and Interdecadal Variabil­ Chaos and Detection ity of Atmospheric Temperature? I report on numerical experiments in which chaotic signals Discrete Hilbert transform was used to obtain instanta­ were reliably detected at signal to noise ratios as low as neous phases from band-pass filtered monthly atmospheric -15dB. The detector was based on a variant of the hidden temperature records from eight European stations (all over Markov models used in speech research. I review theoret­ two hundred years long) as well as from monthly sunspot ical bounds on model performance given in terms of the numbers. A possibility of phase locking between the Fourier power spectrum and the KS entropy of a chaotic sunspots and the temperatures was indicated in a few seg­ system. KS entropy estimates indicate that even better ments of the series, while phase locking between temper­ detection performance is possible. ature records from different locations were also found in Andrew M. Fraser segments not locked with the sunspots. Portland State University Milan Palus Portland, OR 97207 Institute of Computer Science, AS CR E-mail: [email protected] Pod vodarenskou vezi 2 182 07 Prague 8, Czech Republic E-mail: [email protected] CP34 Time-Series Analysis Tools for Nonlinear System Dagmar N ovotna Identification Institute of Atmospheric Physics, AS CR Prague, Czech Republic Deriving ordinary differential equation (ODE) models of nonlinear and chaotic systems from experimental time se­ Ivana Charvatova ries can be extremely difficult, particularly because classi­ Institute of Geophysics, AS CR cal time-series analysis methods are often inadequate when Prague, Czech Republic applied to these types of data sets. VVe have developed a set of nonlinear analysis tools that facilitate the analysis of Abstracts 73

CP34 Department of Mathematics Takens Embedding Theorems for Forced and 233 Widtsoe Building Stochastic Systems University of Utah Salt Lake City, Utah 84112 Takens Embedding Theorem provides the theoretical foun­ dation for most techniques in nonlinear time series analysis. Current versions assume that the time series is generated by an autonomous deterministic system. These conditions CP35 rarely hold in practical applications: most real systems are Interfacial Turbulence at Large Prandtl Number noisy, and even when the noise can be neglected, many The 3D Marangoni-Benard instability problem is studied, deterministic systems are forced, or subject to arbitrary via reduction to a 2D model equation describing the dy­ input signals. We describe extensions of the Embedding namics of the free surface temperature field. The model Theorem to cover these cases. incorporates the energy input at large length scales (pri­ Jaroslav Stark mary instability), thermal dissipation at small scales, and Centre for Nonlinear Dynamics and its Applications nonlinear energy transfer from large to small scales due to University College London the surface advection of the temperature field. Turbulent London, WCIE 6BT, United Kingdom polygonal patterns are obtained as a result of the secondary E-mail: [email protected] instability of the thermal boundary layer occurring at large supercriticality. P. Colinet and J.C. Legros CP34 Universite Libre de Bruxelles, Detecting Singularities of Non-deterministic Dy­ Service de Chimie Physique E.P., namics C.P.165, 50 Av. F.D. Roosevelt, It has been demonstrated that a non-deterministic form 1050 Brussels, Belgium of chaos is possible when Lipschitz conditions at singular e-mail: [email protected] points are relaxed. Detection of such singularities in real life data is made difficult by the inherent limitations of dig­ ital data acquisition. We present the results of several tech­ niques to overcome these limitations,including 1) orthog­ CP35 onal vectors; 2) divergent second derivatives; 3) wavelets; Nonlinear Dynamics of Electrochemical Oscilla­ and 4) recurrences. Both simulated and experimental bio­ tions, Surface Morphology and Corrosion logical data, including arm motion, and ECG's are used. We studied the nonlinear dynamics of electrochemical os­ Joseph P. Zbilut cillations of various "flavors" of bare,coated and ion im­ Dept of Molecular Biophysics & Physiology planted metallic samples.While the inner mechanisms of Rush Medical College, Chicago, IL 60612 these oscillations are still far from being fully understood, E-Mail: [email protected] the behavior of the surface and current oscillations seem to be related. We investigated the correlation between os­ David D. Dixon cillations dynamics and surface morphology. This study University of California, Riverside has several important industrial applications. This project Institute of Geophysics and Planetary Physics is supported by LEQSF and DOE Riverside, CA 92521 Elia V. Eschenazi and Ninja Ballard E-mail: [email protected] Xavier University of Louisiana New Orleans, LA, 70125 Charles L. Webber, Jr. E-mail: [email protected] Department of Physiology Loyola University Chicago Stritch School of Medicine Maywood, IL 60153 CP35 Email: [email protected] Space-Time Modeling and Pattern Formation in Rotating Flows A fluid flow enclosed in a cylindrical container where fluid CP35 motion is created by the rotation of one end wall is studied. Acoustic Turbulence Direct numerical simulations and spatio-temporal analysis have been performed over a broad range of Reynolds num­ We consider Kolmogorov-like spectra for the stochastic mo­ bers. The most striking analogy with experimental results tion of a compressible fluid without vorticity. Unlike to the are obtained by tracking particles in time. The transition turbulence of incompressible fluid, the turbulence spectrum to turbulence scenario, including breakdown ofaxisymmet­ Ek for the compressible fluid can not be obtained only from ric to three-dimensional flow behavior, is addressed. the dimensional considerations, since it can depend on one more dimensional quantity - the sound speed c, in addi­ Erik A. Christensen tion to the fluid's density p, the energy flux (;, and the wave Levich Institute, City College of CUNY, NY 10031 and number k. It is shown that the spectrum Ek is proportional Mechanical Engineering, New Jersey Institute of Technol­ to (;2/3 (as for the Kolmogorov spectrum) if the dimension ogy, NJ [email protected] of the medium is 1, to (;3/5 for 2D media, and to (;1/2 for 3D media; then the dimensional considerations determine the Nadine Aubry other scaling exponents of the corresponding turbulence Mechanical Engineering, New Jersey Institute of Technol­ spectra. The physical implications are discussed. ogy, NJ Alexander M. Balk and Levich Institute, City College of CUNY, NY 74 Abstracts

totic analysis yields an assortment of special solutions for Jens N. Sorensen and Martin O. L. Hansen the equations under consideration including shock(front or Department of Fluid Mechanics heteroclinic orbit) and solitary wave(pulse or homoclinic Technical University of Denmark, DK-2800 Lyngby, Den­ connection) solutions. The existence of homoclinic con­ mark nections and the like is verified both by the use of the stable-unstable manifold and Poincare- Bendixson Theo­ rems, as well as by numerical simulations. Additional, of­ CP35 ten more complex, solutions are obtained by the use of Chaotic Transport in a Chain of Vortices: Anoma­ truncated Painleve singularity analysis. The relevance of lous Diffusion and Levy Statistics the derived solutions to analytic and numerical studies of the model equations with various boundary conditions is Transport in a quasi-geostrophic chain of vortices in shear also discussed. flow is studied using a Hamiltonian model. Particles al­ ternate chaotically between being untrapped, and being S. Roy Choudhury trapped in the vortices. The probability distribution func­ Department of Mathematics tions of trapping/untrapping events are computed and University of Central Florida shown to exhibit power-law scaling behavior. Anomalous Orlando, FL 32816-1364 (non-Brownian) enhanced diffusion is observed and char­ E-mail: [email protected] acterized in terms of Levy (non-Gaussian) probability dis­ tributions. The results are compared with experiments on transport in a rotating annulus (see abstract by Weeks et CP36 al.), and with continuous-time random walk models. Dynamics of a Nonlocal Kuramoto-Sivashinsky Diego del-Castillo-Negrete Equation Scripps Institution of Oceanography In this paper we study the effects of a nonlocal term on University of California, San Diego the global dynamics of the Kuramoto-Sivashinsky equa­ La Jolla, CA 92093 tion. We show that the equation possesses a "family of E-mail: [email protected] maximal attractors" parameterized by the mean value of the initial data. The dimension of the attractor is esti­ mated as a function of the coefficient of the nonlocal term. CP35 Moreover, the impact of the nonlocal term on the bifurca­ Anomalous Transport in an Incompressible, Tem­ tion is discussed. porally Irregular Flow Jinqiao Duan and Vincent J. Ervin We investigate the longitudinal dispersion of passive trac­ Department of Mathematical Sciences ers in a transversely bounded, incompressible, two dimen­ Clemson University sional flow. the flow is assumed to be temporally irregular, Clemson, South Carolina 29634 but to possess smooth largescale spatial structure without E-mail: [email protected] significant small spatial scale motions. We show that, the existence of a region where the longitudinal velocity goes to zero linearly with distance from the wall causes the trac­ CP36 ers to "stick" to the walls. This results in a superdiffusive Space Analyticity on the Attractor Generated dispersion of the tracer. We analyze the growth of the vari­ by the Set of All Stationary Solutions for the ance and show that var(t) '" tV with v = 3/2. We model Kuramoto-Sivashinsky Model this flow using a 2D Hamiltonian system with no KAM sur­ The numerical results obtained by P. Collet, J.-P. Eck­ faces and verify the result for the growth of the variance mann, H. Epstein and J. Stubbe strongly suggest that by numerical simulation. the radius of space analyticity on the attractor for the Shankar C. Venkataramani Kuramoto-Sivashinsky equation with L- periodic bound­ James Franck Institute ary conditions does not depend of the bifurcation param­ 5640, S. Ellis Ave., Chicago, IL 60637 eter L. Using a lower-semicontinuity property of the E-mail: [email protected] analyticity radius we obtain a neighborhood ( on the at­ tractor ) of the set of stationary solutions in which the Thomas M. Antonsen Jr. and Edward Ott conjectured phenomenon is true. Also, the description of Institute for Plasma Research the neighborhood does not explicitely involve L. University of Maryland, College Park, MD 20742 Zoran Grujic Indiana University Department of Mathematics CP36 Bloomington, IN 47405 Analytic Aspects and Special Solutions of Nonlin­ E-mail: [email protected] ear Partial Differential Equations Special analytic solutions are considered for several phys­ ically important nonlinear partial differential equations. CP36 The models treated include a family of Cahn-Hilliard Analytical Properties of PDE Jet Engine Models equations, the phi-4 equation of the family of Klein­ We present a justification for including a viscous term in Gordon equations, the long-wave equations, the Zakharov­ the enhanced compressor model by homogenization. We Kuznetsov equation, as well as a family of reaction­ prove the existence and uniqueness of solutions of that diffusion equations. A combination of techniques includ­ model. Furthermore, we present some linear and nonlinear ing Lie group and direct similarity methods, traveling wave stability results and show that the stability depends on a reductions, phase plane techniques, and nonlinear asymp- certain Lyapunov multiplier. Finally we present a back- Abstracts 75 stepping control scheme for the full p.d.e. model. To our CP37 knowledge, this is the first time backstepping is used on a On Time Optimal Control Trajectories of Biotech­ p.d.e. nological Processes Hoskuldur Ari Hauksson We investigate the time optimal control problem for a class Department of Mathematics of biotechnological processes described by single input non­ University of California Santa Barbara linear control affine systems in dimension three. Applying Santa Barbara, CA 93106 some recent results obtained by Sussmann, Kupka, Bon­ E-mail: [email protected] nard, and Montgomery around the Pontryagin's Maximum Principle and geometric properties of extremals such as the Andrzej Banaszuk existence of abnormal extremals and the classification of University of California Davis singular extremals, we analyze time optimal trajectories Davis, CA 95616 and propose a classification of extremals for the considered particular systems. Bjorn Birnir and Igor Mezic Djamal Atroune University of California Santa Barbara Department of Biological and Agricultural Engineering Santa Barbara, CA 93106 Driftmier Engineering Center The University of Georgia, Athens, GA 30602-4435 E-mail: [email protected] CP36 Higher-Order Bistable Systems in One Space Di­ mension CP37 We present a variational approach to the problem of an­ Stabilization of Uncertain Mechanical Systems Us­ alyzing multitransition solutions of a class of fourth-order ing Dynamic Feedback evolutionary PDE's. One example is the extended Fisher­ Dynamic feedback refers to the specification of control Kolomogorov equation arising in the study of convection forces by dynamical update laws rather than static feed­ in liquid crystals. We show the existence of stable (lo­ back which specifies the force in terms of the instantaneous cally minimizing) solutions with complicated spatial pat­ values of the state variables. This procedure requires an terns. This behavior is due to a loop of Shil'nikov orbits in increase in the dimension of the phase space for the con­ the four-dimensional stationary ODE. Unlike other anal­ trolled system. It will be shown that such methods allow ysis of this situation, no conditions on these orbits, such stabilization of mechanical systems to an arbitrary config­ as transversality, is required. These techniques can be ex­ uration even if the inertial and potential properties of the tended to establish the exponentially slow motion of tran­ system are unknown. sition layers generally in higher-order evolution equations. Ernest Barany and Richard Colbaugh William D. Kalies New Mexico State University Center for Dynamical Systems and Nonlinear Studies, Las Cruces, NM 88003 Georgia Institute of Technology, Atlanta, GA 30332 E-mail: [email protected] E-mail: [email protected]

J. Kwapisz, R.C.A.M VanderVorst, and T. Wanner, Georgia Institute of Technology, Atlanta, GA 30332 CP37 Stabilization of Uncertain Mechanical Systems Us­ ing Bounded Controls CP36 This paper considers the problem of stabilizing uncertain Persistence of Invariant Manifolds for Nonlinear mechanical systems in the presence of constraints on the PDEs available actuator inputs, and proposes both state feedback and output feedback controllers as solutions to this prob­ We prove that under certain stability and smoothing prop­ lem. Each stabilization scheme consists of a nonadaptive erties of the semi-groups generated by certain PDEs, man­ 1 component for gross motion control and an adaptive com­ ifolds left invariant by these flows persist under C per­ ponent to ensure convergence to the desired configuration. turbation. In particular, we extend well known finite­ The controllers are computationally simple, require virtu­ dimensional results to the setting of an infinite-dimensional ally no information regarding the system model, and ensure Hilbert manifold with a semi-group that leaves a submani­ that the configuration error is globally convergent. The ef­ fold invariant. We apply our results to the two-dimensional ficacy of the proposed strategies are illustrated though both Navier-Stokes equations and a perturbation consisting of a computer simulations and hardware experiments. fully discrete approximation of the Navier-Stokes equation. Richard Colbaugh Don A. Jones New Mexico State University Department of Mathematics Department of Mechanical Engineering Arizona State University Box 30001, Dept. 3450 Tempe, AZ 85287, USA E-mail: Las Cruces, NM 88003 [email protected] Ernest Barany Steve Shkoller New Mexico State University Institute of Geophysics and Planetary Physics 0225 Department of Mathematical Sciences Scripps Institution of Oceanography Box 30001, Dept. 3MB University of California, San Diego Las Cruces, NM 88003 La Jolla, CA 92093-0225 Kristin Glass 76 Abstracts

New Mexico State University pendulum that is excited by a six-axis (three translational, Department of Mechanical Engineering three rotational) ship motion simulator. Due to the very Box 30001, Dept. 3450 light damping present (as in real cranes), the dynamics is Las Cruces, NM 88003 primarily transient-dominated, and we empircally inves­ tigate transient-related effects such as low-frequency re­ sponse modulation and transient length. CP37 Michael D. Todd and Sandeep T. Vohra Application of Symbolic Dynamics to Modeling and Naval Research Laboratory Control of an Internal Combustion Engine Optical Sciences Division In this presentation we illustrate the application of sym­ Washington, DC 20375 bolic dynamics to the problem of cyclic combustion varia­ tions in internal combustion engines. Such variations are responsible for loss in fuel efficiency and increased pollutant Frank A. Leban emissions. Using a discrete model of the combustion dy­ Naval Surface Warfare Center namics, we describe our algorithm for adjusting the model Carderock Division parameters to "match" the observed symbolic dynamics. Bethesda, MD 20084 We also find that on-line feedback engine control might be accomplished with highly discretized measurements of cycle-resolved combustion. CP37 C. Stuart Daw, Matthew B. Kennel On a New Route to Chaos in Railway Dynamic Engineering Technology Division A detailed investigation of a speed range with complicated Oak Ridge National Laboratory dynamics of the Cooperrider truck has revealed new bifur­ P.O. Box 2009 cations and the existence of two asymmetric quasiperiodic Oak Ridge TN 37831-8087 solutions. They bifurcate supercritically as unstable solu­ E-mail: [email protected], [email protected] tions and stabilize in a saddle-node bifurcation. A FFT plot shows that one of the two periods has only very little Charles E.A. Finney power. When the speed is lowered, that period undergoes Department of Mechanical Engineering a period doubling sequence, which develops ~ into chaos. University of Tennessee Hans True Knoxville TN 37996-2210 The Technical University of Denmark E-mail: [email protected] Carsten Nordstroem Jensen ES-Consult Vedbaek, Denmark CP37 Nonlinear Regenerative Machine Tool Vibrations Three topics are addressed. First, a regenerative machine MSOI tool vibration model is proposed in the form Spatio-Temporal Chaos in Rayleigh-Benard Con­ vection 2 x"(s) 2K::X'(S) 00 x(s) We present experimental results on spatio-temporal chaos + + 2 = v v in the convection pattern of a fluid heated from below. We show that the stationary states of perfect ordered rolls are --;.k Ih w(a) (x(s - a - VT) - x(s - a)) da + h.o.t .. stable to finite perturbation in the regime where spiral de­ mv 0 fect chaos (SDC) is observed. We will discuss the impor­ Stability charts are presented for different weight functions tance of the observed bistability for the spatio-temporal w which describe different stress distributions on the ac­ evolution of SDC. tive face of length h of the tool. Second, experimental E. Bodenschatz results will be presented for thread cutting which initiate Cornell. University. the third topic, the question of relevant non-linear effects LASSP, 618 Clark Hall, Ithaca, N.Y., 14853 in the model (covered by h.o.t. above), bifurcation and singular perturbation analysis, and the possible existence of chaotic and/or transient chaotic behavior. MSOI Gabor Stepan A Primary Bifurcation to Spatio-temporal Chaos Department of Applied Mechanics Technical University of Budapest We present an experimental study of electroconvection H-1521 Budapest, Hungary (EC) using the nematic liquid crystaI4-ethyl-2-fiuoro-4'-[2- E-mail: [email protected] (trans-4-pentylcyclohexyl)-ethylJ biphenyl. The system un­ dergoes a forward hopf bifurcation from a spatially uniform state directly to a pattern that exhibits spatio-temporal chaos. In principle, the dynamics can be quantitatively CP37 described by coupled complex Ginzburg-Landau equations Dynamical Response of a Spherical Pendulum to that are derived from the fundamental equations of motion Roll and Pitch Excitation for EC. Such a quantitative comparison will provide useful The operability of cranes or crane-like structures placed on insights into the nature of spatio-temporal chaos. ships or other floating platforms is determined by the dy­ Michael Dennin namical response of the crane to the prevailing sea state. Department of Physics and Astronomy As a rudimentary model of a crane structure, we experi­ University of California, Irvine mentally investigate the dynamical response of a spherical Abstracts 77

Irvine, CA 92697-4575 Brown University [email protected] Providence, RI

MSOI MS02 Competition Between Spiral Waves and' Targets in Mathematical Problems in the Theory of Shallow Dictyostelium Discoideum Water This talk describes two studies of the spatio-temporal dy­ We consider two models of shallow water, the so-called lake namics of chemical waves in populations of Dictyostelium and the great lake equations, which were proposed by Ca­ discoideum. We study the competition between pacemak­ massa, Holm and Levermore, and which describe the long­ ers and spirals, showing first that targets dominate at low time motion of an inviscid, incompressible fluid contained population density, but spirals entrain autonomous pace­ in a shallow basin with a slowly spatially varying bottom, makers at high density. Second, we show that chemical a free upper surface and vertical side walls, under the in­ waves can be reset by extrinsic cyclic AMP. They reap­ fluence of gravity and in the limit of small characteristic pear if resetting is early in the signaling stage, but targets velocities and very small surface amplitude. dominate following late resetting. This behavior correlates From a mathematical point of view, the following issues with the wave signal-to-noise ratio. Speculations on these arise: observations are presented. - Are there weak/classical/analytic solutions? - Are the solutions unique? Raymond E. Goldstein - How does bottom topography affect the solutions? Department of Physics - Are the solutions to the shallow water equations in some University of Arizona, Tucson, AZ 85721 sense" close" to solutions of the full three dimensional fluid equation? In other words, can the asymptotic expansion be mathematically justified? MSOI Transitions from Spiral Waves to Chemical Turbu­ The talk will briefly motivate these questions and survey lence in a Reaction-Diffusion System the current state of the subject. We present our experimental results on transitions from Marcel Oliver regular spiral chemical waves to defect-mediated turbu­ Department of Mathematics, lence in a reaction-diffusion system with the Belousov­ University of California at Irvine Zhabotinsky reaction. Three routes leading to the destruc­ tion of the spiral pattern were observed in different control parameter regimes: via Eckhaus instability or convective instability; via spiral meandering instability; and via in­ MS02 Inverse Scattering Analysis of Internal Waves in stability induced by external forcing. The first two insta­ bilities produce point-defects, while third instability gives the Andaman Sea line-source type defects. Internal wave data taken in the Andaman Sea in 1976 are Qi Ouyang revisitedin light of recent advances for the analysis of data using the inverse scattering transform. As is well known NEC Research Institute, Inc. 4 Independence Way, Princeton, NJ 08540 [Osborne and Burch, 1980) these data exhibit solitons with amplitudes up to 110 m and particle velocities up to 2.5 m/s. In the present work I use the periodic inverse scatter­ ing transform in the theta function representation to ana­ MS02 lyze the data. I use both the oscillatory basis (for which the Advection Induced by the Breaking of Vorticity nonlinear modes are cnoidal waves) and the soliton basis. Conservation via Viscous Dissipation. Explicit computation of the nonlinear interactions among Fluid transport afforded by the breaking of vorticity con­ the components is also made. A number of theorems about servation via viscous dissipation is examined. A vorticity theta functions are invoked to isolate the solitons from the conserving base flow is assumed to possess a homo clinic or radiation and to nonlinearly filter the data in a number of heteroclinic orbit. The splitting of this separatrix under novel new ways. I also search for triad interactions among a perturbation permits advection between regimes of dif­ the components of the inverse scattering transform. The ferent characteristic motion. A vorticity dissipating flow results suggest that the periodic inverse scattering trans­ with viscous parameter E and small time dependent forc­ form will serve a new and fundamental role in the under­ ing is shown to maintain an O(E) closeness to the vorticity standing of the nonlinear dynamics of the internal wave conserving flow in a weak sense. A Melnikov theory is de­ field in a wide variety of locations around the world. veloped for such a weak perturbation, and a surprisingly A. R. Osborne simple expression derived for the leading order distance University of Torino, Turin, Italy between the perturbed manifolds, which depends only on the inviscid flow. Heteroclinic orbits generically split apart with no intersection between them, while homoclinic orbits may have only higher order intersections. Viscous dissipa­ MS02 tion therefore yields a retrograde nmechanism for advec­ Dynamical Systems Theory and Dynamical tion, while suppressing chaotic transport severely. Oceanography Sanjeeva Balasuriya It is well known that the discovery by Lorenz of non­ periodic deterministic motion in dissipative dynamical sys­ Department of Mathematics, University of Peradeniya, Sri tems occurred within the discipline of dynamical meteorol­ Lanka ogy, the study of the large-scale dynamics and thermo­ Christopher K. R. T. Jones and Bjoern Sandstede dynamics of the atmosphere. Despite many similarities 78 Abstracts

between the underlying dynamics of oceanic and atmo­ ment will be discussed. spheric motions, which have resulted in the development James J. Collins of the general field of 'geophysical fluid dynamics,' the Department of Biomedical Engineering, Boston University, phenomenon of chaos in low-order systems has received 44 Cummington Street, Boston, MA 02215 relatively less attention in dynamical oceanography, the study of the large-scale dynamics and thermodynamics of the ocean. Some early oceanographic applications of these ideas derived from simple models of forced flow over to­ MS03 pography, in which asymptotic analysis led directly to low­ Stochastic Resonance in Human Muscle Spindles-a order systems that exhibited chaotic behavior. Recently, potential mechanism for fusimotor gain control efforts have been made to apply the techniques of dynami­ Stochastic resonance is a phenomenon in both physical and cal systems theory to the problem of large-scale fluid parcel biological systems where a particular level of random noise dispersion in the ocean. This approach appears most suited improves signal detection. In systems where detection is for the study of coherent features and strongly inhomoge­ near threshold, a random noise input can improve sensi­ neous flows, and in these cases has yielded new insights tivity to small amplitude signals by boosting the receptors into the kinematics of dispersion. Many questions concern­ above threshold. One biological system that could poten­ ing the relation of this picture to the dynamics of critical tially benefit from stochastic resonance is the propriocep­ layers, instabilities, and potential vorticity homogenization tive system, which provides us with the sense of body po­ remain unanswered. sition and movement. Stochastic resonance could enhance Roger M. Samelson proprioception when movements are novel or must be made Woods Hole Oceanographic Institution, extremely precisely. We demonstrate stochastic resonance Woods Hole, MA in human muscle spindle receptors-an important source of proprioceptive information for movement coordination. We hypothesize that stochastic resonance is the mecha­ nism with which the fusimotor system increases the gain MS03 of muscle spindles during novel and precise movements. Augmentation of Sensory Nerve Action Potentials during Muscle Contraction Paul J. Cordo R.S. Dow Neurological Sciences Institute, Portland, OR We previously reported that the median sensory nerve ac­ tion potentials (SNAP's) increased in amplitude during ip­ Sabine M. P. Verschueren silateral abductor pollicis brevis (APB) contraction. The Catholic University of Leuven objectives of the present project were to study the timing Heverlee, Belgium and origin ofthis phenomenon and eliminate the possibility J. Timothy Inglis of local artifact. Ten normal subjects were recruited. The University of British Columbia baseline was established using 10 threshold stimuli which Vancouver, B.C. Canada were delivered to the median nerve at the wrist at 0.2 Hz. Frank Moss Using the same stimulus strength, the SNAP was recorded University of Missouri at St. Louis while the tibialis anterior (TA) was contracted at 25signal St. Louis, MO averaged. Results showed an increase in ipsilateral SNAP Daniel M. Merfeld amplitude between baseline to maximal contraction of 6 R.S. Dow Neurological Sciences Institute +/- 2 uV (SE, p= .004) and contralateral amplitude of 8 Portland, OR +/- 2 uV (SE, p = .01). Statistical analysis was performed James J. Collins with ANOVA for repeated measures and paired t-test. The Boston University effect peaked between 0 to 10 minutes after contraction and Boston, MA lasted from 1.5 to over 20 minutes after muscle relaxation. In conclusion, SNAP's appear to be enhanced during and after muscle contraction. Theories concerning underlying causes for this event including central augmentation and MS03 muscle tension noise enhanced signal transmission are dis­ Noise, Hair Cells, & the Leopard Frog cussed. Auditory synapses are capable of transmitting timing and Faye Y. Chiou-Tan intensity information with unparalleled fidelity. Yet the Kevin N. Magee, Stephen Tuel, Lawrence Robinson, performance of certain auditory functions is best for near­ Thomas Krouskop, Maureen R. Nelson, and Frank Moss threshold stimuli, just where external noise would seem to Department of Physical Medicine and Rehabilitation be most destructive. Curiously, two important steps in the College of Medicine, Huston TX process of sensory transduction in hair cells - the mecha­ no electrical transduction resulting from deflection of the hair bundle and the gating of voltage-dependent calcium ion channels - are described by dynamical models known MS03 to exhibit stochastic resonance. Noise-Enhanced Sensory Function Kurt Wiesenfeld, Noise can enhance the detection and transmission of weak Peter Jung, signals in certain nonlinear· systems, via a mechanism School of Physics, Georgia Institute of Technology, At­ known as stochastic resonance (SR). Here we describe stud­ lanta, GA 30332 ies in which SR-type behavior is demonstrated in: (i) model neurons, (ii) rat cutaneous sensory neurons, and (iii) the Fernan Jaramillo human touch-sensation system. This work suggests that Department of Physiology, Emory University, Atlanta, GA it may be possible to develop a noise-based technique for 30322 lowering sensory detection thresholds in humans. The bio­ engineering and clinical ramifications of such a develop- Abstracts 79

MS05 Rayleigh-Benard convection using a systematic reduction Encoding Information in Chemical Chaos of the equations for the motion of the local spatial structure known as "phase dynamics". The importance of invasive We describe encoding and decoding symbol sequences defects in the chaotic dynamics is proposed. Numerical containing information into chaotic oscillations of the simulations of the spiral state and the role of the vertical Belousov-Zhabotinsky reaction. Encoding is based on con­ fluid vorticity in the dynamics will also be discussed. trolling chaotic oscillations by applying small parameter perturbations and on learning corresponding symbol dy­ M. C. Cross namics produced by the free-running chaotic system. Use California Institute of Technology, Pasadena, CA of small parameter perturbations requires that we respect the grammar of the symbolic dynamics. We present a gen­ eral method for learning the grammar in terms of tran­ MS06 sitions between bins defined by the generating partition. Using Finite-Time Lyapunov Dimensions to Mea­ sure the Dynamical Complexity of Topological De­ Erik M. Bollt fects United States Military Academi, West Point, NY We have defined a finite-time Lyapunov dimension DT based on the evolution of Lyapunov vectors over short time intervals of length T. For the complex Ginzburg-Landau MS05 equation in two spatial dimensions, DT is, on average, lin­ Experimental Control of Chaos for Communication early related to the mean number of defects in the system during the same time interval. The average dimension per The experiments done to demonstrate the use of chaos to defect is about 2 over a wide range of parameter values. In transmit information will be described. The symbolic dy­ an effort to understand whether this result extends to other namics of a chaotic electrical oscillator is controlled to carry defect-dominated chaotic systems, we extend this work to a prescribed message by use of extremely small perturbat­ other states such as spiral defect chaos in a generalized ing current pulses. A movie will be shown demonstrating Swift-Hohenberg equation the whole experimental encoding sequence in real time. D. Egolf Scott Hayes Cornell Theory Center, Cornell, NY Army Research Laboratory, Adelphi, MD

MS06 MS05 Phase Diffusion in Localized Spatio-Temporal Am­ Higher Dimensional Symbolic Encoding plitude Chaos of Parametrically Excited Waves The use of higher dimensional dynamical systems in com­ In simulations of coupled Ginzburg-Landau equations for munication with chaos brings a few potentially technologi­ parametrically excited waves we find fluctuating defects in cally relevant advantages. In particular, it allows for more the form of double phase slips, which eliminate and rein­ practical high-speed symbolic control techniques that could sert a wavelength in the periodic pattern. In large sys­ be used at higher bit rates than an implementation in a tems they lead to spatio-temporal amplitude chaos. De­ low dimensional dynamical system. In this talk a symbolic spite the break-down of the usual phase equation during encoding technique for a chaotic signal to be used in com­ each phase slip the large-scale behavior can be described munication will be discussed. by an effective phase equation. We determine the phase­ Ying-Chen Lai diffusion coefficient explicitly. Its wavenumber dependence University of Kansas, Lawrence, KS leads to stable states consisting of alternating domains of regular and spatio-temporally chaotic dynamics. For this system the phase equation plays a role similar to that MS05 of the KPZ-equation for the Kuramoto-Sivashinsky equa­ Noise Filtering in Communication with Chaos tion. Formally, it can be considered to arise from a spatio­ temporal homogenization of the chaotic state. I will present a method based on fundamental properties of chaotic dynamics for filtering in-band noise of an incoming Glen D. Granzow signal generated by a chaotic oscillator. Initially the 2x Northwestern University, Evanston, IL mod 1 map is used to illustrate the procedure and then the method is applied to recover the message encoded in Hermann Riecke a realistic chaotic signal, after the transmitted signal has Northwestern University, Evanston, IL been contaminated with noise. Epaminondas Rosa, Jr. University of Maryland, College Park, MD MS06 Suppressing Spatiotemporal Chaos Using Time­ Delay Feedback MS06 Predicting Spiral Chaos in Rayleigh-Benard Con­ We introduce a spatially local feedback mechanism for vection: a Phase Dynamics Approach to the Onset stabilizing periodic orbits embedded in spatially extended and Properties of Spatio-Temporal Chaos chaotic attractors. Our method, which is based on a com­ parison between present and past states of the system, does The theoretical prediction of the onset of spatiotemporal not require external generation of a reference state and can chaos and the properties of the resulting dynamic state suppress both absolute and convective instabilities. We from first principles is a daunting task. I describe a first also show that the inclusion of a spatial filter in the feed­ attempt to do this for the spiral spatiotemporal chaos in back loop results in the stabilization of traveling waves in a 80 Abstracts

model of a transversely extended semiconductor laser. The adjustment emission, of inertia-gravity waves by unsteady scheme we study can be implemented in a straightforward vortical motions. The clarification shows (a) which fea­ manner to obtain real-time control of optical systems. Our tures of hamiltonian balanced models, like Hoskins' semi­ results are derived both from exact linear stability analysis geostrophic theory, are special and which are general, and and from numerical simulation of model partial differential how such models can be generalized to arbitrary accuracy equations. without losing hamiltonian structure, (b) how such gener­ Michael E. Bleich alizations can be constructed, at least formally, by inserting Duke University, Durham, NC into the hamiltonian framework any of the highly accurate balance conditions used in recent studies of accurate PV in­ Joshua E. S. Socolar, version, and (c) how, in a certain class of models, canonical Duke University, Durham, NC coordinates, analogous to Hoskins' geostrophic-momentum coordinates, can be found. A quaternionic structure has David Hochheiser been identified within the class of such models. University of Arizona, Tucson, AZ Ian Roulstone UK Meteorological Office, Jerome V. Moloney Berkshire, United Kingdom University of Arizona, Tucson, AZ

MS07 MS07 Statistical-Dynamical Methods in Atmospheric A Leading-Order Singular Hamiltonian Perturba­ Prediction tion Method in Fluid Dynamics Atmospheric prediction has historically been an applied We consider the question of constructing a singular Hamil­ science which has provided early examples of mathemat­ tonian perturbation method for systems with generalized ically interesting phenomena such as low order chaotic Poisson brackets. In partial answer to this question we dynamics and sensitive dependence on initial conditions. have found a leading-order singular Hamiltonian perturba­ Over the past several years, operational forecasting cen­ tion method by combining asymptotic perturbation expan­ ters have initiated research in two major areas which, by sions, based on a slave relation between the fast variables the nature of the posed forecast problems, require the de­ and the slow master variables, with preservation of the velopment of methodologies for dealing with the fact that Hamiltonian formulation. In our method the distinction the atmosphere is a system with a very large number of de­ between fast and slow variables stems from the assumed grees of freedom and which exhibits sensitive dependence time scale separation of the linear (wave) problem. These on both initial conditions and forcing. These two areas leading-order results have been exemplified by a systematic are four-dimensional data assimilation, for the purpose of derivation of the Hamiltonian formulation for the incom­ initial state estimation, and coarse grained probabilistic pressible, homogeneous fluid equations via a Mach-number predictions at the limits of deterministic predictability, for expansion and for the barotropic quasi-geostrophic fluid extended range outlooks. Both problems will be reviewed equations via a Rossby-number expansion. Although these and the current state of the art techniques will be dis­ and many other leading-order Hamiltonian formulations of cussed. The limitations of the currently used methods, various dynamical systems, in particular approximate fluid which are derived from linear theory, will be examined and equations, have been recorded previously, our method pro­ the necessity of nonlinear extension will be suggested. The vides a unified Hamiltonian view on a hierarchy of approx­ position taken will be that in order to reach a level of util­ imations made in (geophysical) fluid dynamics. The for­ ity commensurate with computational expense probabilis­ mulation of a tractable higher-order singular Hamiltonian tic predictions in the short range (for data assimilation perturbation method is still unresolved. However, we have purposes) or the extended range (for climatic anomalypre­ illuminated the link between Dirac's theory of constrained dictions) must be capable of giving probabilistic informa­ Hamiltonian systems and formal manipulation of the slave tion for the situation where a probability density forecast relation within a cosymplectic formulation of the equations becomes multi-modal. A prototypical, simplest example of motion. of such a situation is the planetary-wave regime transition Onno Bokhove which will be examined in detail. Woods Hole Oceanographic Institution Joe Tribbia Woods Hole, MA NCAR, Boulder, CO

MS07 MS07 On Hamiltonian Balanced Models of Atmosphere­ Averaging and Reduced Dynamics in Simple At­ Ocean Eddy Dynamics mospheric Models The general mathematical structure of hamiltonian bal­ We use hamiltonian averaging theory (equivalent to trans­ anced models of vortical atmosphere-ocean dynamics is formation to normal forms) to study reduced, or balance, clarified, at arbitrary accuracy, along with its relation to dynamics in a series of simple hamiltonian models of at­ the concepts of slow manifold and potential-vorticity (PV) mospheric circulations. The existence of higher-order adi­ inversion. Accuracy means closeness to an exact dynam­ abatic invariants in some of these models can be used to ics, meaning a primitive or Euler-equation hamiltonian dy­ construct reduced models which are exponentially accu­ namics, regarded. as the exact 'parent' of the balanced rate in the sense that the unbalanced part of the dynamics model. Arbitrary means limited not by any particular grows in exponentially long timescales. expansion method or approximate formula, or fast-slow Djoko Wirosoetisno scaling assumption, but only by the irreducible, residual University of Toronto, Toronto, Canada inaccuracy or imbalance associated with the spontaneous Abstracts 81

Theodore G. Shepherd Daniel T. Kaplan University of Toronto Department of Mathematics and Computer Science Toronto, Canada Macalester College, St. Paul, Minnesota

MS08 MS09 Markov Chain Monte Carlo Methods in Nonlinear On The Jump Phenomenon in Coupled Oscillators Signal Processing For the study of the jump phenomenon in a system with Dealing with noisy data leads to problems of estimating two degrees of freedom two oscillators, coupled by polyno­ both the system dynamics and the true states in the recon­ mials of the seventh degree are considered. For this system strcuted space. We demonstrate that Markov Chain Monte a normal form is presented in case of non resonance. This Carlo methods provide a powerful approach to solving such normal form is a quartic Kolmogorov system in the plane. problems, particularly when the noise levels are high. Some results of the analysis of this system are presented. M.E. Davies Also an interpretation of these results for a special appli­ Centre for Nonlinear Dynamics and its Applications, cation is given. University College London, Gower St. Adriaan H.P. van der Burgh London WClE 6BT, UK Faculty of Technical Mathematics and Informatics Delft University of Technology PO Box: 5031,2600 GA Delft The Netherlands MS08 Modeling Noisy Chaotic Data Mark Huiskes A generalization of the minimization of the one-step~ prediction error is presented, which allows one to recon­ Faculty of Technical Mathematics and Informatics struct the dynamics underlying chaotic data contaminated Delft University of Technology by large amplitude measurement noise, in an unbiased way. PO Box: 5031,2600 GA Delft Moreover, this new cost function offers a new approach to The Netherlands modeling in the case of noise interacting with the deter­ ministic dynamics. Holger Kantz and Lars Jaeger MS09 Max Planck Institute for Physics of Complex Systems, New Developments in Forced and Coupled Relax­ Bayreuther Str. 40, D 01187 Dresden, Germany ation Oscillations Amersed-Boundary Method Relaxation oscillations form an important class of dynam­ ical systems in electric circuits, biology and other fields. MS08 Forced van der Pol relaxation oscillator provided one of Embedding in the Presence of Dynamical Noise the first examples of deterministic chaos, through the work We present a simple new result in the spirit of Takens's em­ of Cartwright, Littlewood and Levinson. The physical sig­ bedding theorem, but requiring multivariate signals. Our nificance of such chaos is limited by the fact that it unob­ result extends the domain of geometric time series anal­ servable directly: the measure of the corresponding hyper­ ysis to some genuinely stochastic systems, including such bolic invariant set is zero. We will present a class of closely natural examples as related relaxation oscillators which do exibit a physically observable chaos, with the positive measure of chaotic or­ bits. Mark Levi where is some fixed map and the 'f/j are small, LLd. ran­ 4> Rensselaer Polytechnic Institute, Troy, NY 12180 dom displacements. M. Muldoon, J.P. Huke & D.S. Broomhead Maths. Department, UMIST, Manchester M60 1QD, UK MS09 Transition Curves in the Quasiperiodic Mathieu MS08 Equation Processing of Noisy Nonlinear Signals: The Fetal In this work we investigate the following quasiperiodic ECG Mathieu equation: Electrocardiographic (ECG) recordings are characterized by a combination of nonlinear structure on time scales x + (8 + I: cost+ I: coswt) x=O shorter than one cardiac cycle and a stochastic component in the triggering of new cycles. We show that nevertheless, We use numerical integration and Lyapunov exponents to phase space embedding yields a useful parametrization and determine transition curves bounding regions of stability in forms an appropriate framework for the processing of such the 8 - w plane for fixed 1:. In addition, we obtain approxi­ signals. As an example we apply phase space projections mate analytic expressions for these transition curves using to extract the fetal component from noninvasive maternal two distinct methods: regular perturbations and harmonic ECG recordings. balance. Comparison of the results of these methods with Thomas Schreiber and Marcus Richter those of numerical integration shows that the perturbation Physics Department, University of Wuppertal, Wuppertal, method fails to converge in the neighborhood of resonant Germany values of w. The results obtained by harmonic balance do 82 Abstracts not display this undesirable feature. The bridge between the clinical and theoretical aspects will be emphasized. Randolf S.Zounes Cornell University, Ithaca, NY Leon Glass Department of Physiology Richard H.Rand McGill University, Montreal, Canada Cornell University, Ithaca, NY

MS09 MSIO Symmetric and Asymmetric Dynamics of Linearly Spectral Properties of the Tubuloglomerular Feed­ Coupled van der Pol Oscillators back System We present the results of analytical and numerical studies The tubuloglomerular feedback system, which regulates of the dynamics of a pair of van der Pol oscillators with blood plasma processing by the kidney, may exhibit os­ linear diffusive coupling. A critical variational equation, cillations in key variables. Physiologists have used spectral whose stability corresponds to that of the symmetric (in­ analysis to interpret these oscillations. We have formulated phase and out-of-phase) modes, is derived and transformed a mathematical model for this system and investigated its to a Hill's equation to identify a sequence of resonant insta­ spectral properties. The analysis appears to explain salient bilities. Perturbation methods are used to determine the features of experimental power spectra, and the analysis extent of the resonances and a ruled surface, corresponding demonstrates that great care is required in the calculation to zero mean damping, which completes the stability tran­ of spectra from models and in the interpretation of exper­ sition set. Pade approximants arc used to obtain extended­ iments. range approximations of the intersection curves between H. E. Layton the resonances and the mean damping surface. Numerical Duke University, Durham, NC simulations are presented to demonstrate the existence of asymmetric and non-periodic trajectories, and to illustrate E. B. Pitman the effect of detuning. State University of New York, Buffalo, NY Duane W. Storti University of Washington, Seattle, WA L. C. Moore State University of New York, Stony Brook, NY Per G. Reinhall University of Washington Seattle, WA MSIO Modeling the Pupil Light Reflex with Delay Differ­ David M. Sliger ential Equations University of Washington Seattle, WA The pupil light reflex (PLR) is the paradigm of a human neural feedback control mechanism. This reflex is time delayed and the open-loop transfer function is 3rd-order. Considerations of the iris muscle plant, its neural input MSIO and the feedback lead to a model for the PLR expressed Use of Mathematical Methods for Protocol Design in terms of a 3rd-order delay differential equation. The in Cancer properties of this equation are discussed with respect to Analysis of populations undergoing periodic loss process, two important pupil phenomena: the pupil size effect and which is effective only during part of the life-cycle, pin­ "fatigue" waves. points the phenomenon of resonance in population survival. John Milton Resonance arises when the period of the imposed loss pro­ The University of Chicago cess coincides with the inherent reproductive periodicity Chicago, USA of the population. Such a coincidence results in a pref­ erential enhancement of population growth. This general Jacques Belair phenomenon has been verified experimentally and has been Universite de Montreal employed for devising a method, (the Z-Methocf) for mini­ Montreal, Canada mizing the cytotoxic side-effects in cancer chemotherapy. Zvia Agur Tel-Aviv University, Israel MSll Scaling Relations in Chemical Spirals: Selection and Dispersion MSIO We present an experimental survey of spiral waves in the Clinical and Mathematical Aspects of Resetting Belousov-Zhabotinsky reaction using an open spatial re­ and Entraining Reentrant Tachycardia actor. The selected pitch ps and period Ts are measured Reentrant tachycardia is a serious medical problem in for various chemical concentrations; they follow the rela­ which cardiac activity travels in a circuitous pathway. This tion Ps '" T;/2. Taking advantage of the system's sen­ pro blem is sometimes diagnosed and treated by the direct sitivity to light, we impose a longer period on the spiral electrical stimulation of the heart by catheters inserted and thereby determine the dispersion relation at different directly into the heart. The procedure involves applying concentrations, which collapse onto a single dimensionless stimuli in various protocols and sequences. To understand curve when appropriately rescaled. We derive a functional these procedures, it is necessary to consider the resetting form for the curve from a model of the underlying chem- of limit cycle oscillations in partial differential equations. Abstracts 83

istry. http://cochise.biosci.arizona.edu/ art 1 1 Andrew Belmonte , 1, Jean-Marc Flesselles , 2, Vilmos Arthur T. Winfree 2 1 Gaspar , and Qi Ouyang . 3 1 Institut Non-Lineaire de 326 BSW, University of Arizona, Tucson, AZ Nice, Valbonne, France; 2 Kossuth Lajos University, De­ brecen, Hungary MS12 A Unified Approach to Interacting Particle Sys­ MSll tems and Stochastic PDE Using Measure-Valued The Dynamics of Scroll Wave Filaments in the Processes Complex Ginzburg-Landau Equation We present an introduction to the application of measure­ We present an analytical treatment for scroll waves evolv­ valued processes to model interacting particle systems and ing under the Complex Ginzburg-Landau Equation in the lattice systems which arise in the physical and biological limit of small filament curvature and phase twist. Expres­ sciences, their approximation by stochastic partial differ­ sions for the collapse rate, drift rate, and frequency shift ential equations of parabolic type and some examples of are found. Our treatment shows the existence of wavenum­ spatial clumping phenomena which can be understood by ber shifts in the plane of the spirals; a feature not included using these methods. in previous theories of scroll waves. We verify our results Donald Dawson numerically for circular untwisted rings and for twisted and The Fields Institute, Toronto, Canada untwisted sinusoidal filaments. We also display numerical evidence for reconnection of adjacent filaments and of the instability of straight filaments with strong twist. MS12 Michael Gabbay Effect of Noise on Bifurcations in Spatially Ex­ University of Maryland tended Systems College Park, MD The effect of space-time noise in model partial differential Edward Ott equations with a symmetry-breaking transition is studied University of Maryland numerically and analytically. Additive noise is important College Park, MD when the critical parameter is a function of time; the de­ lay of the transition is a function of the log of the noise Parvez Guzdar magnitude. Multiplicative noise makes the statistics non­ University of Maryland Gaussian; its effect is proportional to its magnitude. College Park, MD keywords: stochastic PDEs, multiplicative noise, critical phenomena, symmetry-breaking, pattern formation. Grant Lythe MSll Center for Nonlinear Studies Control of Spiral Territories in Dictyostelium MS-B258 Los Alamos National Laboratory The aggregation stage of the life cycle of Dictyostelium Los Alamos, New Mexico discoideum is governed by the chemotactic response of NM 87545 individual amoebae to excitable waves of cyclic AMP USA (cAMP). Quite often, these waves take the form of large spirals and lead to the formation of aggregation territories with roughly 105 cells each. We model this process via a recently-introduced hybrid automata-continuum scheme MS12 and use computer simulation to unravel the role of specific Width of Wavefronts for Random Travelling Waves components of this complex developmental process. Our We consider the super-critical super-Brownian motion in 2 results indicate an essential role for positive feedback be­ or more dimensions. Under certain initial conditions, this tween the cAMP signaling and the expression of the genes process has a random travelling wave. We give a defini­ encoding the signal transduction and response machinery. tion for the width of the wave, and show that the width Herbert Levine is stochastically bounded. Roughly speaking, we examine U niv. of California, San Diego the process along a line which is perpendicular to the wave­ La Jola, CA 92093 front, and measure the wavefront as it intersects this line. We do not expect the entire wavefront to have finite width. Although our results are limited to this model, we believe that the same phenomenon holds for a large class of random MSll growth models. To prove our result, we exploit the connec­ Optical Tomography of Chemical Waves in Three tion between the super-critical super-Brownian motion and Dimensions the Kolmogorov-Petrovskii-Piscuinov equation. Using the Numerically solving the partial differential equations of Feynman-Kac formula, we study travelling waves for the excitability revealed topologically distinct persistent solu­ Kolmogorov-Petrovskii-Piscuinovequation. Using this in­ tions like linked and knotted vortex rings (Science 371, formation, we then deduce our result for the super-critical 233 [1994]; Physica D 84, 126 [1995]). No attempts to super-Brownian motion. determine whether these also occur in nature have suc­ Carl Mueller ceeded, for want of a way to capture moving transpar­ Department of Mathematics ent color patterns 3-dimensionally. My 1992 plan for such University of Rochester a device is now a working reality: (Chaos 6(4) [1996]); Rochester, NY 14627 Roger Tribe 84 Abstracts

Mathematics Institute MS13 University of Warwick Spatial Structure in Oceanic Plankton Populations Coventry CV4 7 AL The coupling of nonlinear interactions in aquatic commu­ England nities to periodic chemical and physical forcing can cause numerous spatial and spatio-temporal patterns. A sim­ ple model of phytoplankton-zooplankton dynamics in space MS12 and time is presented as an example. The temporal and Front Propagation in Noisy Burgers Type Equa­ spatial patterns in natural nutrient-plankton-fish systems tions are interpreted as transient and stationary nonequilibrium It is well-known that traveling front solutions to scalar con­ solutions of dynamical nonlinear interaction-diffusion­ servation laws with convex flux are stable up to a phase advection models. Nutrients and planktivorous fish are shift for large times if the initial perturbations have enough treated as control parameters, varying in space and time. spatial decay. When the initial perturbation is a station­ For constant parameters, local properties as multiple sta­ ary random process or the convex flux is spatially random, bility, oscillations and excitability are found. Periodic forc­ we show for Burgers type equations that front structures ing through temperature, light or nutrient variability as well as periodic fish invasions can drive the system from persist and front location obeys the law ~ mean velocity regular predator-prey limit-cycle oscillations to determin­ times t plus Gaussian noise. istic chaos via quasi-periodic torus oscillations and their Jan Wehr destruction. It is shown, that the external forcing does Department of Mathematics not necessarily lead to unpredictable chaotic dynamics but University of Arizona can even stabilize the system's behaviour. Having regard Tucson, AZ 85721 also to diffusion and advection, the emergence of stand­ [email protected] ing and travelling population patches is obtained. These Jack Xin waves appear after diffusion-induced (Turing) or diffusion­ Department of Mathematics advection-induced (DIFII) instability of a spatially uniform University of Arizona population distribution, considering phytoplankton as the Tucson, AZ 85721 activating and zooplankton as the inhibiting species. Un­ [email protected] der relative physical uniformity, the latter mechanisms are suggested as possible ways to generate the patchy plank­ ton patterns, observed in natural systems. Compared to MS13 the Turing instability, the differential-flow-induced insta­ Diurnal Vertical Migration, A Mechanism for bility (DIFII) is a much more general mechanism of spatio­ Patchiness in Shear Flows temporal pattern formation because it does not depend on the condition of considerably different diffusion coefficients Diurnal vertical migration is a common behavioural charac­ but only on the condition of different species velocities, teristic of (herbivorous) zooplankton (Z), contrasting with regardless of which one is faster. the photosynthesising phytoplankton (P), which behave much more like passive suspensions in the ambient fluid Professor Horst Malchow flow. When shear is present, this different behaviour im­ Inst. of Environmental Systems Research plies different advective effects for P and Z. We use simple Dept. of Math & Computer Science, Univ. of Osnabrueck, models to demonstrate that one consequenceis the devel­ Artilleriestr 34, D-49069, opment of strong patchiness in the populations of P and Z Osnabrueck, Germany on scales dictated by shear strength and biological param­ eters. John Brindley MS13 University of Leeds, Leeds, UK Chaotic Advection and Plankton Patchiness We discuss plankton distributions and interactions arising Louise Matthews in velocity fields driven by wind or thermohaline effects University of Leeds, Leeds, UK (e.g. Langmuir circulation, thermohaline convection). We show that the Unsteady velocity field affects the distribu­ tion of plankton by chaotic advection, provides a mech­ MS13 anism of replenishment of Stommel Retention Zones and Vertical Migration in Phytoplankton-Zooplankton an explanation for the observed disappearance of patchi­ Interactions ness of plankton population under high-wind conditions. We discuss the effects of different plankton population dy­ Based on empirical evidence documenting the importance namics and convective velocity fields on asymptotic spatio­ of size classification in phytoplankton-zooplankton inter­ temporal states. actions, various models have been proposed which gener­ alise the aggregate NPZ model to include such classifica­ Igor Mezic tions. However, different sizes and species of zooplankton Dept. of Mech. & Environ. Eng. differ markedly in behaviours such as vertical migration. In University of California,USA this talk I present a simple 2-class food chain model which incorporates vertical migration of the larger zooplankton class, and study the effects this migration has on the dy­ MS14 namics of the populations. A Finite Element Method for the Frobenius-Perron Kathleen Crowe Operator Equation University of California, USA We present a general class of finite element methods to solve numerically the absolutely continuous invariant mea- Abstracts 85

sure problem associated with a multi-dimensional nonsin­ light Ulam's method is seen to have a natural extension to gular transformation. This class includes the method of the set-valued case-giving discrete stochastic approxima­ Markov finite approximations and the projection method, tions of Markov processes by Markov chains. in particular the original Ulam method. The convergence Walter Miller of the method will be proved for the class of piecewise ex­ Howard University panding maps with the help of the bounded variation tech­ Washington D.C. nique. The convergence rate analysis will also be given. Jiu Ding The University of Southern Mississippi MS14 Hattiesburg, MS Cone conditions and error bounds for Ulam's method Aihui Zhou Institute of Systems Science Fundamental to error bounds for Ulam's Markov approx­ Academia Sinica imation scheme for approximating invariant measures are Beijing 1000080, China estimates of the rate of mixing for the dynamical system. Liverani 4 has recently shown how certain cones of func­ tions can be used to good mixing estimates. Using ex­ panding circle maps as an example, I will show how the MS14 continuous structures can be "discretized" to get explicit Computing Physical Measures of Mixing Multidi­ quantitative bounds for the approximation error in Ulam's mensional Systems with an Application to Lya­ method5. Generalizations will be mentioned briefly. punov Exponents Rua Murray We present new results on the rigorous computation of Dept. Pure Math. and Math. Stats. physical invariant measures of higher dimensional mix­ University of Cambridge ing transformations, including a new method of proof of 16 Mill Lane, Cambridge CB2 ISB, UNITED KINGDOM Ulam's conjecture. The exponential mixing property guar­ EMAIL: [email protected] antees that the system is sufficiently insensitive to errors in the approximation algorithm. Our method has the ad­ vantages of having very relaxed conditions on its numeri­ cal implementation, being applicable to higher dimensional MS15 systems, and potentially applicable to a wide class of maps. Particle Motion in Capillary Surface Waves The numerical approximation of the physical measure is The motion of drifters near the sea surface has been a cru­ used directly to estimate the Lyapunov exponents of the cial element in probing oceanic turbulence. In the geo­ system. The exponents are calculated as an integral of a physical dynamical regime, the drifter motion possesses special "stretching function" over phase space (a space av­ significant persistence, showing that the motion is far from erage), rather than by following a single long orbit (a time Brownian. A similar persistence is observed in laboratory average). experiments on capillary surface waves. We obtain results Gary Froyland for the self-diffusivity as well as for the relative diffusivity, The University of Western Australia and compare our measurements with results obtained from Perth, AUSTRALIA turbulence theories and with observations from geophysical turbulence. Preben Alstr(1lm, MS14 Mogens T. Levinsen, Approximating Attractors and Chain Transitive and Elsebeth Schroder Invariant Sets with Ulam's Method Niels Bohr Institute, Copenhagen, Denmark In a departure from its customary application Ulam's method is used for maps where the physical measure is possibly singular with respect to Lebesgue measure. The method can be used to approximate a large class of Lya­ MS15 pounov stable attractors and more general sets associated Motion of Floating Particles on a Turbulently with the long time dynamics of the map. Driven Fluid Fern Hunt We have studied the motion of small particles floating on National Institute of Standards and Technology the surface of a turbulent fluid and in addition, the turbu­ Gaithersburg, Maryland lence in the underlying fluid that drives the surface particle motion. The experiment is aimed at illuminating the con­ nection between the two. Attention was focused on mea­ suring the R-dependence of the instantaneous root mean MS14 Markov Finite Approximation of Frobenius-Perron square velocity difference 17 = .J < 8v(R)2 >. In one ex­ Operators and Invariant Measures for Set-Valued periment R is the separation of the surface particles; in the other, is the separation of seed particles in the bulk Dynamical Systems R fluid. The results of the study will be compared with with We view a method proposed by S. Ulam as a discrete theoretical expectations and with analogous high Reynolds stochastic approximation of a dynamical system by a ran­ dom perturbation modelled by a Markov chain induced by a partition of the state space. This leads to an alternative view of a set-valued amp as an aggregate of random pertur­ 4C Liverani. Decay of Correlations. Annals of Mathematics bations from which notions of Frobenius-Perron operator 142 (1995) and invariant measure for set-valued maps follow. In this 5Joint work worth M S Keane and L-S Young. 86 Abstracts number experiments carried out in the ocean. MS16 Phase Synchronization of Chaotic Systems Walter 1. Goldburg and Cecil Cheung University of Pittsburgh Cooperative behaviour of chaotic dynamical systems and Pittsburgh, PA in particular synchronization phenomena have received much attention recently. Generally, synchronization can be treated as an appearance of some relations between functionals of two processes due to interaction. The clas­ MS15 sical case of periodic oscillators was already described by Drifter Trajectories in Geophysical Flows Huygens in 1673. We have recently found the new effect Some of the properties of Lagrangian trajectories in large­ of phase synchronization of coupled chaotic systems. To scale geophysical flows are discussed. Velocity statistics characterize this phenomenon, we use the analytic signal and single-particle dispersion are considered in detail, and approach based on the Hilbert transform. In the synchro­ the issue of relating Lagrangian and Eulerian measure­ nized regime the phases are locked, whereas the amplitudes ments is adressed. In particular, I review some recent re­ vary chaotically and are practically uncorrelated. A rela­ sults on the behavior of freely-drifting subsurface buoys in tion between the phase synchronization and the properties the North Atlantic and compare them with the analogous of the Lyapunov spectrum is studied. This effect has been results obtained for simple barotropic models such as two­ found for different kinds of models, e.g. coupled Roessler dimensional turbulence and point vortices. or Lorenz attractors, periodically forced chaotic systems, Antonello Provenzale and large ensembles of non-identical chaotic oscillators. Istituto di Cosmogeofisica, Corso Fiume 4, 1-10133 Torino, Jurgen Kurths University of Potsdam, Germany Italy

MS16 MS16 Synchronization Transitions in Coupled Chaotic Long-Range Rotating Order in Extensively­ Oscillators: From Phase to Compete Synchroniza­ Chaotic Systems tion After a brief presentation of non-trivial collective be­ Complete, generalized and phase synchronization (CS, GS, havior in extensively-chaotic systems, or, in more tradi­ and PS) of chaotic oscillators have been described in the lit­ tional terms, long-range order arising from spontaneous erature. The relation between these different types of syn­ symmetry-breaking in large, chaotic, dynamical systems, chronization and the scenarios of transitions to or between I will focus on the particular case of long-range rotating them have not been addressed yet. We study synchroniza­ order (LRRO), where a continuous global "phase" variable tion of symmetrically coupled non-identical self-sustained can describe the collective motion. I will show how "noisy oscillators. We demonstrate that with the increase of cou­ synchronization" of chaotic oscillators is but a particular pling first the transition from non-synchronous state to PS case of LRRO. Universal features, the connections to the occurs; it means entrainment of phases of chaotic oscilla­ Kardar-Parisi-Zhang and the complex Ginzburg-Landau tors, whereas their amplitudes remain chaotic and non­ equations will be discussed in light of recent numerical re­ correlated. For larger couplings a new regime which we sults on various systems. call lag synchronization (LS) is observed. LS appears Hugues Chate as coincidence of shifted in time states of two systems, Centre d'Etudes de Saclay, France Xl(t + 7"0) = X2(t). This means that the relation appears not only between the phases, but also between the ampli­ tudes. Finally, with further increase of coupling, the time shift 7"0 -t 0 and this regime tends to CS. The transi­ MS16 tions in the type of synchronization can be related to the Spiral Waves in Chaotic Systems transitions in the Lyapunov spectrum. Spiral waves can exist in systems whose local dynamics M. Rosenblum is chaotic or supports other types of complicated periodic Max-Plank-Arbeitsgruppe "Nichtlineare Dynamik" behavior. The spatio-temporal structure of such systems an der Universitiit Potsdam, Potsdam, Germany; is considerably more complex than that of simple period- permanent address: Mechanical Engineering Research In­ 1 oscillatory media. This structure has been studied in stitute, systems where a chaotic attractor arises from a period­ Russian Acad. Sci., Moscow, Russia doubling cascade or intermittency, as well as in systems displaying other types of complex periodic behavior. In the case of period doubling, as one moves from the core of the spiral wave, the local dynamics takes the form of per­ MS17 turbed, period-doubled orbits whose character varies with Synchronous and Asynchronous States in a Net­ spatial location. Topological featues that underlie the orga­ work of Spiking Neurons nization of the medium will be discussed. Different scenar­ We study the dynamics in a fully connected network of ios for the spatio-temporal organization will be described spiking neurons. As a neuron model, we use the spike for chaotic and complex-periodic dynamics that arises by response model, a generalized version of the integrate-and­ other mechanisms. fire model. The stability of asynchronous firing and of col­ Raymond Kapral lective network oscillations is analyzed. The result can University of Toronto, Canada be discussed in terms of biological paramters, like the sig­ nal transmission delay, the postsynaptic potential, and the spike-afterpotential. Wulfram Gerstner Swiss Federal Institute of Technology Abstracts 87

EPFL,IN-J which is applied to segmenting real, gray-level images. CH-I015 Lausanne David Terman SWITZERLAND Ohio State University, Columbus, Ohio

DeLiang Wang MS17 Ohio State University Saddle-Node Bifurcations and Wave Propagation Columbus, Ohio in Model Cortical Column Structures A canonical model for a saddle-node bifurcation on a limit cycle will be described and related to column structures MS18 in the neocortex. A multi-layer network of such canoni­ Singular Limits of Phase Equations for Patterns cal models will be formulated and analyzed for the exis­ Far from Threshold tence and stability of steady progressing wave solutions. We introduce a hyperviscous regularization of the phase Relations of these results to the propagation of activity in diffusion equation which models the behaviour of patterns models of cortical barrel structures will be discussed. in extended systems far from threshold. In analogy to vis­ Frank C. Hoppensteadt cosity limits of Burgers' equation one finds that the associ­ System Science and Engineering Research Center ated viscosity limits of our regularization can produce weak Arizona State University, Tempe, AZ 85287-7606 solutions of the unregularized equation which effectively are single-valued truncations of the multi-valued phase dif­ fusion solutions. We describe some generic types of defects that are found in these weak solutions. A complete de­ scription of the limits can be given when the boundary MS17 data is single-valued. We also describe some extensions Rhythms and "Lurching" Waves in the "Sleeping" to multi-valued boundary data, incorporating twist, which Thalamic Slice are needed to realize disclinations in the viscosity limit. Sleep spindle rhythms are generated in the thalamus. They Nicholas M. Ercolani can also occur spontaneously in an isolated "slice" of tha­ University of Arizona, Tucson, AZ lamic tissue, where interesting wave-like behavior is also observed. Current hypotheses for the neural basis of these Robert Indik rhythms will be outlined, and neural network models that University of Arizona, Tucson, AZ simulate the rhythms will be presented. The nonlinear dy­ namics of single cells, and of the synaptic coupling between Alan Newell the excitatory (thalamocortical relay) and inhibitory (retic­ University of Warwick, ENGLAND ular nucleus) cells, will be described for these Hodgkin­ Huxley-like models coupled in a I-D chain architecture. In Thierry Passot this system, coupling via synaptic inhibition plays a key role in rhythmogenesis and in wave propagation. The net­ Observatoire de Nice, FRANCE work has a stable resting state but a localized stimulus can initiate oscillatory activity. The activity spreads as a front with a "lurching" type propagation; in each cycle a discrete MS18 number of silent cells are recruited to join the population Self-Replicating Spots rhythm. The self-replicating spot is a particle-like solution that John Rinzel occurs in reaction-diffusion systems. A spot grows until National Institutes of Health, Bethesda, MD reaching a critical size at which time it divides in two. The two resulting spots again grow and divide. They propagate Xiao-Jing Wang and deform on a slow time scale. Eventually they become Brandeis University unstable to perturbations on the fast time scale and divide. Waltham, MA I will present an asymptotic analysis reflective of the above discussion. David Golomb John Pearson Ben-Gurion University [email protected] Beer-Sheva, Israel Los Alamos National Laboratory, Los Alamos, NM

MS17 MS19 Image Segmentation Based on Oscillatory Correla­ Critical Behavior of the Distribution of Optimal tion Paths We consider a recent model for image segmentation based The distribution of paths for large fluctuations away from on neural oscillations. The model consists of a locally ex­ a stable state is investigated. Singular features of the pat­ citatory globally inhibitory oscillator network. We show tern of most probable (optimal) paths in non-equilibrium both analytically and numerically that the network rapidly systems are revealed. We demonstrate critical broadening achieves both synchronization within blocks of oscillators of the distribution of the paths coming to a cusp point that that are stimulated by connected regions and desynchro­ represents the simplest generic singularity of the optimal nization between different blocks. We further extend the paths pattern. The critical behavior is shown to be de­ model so that it is able to remove noisy regions from an im­ scribed by a Landau-type theory. The problem of optimal age. The analysis naturally leads to a computer algorithm, 88 Abstracts control of large fluctuations is discussed. derstanding global bifurcations involving the heteroclinic interaction of two-dimensional manifolds of different equi­ Mark 1. Dykman and Vadim N. Smelyanskiy libria and/or limit cycles in a variety of three-dimensional Michigan State University, East Lansing, MI systems. Mark E. Johnson MS19 Princeton University, Princeton, NJ Periodic Modulation of the Rate of Noise-Induced Escape through an Unstable Limit Cycle Michael S. Jolly Indiana University, Bloomington, IN Suppose that a bistable, one-dimensional continuous dy­ namical system is periodically driven in such a way that Ioannis G. Kevrekidis its attractors become distinct limit cycles: stable states of Princeton University, Princeton, NJ periodic vibration. Suppose that the system is addition­ ally driven by weak noise. The noise may induce tran­ John Lowengrub sitions between the two limit cycles. In the limit as the University of Minnesota, Minneapolis, MN noise strength tends to zero, to leading order the frequency of these transitions falls off exponentially. We show that asymptotically, the transition rate includes a multiplica­ MS20 tive factor that is periodic in the logarithm of the noise strength. Our analysis is based on a semiclassical approx­ Computing Invariant Manifolds of Saddle-Type imation to the stationary probability density of the driven The computation of stable and unstable manifolds has system, and applies both to the case of white noise and mainly been done for hyperbolic fixed points. However, Markovian dichotomous noise. normally hyperbolic invariant manifolds also have stable and unstable manifolds, the knowledge of which promises Robert S. Maier much insight in the dynamics. Using an adaptation of the University of Arizona, Tucson, AR graph transform technique, we compute normally hyper­ bolic invariant manifolds of saddle-type and, subsequently, their stable and unstable manifolds. MS19 Signal Enhancement by Large Fluctuations Hinke Osinga The Geometry Center The role of large fluctuations in mediating signal/noise en­ 1300 South Second Street hancement of weak periodic signals in two different con­ Minneapolis, MN 55454 texts will be discussed: heterodyning, with applications to electronics and nonlinear optics; and underdamped zero­ dispersion systems, with applications to SQUIDs (super­ conducting quantum interference devices), relativistic os­ MS20 cillators and particle accelerators. In each case, large fluc­ Computing Stable Sets of Noninvertible Mappings tuations can give rise to pronounced amplification of both For diffeomorphisms, every point has a unique preimage the signal and the signal/noise ratio, much as occurs in and it is straightforward to compute one-dimensional stable classical stochastic resonance. manifolds of periodic points. For non-invertible mappings, Peter V.E. McClintock however, some points have multiple preimages; others may Lancaster University, Lancaster, UK have no preimages. This makes the computation of sta­ ble sets (the generalization of stable manifolds) difficult, because accurate computations require global knowledge about the way the mapping folds phase space. In this talk MS19 we geometrically analyze the folding and use this informa­ The Stochastic Manifestations of Chaos tion to construct an algorithm for computing stable sets. Many aspects of the dynamics of random walks and Brown­ Frederick J. Wicklin ian motion can be understood using techniques of classical Chia-Hsing Nien and quantum chaos theory. Symmetry breaking boundaries Dept. Mathematics Department and Geometry Center, U. or coupling between degrees of freedom can lead to mutual Minnesota repulsion of decay rates and a change of rate of approach to equilibrium. Whether or not these effects manifest them­ selves in the stochastic system depends on the size of the diffusion coefficient. MS20 Stable and Unstable Manifolds of Halo Orbits in Linda E. Reichl the Circular Restricted Three-Body Problem University of Texas, Austin, TX Recently, the space science community has shown an in­ terest in sending scientific missions to the vicinity of the collinear Earth-Sun libration points. A fuller understand­ MS20 ing of the geometry of the phase space of the circular re­ Computation of Heteroclinic Two-Dimensional In­ stricted three-body problem could provide new possibili­ variant Manifold Interactions ties for baseline trajectory design. For this purpose, we We present and discuss alternative approaches to comput­ compute families of halo orbits in the circular restricted ing two- dimensional (un)stable manifolds of steady states three-body problem. The stable manifolds of these orbits and limit cycles of saddle-type; these are partially moti­ provide low energy trajectories to the orbits. We present vated from fluid interface evolution computations. We il­ an algorithm for the computation and visualization of these lustrate such techniques by computing, visualizing, and un- Abstracts 89

orbits and their stable and unstable manifolds. that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate mul­ Robert Thurman tiphase solutions for the vortex filament lfow (also known as University of Minnesota, Minneapolis, MN the Losalized Induction Equation). In doing so, we obtain initial data for multi phase solutions ofthe LIE representing Patrick Worfolk a large number of knots types. University of Minnesota, Minneapolis, MN Annalisa Calini, Department of Mathematics, University of Charleston, MS21 66 George st., Dynamical Properties of Two-Dimensional Charleston SC 29424 J osephson-J unction Arrays* Thomas I vey, Two-dimensional arrays of Josephson junctions are poten­ Department of Mathematics, tially useful as high-frequency oscillators, as well as pro­ Case Western Reserve University, viding an experimental system for studying the stability Cleveland OH 44106 of coupled non- linear oscillators. Our experiments and simulations show that the stability of arrays is strongly in­ fluenced by parameters of the junctions in the array and the load-coupling circuit. In our best designs, a significant MS22 Homotopies and Knot Types of Elastic Rods fraction of the theoretical maximum power has been cou­ pled to a detector. Elastic rods can be characterized either as space curves Work supported in part by the Air Force Office of Scientific which are critical for some combination of length, total Research under Grant No. F49620-92-J-0041. torsion, and elastic energy, or as solutions of the Localized Induction Equation that move by a screw motion. Based A.B. Cawthorne, P. Barbara, and C.J. Lobb on work of Langer and Singer, we show that every torus University of Maryland College Park, MD knot type is realized by an elastic rod. Furthermore, there exist one-parameter families of elastic rods connecting the m-covered circle with the n-covered circle whenever m and n are coprime. MS21 Generation of Submillimeter Wave Radiation Us­ Thomas Ivey, ing Linear Phase-Locked Josephson Junction Ar­ Department of Mathematics, rays Recent results on the generation of submillimeter Case Western Reserve University, Cleveland OH 44106 wave radiation using Josephson junctions arrays are dis­ cussed. Power levels approaching 1/2 milliwatt for fre­ quencies above 400 GHz have been achieved. It appear that instabilities arising in the individual junctions are an MS22 important limitation on the junction's maximum critical Geometric Realizations of Fordy-Kulish Integrable current and thus on the array power. Systems, Sym-Pohlmeyer Curves, and Related Variation Formulas. Part I James Lukens State University of New York A method of Sym and Pohlmeyer, which produces geo­ metric realizations of many integrable systems, is applied to the Fordy-Kulish "generalized non-linear Schrodinger equations" associated with Hermitian symmetric Lie al­ MS21 gebras. Such equations are seen here to correspond to Josephson Junction Arrays: Practical Devices and arclength-parametrized curves evolving in Euclidean and Non-linear Dynamics Experiments, numerical simula- other spaces, generalizing the "localized induction model" tions and analysis are used to investigate dynamical states of vortex filament motion. In this context, a "geometric re­ in arrays of Josephson junctions which are governed by cursion operator" is introduced, and a compact formula for the discrete sine-Gordon equation. We will discuss their the variation of " natural curvatures" in terms of this oper­ potential for microwave applications as well as for models ator is derived. Applying the above formalism, hierarchies of coupled physical systems, and the types of experiments of evolution equations are obtained for curves evolving in and systems that have been studied. Experimental evi­ n-dimensional Euclidean space, with curvatires satisfying dence will be shown for dynamical effects such as paramet­ an equation in a "generalized m-KdV hierarchy". A spe­ ric instabilities of whirling modes, phase-locking of kinks to cial case appears to be related to recent constructions of linear waves, phase-locking between rows, and dynamical Doliwa and Santini, and may help illuminate certain fea­ checkerboard states. tures of the latter. Terry P. Orlando Joel Langer, Department of Mathematics, Case Western Massachusetts Institute of Technology Reserve University, Cleveland OH 44106 Cambridge, MA Ron Perline, Department of Mathematics and Computer Science, Drexel University, Philadelphia PA 19104 MS22 Backlund Transformations of Knots of Constant Torsion MS22 Geometric Realizations of Fordy-Kulish Integrable The Biicklund transformation for the sine-Gordon equation Systems, Sym-Pohlmeyer Curves, and Related can be adapted to give a transformation on space curves 90 Abstracts

Variation Formulas. Part II Discrete Neural Networks with Synaptic Depres­ sion A method of Sym and Pohlmeyer, which produces geo­ metric realizations of many integrable systems, is applied We discuss the oscillatory activity of a random neural net­ to the Fordy-Kulish "generalized non-linear Schrodinger work with purely excitatory connections and use dependent equations" associated with Hermitian symmetric Lie al­ synaptic depression. The work is motivated by measure­ gebras. Such equations are seen here to correspond to ments in slice cultures of embryonic rat spinal cord. Mod­ arclength-parametrized curves evolving in Euclidean and eling two symmetrically coupled networks, one observes other spaces, generalizing the "localized induction model" inphase, antiphase, quasiperiodic or phase-trapped oscil­ of vortexfilament motion. In this context, a "geometric re­ lations. These different activity patterns are obtained by cursion operator" is introduced, and a compact formula for adapting the time constant of the synaptic depression. The the variation of "natural curvatures" in terms of this oper­ mathematical methods we use are bifurcation theory and ator is derived. Applying the above formalism, hierarchies average phase difference theory. of evolution equations are obtained for curves evolving in Walter Senn n-dimensional Euclidean space, with curvatires satisfying Institute of Informatics, University of Bern, Switzerland an equation in a "generalized m-KdV hierarchy". A spe­ cial case appears to be related to recent constructions of J iirg Streit Doliwa and Santini, and may help illuminate certain fea­ Department of Physiology, tures of the latter. University of Bern, Switzerland Joel Langer, Department of Mathematics, Case Western Reserve University, MS23 Cleveland OH 44106 Modeling Cortical 40 Hz Oscillations: Pacemaker Ron Perline, Neurons and Synaptic Synchronization Department of Mathematics and Computer Science, Drexel University, Philadelphia PA 19104 Fast neuronal oscillations (gamma, 40 Hz) have been ob­ served in the neocortex and hippocampus during behav­ ioral arousal. In this talk I shall review experimental re­ sults and present thereotical studies that suggest how such MS23 a rhythm may emerge in cortical networks. It is shown Synaptically Induced Bistability and Transition to that the rhythm can be generated by pacemaker neurons, Repetitive Activity which either display subthreshold membrane oscillations or Many model cortical cells undergo a transition from rest bursting at around 40 Hz. Then, I shall discuss synaptic to repetitive activity as some parameter varies via a saddl­ mechanisms for network synchronization of this rhythm, node loop bifurcation. We examine how the temporal prop­ with an emphasis on recurrent inhibition. erties of excitatory synapses influence the transition from Xiao-Jing Wang a single transmitted pulse to repeated activity. The transi­ Physics Department and Center for Complex Systems, tion behavior can be understood via simplified dynamical Brandeis University, Waltham, MA 02254 equations and an analysis of these equations. Two cases are considered: (i) self-excitation and (ii) a pair of mutu­ ally coupled cells. A map is numerically computed showing an apparent chaotic transition between states. MS24 ODE Architect: CODEE's Multimedia Package Bard Ermentrout and The introductory DE course is evolving very rapidly and Jim Uschock relying more and more on technology. Powerful com­ Department of Mathematics puter platforms and software make it possible for students University of Pittsburgh to explore the properties of dynamical systems and have Pittsburgh, PA 15260 fun doing it. ODE Architect, an inexpensive multime­ dia solver/modeler, developed by the CODEE consortium, runs on a PC and teaches students modeling concepts in a supportive, stress-free environment. We explore its fea­ MS23 tures via modeling modules that overlay a state-of-the-art The Role of Spike Adaptation in Shaping Spa­ solver. tiotemporal Patterns of Activity in Cortex Robert L. Borrelli We study analytically and numerically cortical network Harvey Mudd College, Claremont, CA models to show that synchronized bursts or traveling pulses of activity can emerge in cortex from the "conspiracy" be­ tween spike adaptation and cortical excitatory feedback. The bifurcations leading to these states are studied. The MS24 properties of the emergent spatiotemporal activity patterns How to Make a Pendulum Do Anything You Want are analyzed. In particular, the bursting period is shown to Although the forced pendulum is one of the most standard depend mainly on the adaptation time constant. Possible differential equations one encounters in calculus classes, the functional roles of the traveling pulses are examined. actual behavior it predicts can be most complicated. We'll D. Hansel, Centre de Physique Theorique, CNRS-UPR014, show that for appropriate forcing and damping, we can find Ecole Polytechnique, 91128, Palaiseau, France initial conditions that lead to any preassigned sequence of gyrations. The instability of these motions has a counter­ part - the pendulum is highly controllable. MS23 John H. Hubbard Pattern Generation by Two Coupled Time- Cornell University, Ithaca, NY Abstracts 91

MS24 Visualization and Verification, with Mathematica, of Solutions to Differential Equations Dan Schwalbe and I have written a comprehensive Mathe­ MS25 matica package for the visualization of solutions to DEs. Removing the Stiffness of Curvature in 3-D Fila­ Innovative ideas include shaded nullcline regions,curved ment Calculations fish-like shapes to generate flow fields. In addition to pre­ We present a new formulation for computing the motion of senting the package, I will show how Mathematica can be 3-D filaments with curvature effect. The curvature effect used to check on the correctness of mimerica I solutions, and introduces high-order terms into the dynamics. This leads discuss some complicated examples, such as the unstable to severe stability constraints for explicit time integration motion of a book thrown into the air with certain initial methods and makes the application of implicit methods spins. difficult. Our new formulation completely removes the se­ Stan Wagon vere time stability constraint, and gives an efficient implicit Macalester College, St. Paul, MN method. Applications to motion by mean curvature and some elastic rod model problem will be given. Thomas Y. Hou MS24 Applied Mathematics, Caltech Teaching ODE's with the WWW and a Calculator Based Laboratory Issac Klapper Dept. of Mathematics Teaching Differential Equations using the World Wide Web with a combination of hands-on experimentation (using the UCLA Texas Instrument Calculator Based Laboratory), JAVA ap­ Hui Si Applied Mathematics plets, and a Computer Algebra System - all orchestrated by a browser like Netscape. Caltech Frank Wattenberg Carroll College and Montana State University, Helena, MT MS26 The Role of Attractors in Stochastic Bifurcation Theory MS25 Curve Dynamics and Gene Regulation Inspite of great efforts in recent years, bifurcation theory of dynamical systems in the presence of noise is still in its The motion of DN A plays an important role in gene regula­ infancy. There are few rigorous general theorems and cri­ tion. We present three complementary mathematical mod­ teria, and many phenomena have been "proved" only by els for the initiation of transcription, in which DNA motion compl,lter simulations, or for particular models. We report is incorporated into the model using Langevin dynamics, on progress made by utilizing the concept of a random at­ Monte-Carlo simulations and from energy considerations. tractor recently introduced by Crauel, Debussche, Flandoli Gadi Fibich and SchmalfuB. Its existence can be assured under quite UCLA, Los-Angeles, CA general conditions. Since a random attractor carries all invariant measures (Crauel), the bifurcation of the attrac­ Danny Petrasek tor gives us information about the bifurcation of invariant UCLA, Los-Angeles, CA measures. Several examples in dimension one and two are considered. Isaac Klapper Ludwig Arnold Montana State University, MO Institute for Dynamical Systems University of Bremen

MS25 Nonlinear Dynamics of Elastic Filament MS26 The Kirchoff model provides a well established framework Bifurcation Theory for Stochastic Differential to study the dynamics of thin elastic filament. A new Equations perturbation scheme is introduced to study the stability We study a parametrised family of multidimensional non­ and dynamics of static solutions after bifurcation. A lin­ linear stochastic differential eql,lations fixing O. Suppose ear analysis of the static solutions enables us to describe the parameters are varied in such a way that the Lyapunov the instabilities that filaments undergo when twist or ten­ exponent >. for the associated linear system changes sign sion is varied. At the bifurcation point where the filament from negative to positive. The stable fixed point 0 be­ becomes unstable, a weakly nonlinear analysis can be car­ comes unstable and a new stationary probability measure ried out and an amplitude equation for the post-bifurcation p" with 'most' of its mass 'near' 0, appears. We give some dynamics is obtained. The static solutions of a variety of quantitative results on p, valid for small positive values of problems, including the twisted planar ring, the straight A. rod and helical filaments with intrinsic curvature, are ana­ lyzed by this method. Peter H. Baxendale, Department of Mathematics, Alain Goriely University of Southern California, University of Arizona, Tucson USA Los Angeles, CA 90089-1113, U.S.A. Michael Tabor University of Arizona, Tucson USA 92 Abstracts

MS27 in these examples exhibit singularities that we interpret as Some Experiments on Nonstationary Vibrations in evidence of new nonlinear scalings and also show no evi­ a Single Degree-of-freedom Nonlinear System dence of being finitely determined. This research is being An experimental rig was constructed to observe the re­ pursued in collaboration with KT.R. Davies and A. Ja­ yaraman. sponse of a thin cantilever beam system undergoing non­ stationary excitation. Initially, the beam was configured J.D. Crawford so that under stationary, sinusoidal excitation, chaotic re­ Dept. of Physics sponses were observed in a region of frequencies. The beam University of Pittsburgh was also subjected to swept sinewave excitation, sweeping Pittsburgh, PA 15260 through this chaotic region, in order to discover how the [email protected] sweep rate affected the response characteristics. The faster the sweep rate, the smaller the frequency region over which chaotic behavior was observed. Oscillations in the response MS27 amplitude, similar to those found in similar experiments on Weakly Unstable Systems in the Presence of Neu­ linear systems, were also observed. trally Stable Modes Anil K Bajaj, Mark Hood and Patricia Davies Near the onset of a linear instability, the nonlinear evo­ School of Mechanical Engineering lution of the mode amplitudes can be described using an Purdue University expansion in the amplitude of the unstable modes. Since W. Lafayette the growth rates are very small near onset, nonlinear ef­ Indiana 47907 fects often act to saturate the instability before the ampli­ tudes grow appreciably; for this reason such expansions have proved a powerful tool for studying the nonlinear MS27 states emerging from the bifurcation. In addition, the pro­ Some Experiments on Nonstationary Vibrations in totypical dissipative examples share another characteristic: a Single Degree-of-freedom Nonlinear System except for the unstable modes, all other linear modes are An experimental rig was constructed to observe the re­ exponentially damped. In particular, there are no neu­ sponse of a thin cantilever beam system undergoing non­ trally stable modes with nonlinear couplings to the unsta­ stationary excitation. Initially, the beam was configured ble modes. If such neutral modes are present, the ampli­ so that under stationary, sinusoidal excitation, chaotic re­ tude equations for the unstable modes, i.e. normal forms sponses were observed in a region of frequencies. The beam on the unstable manifold, can have very different features. was also subjected to swept sinewave excitation, sweeping We discuss several illustrative examples: the normal form through this chaotic region, in order to discover how the for a Hopf/saddle node mode interaction, the Vlasov equa­ sweep rate affected the response characteristics. The faster tion from plasma physics, and a continuum model for the the sweep rate, the smaller the frequency region over which Kuramoto system of phase oscillators. The normal forms chaotic behavior was observed. Oscillations in the response in these examples exhibit singularities that we interpret as amplitude, similar to those found in similar experiments on evidence of new nonlinear scalings and also show no evi­ linear systems, were also observed. dence of being finitely determined. This research is being pursued in collaboration with KT.R. Davies and A. Ja­ Anil K Bajaj, Mark Hood and Patricia Davies yaraman. School of Mechanical Engineering Purdue University J.D. Crawford W. Lafayette Dept. of Physics Indiana 47907 University of Pittsburgh Pittsburgh, PA 15260

MS27 Weakly Unstable Systems in the Presence of Neu­ MS27 trally Stable Modes Accurate Phase After Slow Passage Through Sub­ harmonic Resonance Near the onset of a linear instability, the nonlinear evo­ lution of the mode amplitudes can be described using an The slow passage through subharmonic resonance is an­ expansion in the amplitude of the unstable modes. Since alyzed for a periodically forced conservative system with the growth rates are very small near onset, nonlinear ef­ weak damping. Multiphased averaging fails, requiring a fects often act to saturate the instability before the ampli­ subharmonic resonance layer. By matching to sufficiently tudes grow appreciably; for this reason such expansions high order, a new more accurate jump in the average en­ have proved a powerful tool for studying the nonlinear ergy across a subharmonic resoance layer is obtained. It is states emerging from the bifurcation. In addition, the pro­ shown that the correct phase of the strongly nonlinear os­ totypical dissipative examples share another characteristic: cillator after subharmonic resonance may be obtained using except for the unstable modes, all other linear modes are a time shift and a constant phase adjustment. exponentially damped. In particular, there are no neu­ Jerry D. Brothers trally stable modes with nonlinear couplings to the unsta­ Ftaytheon E-Systems ble modes. If such neutral modes are present, the ampli­ Garland, TX tude equations for the unstable modes, i.e. normal forms on the unstable manifold, can have very different features. Richard Haberman We discuss several illustrative examples: the normal form Southern Methodist University for a Hopf/saddle node mode interaction, the Vlasov equa­ Dallas, TX tion from plasma physics, and a continuum model for the Kuramoto system of phase oscillators. The normal forms Abstracts 93

MS27 deformation, and the fact that the metric is Riemannian Slow Evolution Near Falllilies of Stable Equilibria is a consequence of a reciprocity theorem for the Stokes of Hallliltonian Systellls equations. The problem of slow evolution in Hamiltonian systems is Richard Montgomery formulated as a singularly perturbed initial-value prob­ UCSC, Santa Cruz, CA lem. The conditions for a family of equilibria to repre­ sent asymptotic solutions to the full evolution equations are given. These conditions break down at a critical point MS28 for orbits of the unperturbed system. Examples from fluid Self-Propulsion of Spherical Swillllllers Using Sur­ dynamics and astrophysics are given to motivate consid­ face Deforlllations eration of particular kinds of critical points and normal forms, and evolution through families containing these crit­ Strategies for swimming adopted by microorganisms must ical points is discussed. conform to the physical principles of self-propulsion at low Reynolds numbers. Here we present an analysis that re­ Norman R. Lebovitz lates the translational and rotational speeds to the surface University of Chicago, Chicago, IL motions of a swimming object, and, for swimming spheres, makes evident novel constraints on potential mechanisms for propulsion. We consider strategies for the swimming MS28 of a cyanobacterium, an organism whose motile mecha­ Discovering the Mysteries of Swilllllling Microor­ nism is unknown, by considering incompressible streaming ganisllls on the Beaches of Rio of the cell surface and oscillatory, tangential surface defor­ mations. Finally, the efficiency of swimming is discussed. Determining the mechanism for self propulsion is funda­ mental in the understanding of the relationship between Aravi Samuel motile bacteria and their environment. The fine structures Harvard, Cambridge, MA and detailed motions of many bacteria are too small to be observed under a light microscope. Clever experiments Howard Stone are ultimately necessary to determine the mechanism. We Harvard, Cambridge, MA present some mathematical models and predictions which help biologists in designing such experiments. Howard Berg Harvard, Cambridge, MA Kurt Ehlers L.N.C.C., Rio de Janeiro

MS29 On the Influence of Viscosity on Rielllann Solutions MS28 of Nonlinear Conservation Laws A Spectral Method for Stokes Flows and Applica­ tions to Microswilllllling In this talk I will show how existence, uniqueness and con­ struction of Riemann solutions are affected by the precise We present the spectral properties of the propulsion oper­ form of viscosity which is used to select shock waves ad­ ator P : tJ -> ft, relating velocity fields to surface force mitting a viscous profile. The results are obtained in the fields along the boundary of a domain (possibly multiply­ context of nonlinear 2 x 2 systems of conservation laws connected). Numerical methods based on the boundary that change type. The approach is based on a study of integral approach are discussed, with examples. For mi­ codimension-l bifurcations that distinguish between Lax croswimming applications, it is important to relate the shock waves with and without a profile. I will present eigenvalues .>..,." and '>"n to the curvature coefficients FTTtn • "generic" situations in which viscous Riemann solutions We explain our conjectures and how they are related to the differ from Lax solutions. question of optimizing the efficiency. Suncica Canie Jair Koiller, L.N.C.C. Department of Mathematics Laboratorio Nacional de Computacao Cientifica, Brazil Rio Iowa State University de Janeiro, Brazil Ames, IA 50011 Joaquin Delgado U.N.A.M., Mexico City, Mexico MS29 Marco Raupp An Application of Vectorfield Bifurcation to Con­ Laboratorio Nacional de Computacao Cientifica, Brazil Rio servation Laws that Change Type de Janeiro, Brazil Viscous perturbations are used in conservation law theory to validate shock admissibility criteria. In this context, the problem of determining which shocks are admissible re­ MS28 duces to finding heterocIinic orbits in a related dynamical Geollletry of Microswilllllling system. It is instructive to apply this method to systems which change type (are not hyperbolic for all states). One Shapere and Wilczek observed that low-Reynolds number example of such a system is a standard model for steady swimming can be described in terms of the holonomy of transonic flow; change of type also arises in models for un­ a certain connection (gauge field), the Stokes connection. steady flows. When a perturbation based on physical vis­ We show that this connection is in fact a "mechanical con­ cosity is applied to the steady transonic flow equations, the nection" :it can be defined solely in terms of the "physically dynamics are described by a codimension one bifurcation correct metric" on configuration space. The quadratic form near the sonic line. The standard admissibility condition is for this metric measures the power output of a given shape recovered in this case. However, a standard, and plausible, 94 Abstracts

choice for viscous perturbation of unsteady systems leads Stony Brook, NY to a variant of the codimension two Takens-Bogdanov vec­ torficld bifurcation. One conclusion is that admissibility conditions in this case are sensitive to the coefficients in MS30 the viscous perturbation. Low Frequency Variability in Ocean Circulation Barbara Lee Keyfitz Models Driven by Constant Forcing Department of Mathematics An example is given of low frequency variability in an ocean University of Houston circulation model driven by constant surface forcing. Par­ Houston, TX 77204-3476 ticular attention is paid to the role of the internal wave dy­ namics and the dependence of the variability on the mean state. MS29 Richard J. Greatbatch The Role of Polycycles in Nonuniqueness of Rie­ Department of Physics and Physical Oceanography mann Solutions Memorial University of Newfoundland We discuss how homo clinic orbits, two-saddle cycles, and St. John's, Newfoundland, Canada, AlB 3X7 three-saddle cycles in the planar dynamical systems asso­ ciated with traveling waves for viscous conservation laws lead to nonuniqueness of solutions of Riemann problems. MS30 The bifurcation analysis involves co dimension three phe­ Mathematical and Computational Issues in the nomena, including degenerate nilpotent singularities of the Double Gyre Model ~addle type, studied by Dumortier, Roussarie, Sotomayor, Zoladek, and Rousseau. We demonstrate that these poly­ Although the oceanic double-gyre model has been the sub­ cycles occur for a class of conservation laws important in ject of continuous study for over two decades, a number of petroleum reservoir simulation. basic mathematical and computational issues remain un­ resolved. I will review and discuss issues related to the Arthur Azevedo consistency and convergence of models with free- or partial­ Universidade de Brasilia slip boundary conditions, and will provide examples of the Brasilia, Brazil effect of alternate discretization strategies and alternate domain geometries on the structure of the western bound­ Dan Marchesin ary layer. I will conclude with a presentation of very high Instituto de Matematica Pura e Aplicada resolution results with special attention paid to resolution Rio de Janeiro, Brazil of the vorticity discontinuity of the free jet, and its influ­ ence on the existence of multiple stationary states in the Bradley Plohr system. Departments of Mathematics and of Applied Mathematics and Statistics John D. McCalpin University at Stony Brook Silicon Graphics Computer Systems Stony Brook, NY 11794-3651 Mountain View, CA

Kevin Zumbrun Indiana University MS30 Bloomington, IN Low-Frequency Variability of Wind-Driven Ocean Gyres We consider the equilibrium dynamics of a mid-latitude MS29 ocean in an enclosed basin driven by a steady surface wind Riemann Problem Solutions of Co dimensions 0 and stress. The Reynolds number Re associated with the hori­ 1 zontal (eddy) viscosity can be viewed as a control param­ A Riemann solution is structurally stable if any perturba­ eter for a bifurcation sequence that begins with a steady tion of the left state, right state, and flux function yields a circulation. The first bifurcation is due to a mesoscale in­ nearby Riemann solution with the same sequence of wave stability of this circulation, giving rise to the a broad-band types. For systems of two conservation laws in one space spectrum of the now familiar mesoscale eddies as Re in­ dimension, we describe a large class of structurally stable creases further. A further bifurcation occurs due to fluc­ Riemann solutions. We also describe how to study sys­ tuations with even longer periods and larger spatial scales tematically the codimension one Riemann solutions that (than the mesoscale), which can therefore be consider an separate them. They give rise to various folds, "joins," aspect of climate variability for the Earth. We analyze this and frontiers of the Riemann solution manifold. latter class of fluctuations, using Principal Orthogonal De­ compositions, and diagnose their essential dynamical bal­ Stephen Schecter ances. Department of Mathematics North Carolina State University Pavel S. Berloff Raleigh, NC 27695-8205 IGPP, UCLA, Los Angeles, CA 90095-1567

Dan Marchesin James C. McWilliams Instituto de Matematica Pura e Aplicada IGPP, UCLA, Los Angeles, CA 90095-1567 Rio de Janeiro, Brazil

Bradley Plohr MS30 University at Stony Brook Low Frequency Variability in the Reduced-Gravity Abstracts 95

Shallow Water Model survey will be presented of new (and old) attempts to over­ come this challenge. Century-scale simulations of the steady wind-forced dou­ ble gyre circulation using a reduced-gravity shallow water Larry Sirovich model reveal significant levels of low-frequency variability. Div. Appl. Math, Brown U. & Lab. Appl. Math, We will discuss various methods of analyzing the result­ CUNY /Mt Sinai ing short and noisy time series that take into account the different physical processes that are represented. Balu Nadiga MS31 Analysis and Control of Spatio-Temporal Chaos in Len Margolin Reaction-Diffusion Process Karhunen-Loeve decomposition is done on a chaotic spatio­ Darryl Holm temporal solution of a reaction-diffusion model exhibiting Los Alamos National Laboratory, Los Alamos, NM 87544 bursting. It is shown that unstable states can be main­ tained by performing fluctuations of the concentration at the boundaries, while monitoring the dynamics of the time MS31 series obtained by projecting the main Karhunen-Loeve New Measures of Coherence in Disordered and mode on the solution as the reaction-diffusion equation is Noisy Arrays of Coupled Phase Oscillators solved in time. We also discuss the bifurcations of a low­ dimensional model for the same reaction-diffusion process Arrays of coupled phase oscillators are used to describe the obtained by using a Galerkin projection of the dominant dynamics of systems arising in such diverse areas as elec­ Karhunen-Loeve modes back onto the nonlinear partial dif­ tronics, biology and chemistry. Of importance is whether ferential system. the oscillators are phase locked, their phase relationship, and the effects of oscillator disorder and noise. We have Ioana Triandaf applied bi-orthogonal decomposition methods to the data Science Applications International, Corp, McClean, VA resulting from the arrays and developed new measures to describe the dynamics. Compared to more traditional mea­ sures, in many cases our results show increase sensitivity MS32 to changes in the system dynamics. Stability of the Stationary States of the Elastic Rod Thomas W. Carr Model Under an Applied External Force that Rep­ Department of Mathematics resent Configurations of DNA in Living Cells Southern Methodist University Solutions for the stationary states of the elastic rod model Dallas, TX 75275-0156 under an externally applied bending force have been de­ rived. The solutions represent the helical (llnm fiber) and Ira B. Schwartz superhelical (30nm fiber) configurations of DNA. In vivo Special Project for Nonlinear Science the bending force arises from electrostatic interactions be­ Naval Research Laboratory tween the histone octamer and DNA; thus, electrostatic Washington, D.C. 20375 interactions that produce the bending forces have been de­ termined. The interactions are chosen so as to stabilize the llnm and 30nm stationary state solutions of the elas­ MS31 tic rod. Analysis of Extensive Chaos by the Karhunen­ Thomas Connor Bishop Loeve Decomposition and by A Hierarchy of U n­ University of California, Berkeley, CA stable Periodic Orbits An important challenge is to develop theoretical, and hope­ John E. Hearst fully practical, ways to characterize the rich spatiotem­ University of California, Berkeley, CA poral dynamics often found in large, nontransient, high­ dimensional nonequilibrium systems. I review recent work (in collaboration with Scott Zoldi) in which the Karhunen­ MS32 Loeve decomposition and the hierarchy of unstable periodic A Combined Wormlike-Chain and Bead Model for orbits have been investigated to determine what kinds of Dynamic Simulations of Long DNA insights they can provide about EXTENSIVE CHAOS, for which the attractor dimension grows in proportion to the A new model for simulating dynamic properties of long physical volume of the system. DNA is presented, combining features of both wormlike chain and bead models. Our goal is to use the model for Henry Greenside Langevin dynamics simulations of both linear and closed Duke University, circular DNA. The energy of the model chain includes Durham, NC stretching, bending, twisting and electrostatic components. Beads are associated with each vertex of the chain and specify hydrodynamic properties of the DNA. Careful pa­ MS31 rameterization of the model is presented. In addition, we Modeling The Dynamics of Wall Bounded Tur­ modified the second-order Brownian dynamics (BD) algo­ bulece rithm to make it more efficient for our model. Equilibrium properties obtained from our dynamic simulations are com­ Turbulent channel flow, which is homogeneous in two spa­ pared with results of Monte Carlo simulations. Very good tial dimensions, is· probably the simplest example of gen­ agreement is obtained for distributions of writhe and the uine turbulence. It is nevertheless too complex to be ad­ radius of gyration. Our BD results for the translational equately described low dimensional dynamical systems. A diffusion constant agree very well with the experimental 96 Abstracts

results as well. We also obtained very good agreement be- University of Maryland, College Park, MD tween corresponding rotational diffusion constant. Finally, an illustrative dynamic result on DNA slithering is prc- John H. Maddocks scnted. University of Maryland, College Park, MD Hongmei Jian New York University, New York, NY MS33 Alexander Vologodskii Propagation of Nonsoliton Pulses in Long Optically New York University, New York, NY Amplified Transmission Lines Current generation telecommunications systems require Tamar Schlick single channel data rates of 5-20 GB/s and may be as long New York University, New York, NY as 9000 km. There may be many channels at different The Howard Hughes Medical Institute wavelengths on a single fiber. The digital data may be encoded in either soliton or nonsoliton pulses. One thing all these systems have in common is that their success de­ MS32 pends critically on our understanding of how the interplay Modeling Stable Configurations of Protein-bound of dispersion and nonlinearity in the fiber determines the DNA Rings evolution of the pulses. The evolution of these signals is The precise geometry of protein-DNA complexes has been governed by the nonlinear Schrodinger equation. We will shown to be a significant factor in the proper function and discuss the results of recent experiments where nonsoliton regulation of the genetic machinery. However, the effect of pulses were propagated in a transmission line with periodic nucleosome formation, a packaging motif which entails the dispersion changes. superhelical wrapping of DNA about a core of eight histone Stephen Evangelides Jr. proteins, on the configuration of the rest of the DNA chain AT&T Research is not clear. Precise three-dimensional images of long DNA Holmdel, N J 07733 chains bound to histone-type proteins are difficult to ob­ tain by experimental means. Presently, to visualize DNA minichromosomes, constrained by the presence of nucle­ Nonlinear Optical Phenomena in Periodic Struc­ osomes, we have stochastically determined the minimum tures energy configurations based upon a library of equilibrium structures. Here, the behavior of the protein-free segments We consider optical waveguides that have a periodic spa­ of each DNA chain is likened to that of an elastic rod with tial modulation of the refractive index. The inclusion of equivalent end constraints. Our results suggest that these an intensity-dependent refractive index leads to such phe­ interactions may play a significant role in activating ge­ nomena as pulse compression and soliton propagation. For netic events by dramatically altering the configuration of active periodic structures, Le., those with gain, we find an the duplex through minor adjustments in the path of the instability that leads to very high frequency self-pulsations. DNA on the protein core, the shape of the core itself, and Herbert G. Winful the location of the complex along the chain contour. EECS Department Jennifer A. Martino. University of Michigan Rutgers University, Piscataway, NJ Ann Arbor, MI 48109

MS32 MS33 Stability in Continuum Models of DNA Minicircles Modelocked Fiber Laser Simulation and Modeling Stability results based on a constrained variational prin­ We discuss models of actively and passively modelocked ciple will be presented for elastic loops loaded with vary­ lasers for communication applications. Comparison be­ ing amounts of imposed twist. The underlying variational tween the models and experiment are developed and an­ structure is employed to predict exchanges of stability alytical results are given. based on the shape of the solution branch in a 'distin­ J. W. Haus and G. Shaulov guished bifurcation diagram', namely a plot of twist mo­ Physics Department ment vs. end angle. The shapes of the solution branches Rensselaer Polytechnic Institute in this distinguished diagram, and therefore the stability Troy, NY 12180-3590 properties of equilibria, depend upon the material parame­ J. Theimer ters of the rod. These stability results are used as part of a Rome Laboratory RLIOCPA study of DNA mini-circles using an elastic rod model with 25 Electronics Ave. experimentally motivated parameters for bending stiffness, Rome, NY 13441-4515 twisting stiffness, and the curved unstressed shape. Com­ putations with the boundary value problem solver AUTO, the visualization tool PCR, and a numerical implementa­ tion of an appropriate conjugate point test allow the de­ MS33 termination of the stability properties and index of each The Whitham Equations for Optical Communica­ cyclized equilibrium of the DNA. tions in NRZ Format Kathleen A. Rogers We present a model of an optical communication system University of Maryland, College Park, MD using high-bit-rate data transmission in the nonreturn-to­ zero (NRZ) format over transoceanic distances. The sys­ Robert S. Manning tem operates in the small group dispersion regime, and the model equation is given by the Whitham equations Abstracts 97

describing the slow modulation of multi-phase wavetrains Berkeley, CA 94702 for the (defocusing) nonlinear Schrodinger (NLS) equation. The model equation has a hyperbolic nature, and certain initial NRZ pulse develops a shock. We then show how MS34 one can regularize solutions by choosing an appropriate Crises: The Dynamics of Invariant Sets as a Pa­ Riemann surface where the Whitham equation is defined. rameter is Varied The present analysis may be interpreted as an alternative to the method of inverse scattering transformation for the "Basic sets" are sets that are invariant and have a dense NLS solitons. trajectory (and are not contained in a larger such set). They can include attractors and the Cantor sets of horse­ Yuji Kodama shoes and periodic orbits. This talk will describe how basic Department of Electronics and Information Systems sets change and collide and merge as parameters are varied. Osaka University This is joint work with Kathleen Alligood. 2-1 Yamade-Oke Suita Osaka 565, Japan James A. Yorke LP.S.T. University of Maryland College Park, MD 20742-2431 MS34 A Case Study of Topology Preserved in an Envi­ ronment with Noise: Fluid Flowing Past an Array of Cylinders MS35 Mathematical Models of Stochastic Ratchets Standard dynamical systems theory is based on the study of invariant sets. For area preserving models that incorpo­ We describe some simple mathematical models of directed rate noise, there are no bounded invariant sets. Our goal is motion resulting from the interplay of undirected random to study the fractal structure that exists even with noise. forces in an anisotropic environment. While some more The problem we investigate is fluid flow past two or more elaborate versions of these ideas are used to model molecu­ cylinders with small random disturbances. lar motors in subcellular and molecular biology, the essen­ tial concepts are also being developed for possible appli­ Judy A. Kennedy Dept of Math. Sci. cations in materials processing technology. In this talk we Univ. of Delaware will show how even some simple mechanisms can generate Newark DE complex, nonintuitive motions, and how their analysis can 19716-0001 present interesting mathematical challenges. Charles R. Doering University of Michigan, Ann Arbor MI MS34 Horseshoes and the Conley Index Spectrum Let f:X to X be a continuous map. The Conley Index is defined for an isolating neighborhood. The talk will discuss MS35 how and when the Conley Index can be used to conclude The Bacterial Flagellar Motor: A New Mechanism that some power of f can be represented by the full shift on for Energy Transduction two symbols. In other words that f exhibits the dynamics of a horseshoe. The bacterial flagellar motor is driven by a flux of ions between the cytoplasm and the periplasmic lumen. Here Konstantin Mischaikow we show how an electrostatic mechanism can convert this addressSchool of Math ion flux into a rotary torque. We demonstrate that with Georgia Tech. reasonable parameters, the model can reproduce many of Atlanta Georgia the experimental measurements. 30332-0001 Timothy C. Elston Center for Nonlinear Studies, MS-B258 Los Alamos National Laboratory MS34 Los Alamos, NM 87545 Stable Ergodicity and Stable Accessibility [email protected] Boltzmann's ergodic hypothesis - that ergodicity is a generic property of volume preserving dynamical systems - George Oster underlies statistical mechanics and much of physical think­ Department of Molecular and Cellular Biology ing. Yet, in 1954, Kolmogorov showed that Boltzmann University of California, Berkeley was wrong. The twist map is an area preserving diffeo­ Berkeley, CA 94720-3112 morphism of the plane that is not ergodic, nor is any area [email protected] preserving diffeomorphism that C 4 approximates it, due to the persistence of the invariant KAM circles. Nevertheless, it is our contention that Boltzmann was more right than MS35 wrong: in the joint presence of hyperbolicity and KAM dy­ Is the Speed of a Molecular Motor Influenced by namics, hyperbolicity can overwhelm the KAM dynamics its Elasticity? and produce robust statistics in the form of ergodicity. - joint work with Mike Shub Biomolecular motors typically have elastic components, and the question arises whether such elasticity is important Charles C. Pugh to the effective operation of the motor. This question will Mathematics Department be considered within the framework of Brownian ratchet Univ of California, Berkeley molecular motors. An idealized example will be discussed 98 Abstracts in which a perfect Brownian ratchet motor pulls a "heavy" of topological properties of the phase space it is possible load, i.e., a load with a much smaller diffusion coefficient to derive a renormalization group equation and express in than that of the motor itself. The load is connected to the an explicit form characteristic exponents for the probabil­ motor by an elastic tether, and a comparison is made of ity density distributions of the exit time and the Poincare the two limiting cases of an infinitely stiff connection and recurrence time. an infinitely compliant one. An analogous (but unsolved) G.M. Zaslavsky problem for the case of chromosome transport will also be New York University, New-York, NY briefly discussed. Charles S. Peskin Courant Institute of Mathematical Sciences MS37 New York University (New York, NY 10012) Bifurcations to Periodic Solutions in Diode Lasers ([email protected]) Subject to Injection For a diode laser with external optical injection, we derive a third- order differential equation for the phase of the field MS36 using asymptotic arguments. The bifurcations and isolated Lagrangian and Eulerian transport: Are they re­ branches of time-periodic solutions we find, are compared lated? with numerical results and experiments. In experiments with fluid flows one often observes the mo­ A. Gavrielides, V.Kovanis tion of extended Eulerian structures, e.g., jets, eddies, or Nonlinear Optics Center vortices. One has the general intuition that the trans­ Phillips Laborotry PL/LIDN port of Eulerian structures is strongly related to particle 3350 Aberdeen Ave SE transport. In this talk I describe the geometry of two­ Kirtland AFB, NM 87117-5776 dimensional Eulerian transport and show that, in general, T. Erneux, G. Lythe there is no relationship between Eulerian and Lagrangian Optique Nonlineaire Theorique transport. However, there exists such a relationship in the Universite Libre de Bruxelles Campus Plaine CP 231 case of adiabatic flows. I discuss the application of these BId du Triomphe results to eddy-jet interactions. B-1050 Brussels George Haller Belgium Brown University, Providence, RI

MS37 MS36 Controlling Temporal and Spatio-Temporal Dy­ Transport in Quadratic Volume Preserving Maps namics in Semiconductor Lasers by Delayed Op­ By analogy with the Henon map of the plane, we find the tical Feedback normal form for quadratic volume preserving maps in 'RfJ. Delayed optical feedback may cause (spatio-) temporal in­ We investigate the set of bounded orbits, using the tran­ stabilities, but also allows a succesful control of coher­ sit time decomposition, following our work in Meiss, J. D. ent regimes as well as stabilization of periodic and quasi­ (1996). " Average Exit Time for Volume Preserving Maps." periodic regimes in otherwise temporally and/or spatially Chaos in press. chaotic semiconductor lasers. J.D. Meiss Christian Simmendinger and Ortwin Hess University of Colorado, Boulder, CO Theoretical Semiconductor Quantum Electronics Institute of Technical Physics, DLR Pfaffenwaldring 38-40 MS36 D-70569 Stuttgart Degeneracies, Instabilities and Transport Germany In many applications underlying symmetries cause strong degeneracies which are only slightly broken in the real set­ ting. These degeneracies lead to strong instabilities even MS37 in low dimensional systems in which usually partial barri­ Control of Optical Turbulence in High Brightness ers/barriers slow down/inhibit transport. Thus, the study Semiconductor Lasers of these singular mechanisms of instabilities is fundamental In the contexxt of a complex Swift-Hohenberg equation we for understanding transport in real systems. Examples of discuss a robust control strategy, combining spatially and this phenomena arising in an atmospherical model and in a spectrally filtered optical feedback, that simultaneously al­ Plasma model (a model describing the motion of a charged lows temporal and beam steering control of the emmitted particle) will be presented. light. Vered Rom-Kedar J.V. Moloney and David Hochheiser Weizmann institute of Science, Rehovot, Israel Arizona Center for Mathematical Sciences Department of Mathematics University of Arizona MS36 Tucson, AZ 85721 Scaling Properties of Distributions of Exit Time and Poincare Recurrences For the Hamiltonian dynamical systems with chaotic dy­ MS37 namics transport is anomalous in general. On the basis Bifurcation Analysis of a Phase-Amplitude Model Abstracts 99

of an Injected Diode Laser results arc illustrated by analytical and numerical exam­ ples. Relaxation oscillations in an optically injected laser can be studied by means of a phase-amplitude equations, which Ljupco Kocarev define a vector field on a half-cylinder. We use bifurcation Faculty of Electrical Engineering, St. Cyril and Methodius theory to learn about its dynamics with an emphasis on University possible multi-stable regimes. Skopje, PO Box 574, Republic of Macedonia Bernd Krauskopf, Daan Lenstra, Wim van der Graaf Bernd Krauskopf Dept. of Physics and Astronomy MS38 Vrije Universiteit Multiplexing Chaotic Signals using Synchroniza­ De Boelelaan 1081 tion The Netherlands Chaotic systems typically generate broadband signals. To achieve synchronization between two chaotic systems, a wide communication channel is usually required. However MS38 in real-world communication systems, channel bandwidth Synchronizing Hyperchaotic Volume-Preserving is limited. In this talk, we discuss various methods to in­ Map Circuits crease the channel capacity via simultaneous transmission of multiple chaotic signals through the channel. Exploiting Most plans for using chaotic signals for communication as­ the stability properties of chaotic attractors allows for mul­ sume that the chaotic signal will be used in a similar man­ tiplexing chaotic signals either in frequency/time domain ner to the way pseudo- random binary signals are already or directly in the state space without destroying chaotic used. Many continuous dissipative chaotic systems have synchronization. We demonstrate the principal possibility power spectra with strong features that make them poor of multiplexing chaotic signals using synchronization both candidates for noise generators. We show a simple 3-D for iterated maps and ordinary differential equations. volume- preserving map circuit that has a flat power spec­ trum and a near delta function autocorrelation. The cir­ Lev S. Tsimring cuit is based on a linear map, so it is easy to design both Institute for Nonlinear Sciences, University of California the circuit and synchronizing subsystems. A transforma­ San Diego, La Jolla, CA 92093-0402, USA tion technique is used to create a synchronizing response system, since no natural subsystem exists. N.F.Rulkov, M.M.Sushchik, H.D.I.Abarbanel Institute for Nonlinear Sciences, University of California Thomas L. Carroll San Diego, La Jolla, CA 92093-0402, USA Naval Research Laboratory, Code 6343 Washington DC 20375, USA

MS39 Transition from Slow Motion to Pinning in Lattice MS38 Equations Synchronizing Hyperchaos for Communications Certain lattice differential equations from materials science Recent work pioneered by Pecora and Carroll has consid­ exhibit slow motion of phase boundaries across many lat­ ered the possibility of exploiting the phenomenon of chaos tice sites when the ratio of intersite distance to interaction synchronization to achieve secure communication. But, length is sufficiently small but undergo pinning of those theoretical and experimental models studied thus far have transition layers when this ratio is increased. Analysis of been limited to low dimensional systems with one posi­ the equilibria of these equations shows that the way in tive Lyapunov exponent. Consequently, messages masked which this transition takes place may be surprisingly com­ by such simple chaotic processes are shown to be eas­ plex. We will present both analytical and numerical results. ily extracted in some cases. In this contribution we in­ vestigate high dimensional implementation of the Pecora­ Christopher P. Grant Carroll synchronization paradigm. In particular, in regard Brigham Young University, Provo, UT to potential applications to communication, we address the question of whether by transmitting just one scalar signal one can achieve synchronism in chaotic systems with two MS39 or more positive Lyapunov exponents (hyperchaos). Us­ Modeling Biological Systems by means of Nonlin­ ing both numerical and analytical examples we argue that ear Electronic Circuits under very general conditions the answer to the above ques­ tion is affirmative. We present some results obtained in the framework of reaction-diffusion systems that may be usefull for under­ Mingzhou Ding standing processes taking place in biological systems. In Center for Complex Systems, Florida Atlantic University particular three mechanisms will be described: i) 1D model Boca Raton FL 33431, USA of the cardiac pulse propagation [1], ii) synchronization waves in 1D and 2D arrays of driven chaotic cells [1, 1] (related to propagation of information in nerve cells) and MS38 iii) description of 1D systems that exhibit periodic or On Generalized Synchronization chaotic behavior depending on an external perturbation (phenomenon similar to that observed during epileptic Conditions for the occurrence of generalized synchroniza­ episodes) [1]. All the results here presented were obtained tion of unidirectionally coupled dynamical systems are using as a basic cell a Chua's circuit that has been shown given in terms of asymptotic stability. The relation be­ to exhibit a great variety of bifurcation and chaotic phe- tween generalized synchronization, predictability, and eqi­ valence of dynamical systems is discussed. The theoretical 100 Abstracts

nomena [?]. Ui,j E {-l,a,l} at each (i,j) E :;z2. This in turn leads to A.P. Munuzuri and L.a. Chua explicit combinatorial criteria for the existence and stabil­ Department of EECS, ity of these equilibria. Rigorous upper and lower bounds University of Berkeley, USA on the spatial entropy of such stable mosaic solutions are obtained for a wide range of the coupling coefficients. In addition, a numerical method for estimating the spatial entropy is proposed and numerical results are presented. MS39 Traveling Waves in Lattice Dynamical Systems John W. Cahn NIST, Gaithersburg, MD The talk will discuss the existence, stability, and structure of traveling wave solutions in lattice dynamical systems, Shui-Nee Chow in particular, in coupled map lattices and lattice ordinary Georgia Institute of Technology, Atlanta, GA differential equations. The transition from standing waves to traveling waves will also be considered. John Mallet-Paret Wenxian Shen Brown University, Providence, RI Auburn University, Auburn, AL Erik S. Van Vleck Colorado School of Mines, Golden, CO MS39 Information Propagation, Pattern Formation and Reaction-Diffusion with Cellular Neural Networks. MS40 From High Dimensional Chaos to Stable Periodic Cellular Neural Networks are arrays of simple nonlinear Orbits: the Structure of Parameter Space dynamical systems in which each cell is connected to its nearest neighbors only. The local range of the connections Regions in the parameter space of chaotic systems that cor­ is needed for their hardware implementation as analog cir­ respond to stable behavior are often referred to as windows. cuits. Their applications mainly lie in image preprocessing We elucidate the occurrence of such regions in higher di­ tasks, for which it is essential to understand how the infor­ mensional chaotic systems. We describe the fundamental mation (image), introduced as initial condition, propagates structure of these windows, and also indicate under what in the network: either by local diffusion between neigh­ circumstances one can expect to find them. These results boring cells only, or global propagation through the entire are applicable to systems that exhibit several positive Lya­ array. We will study the dynamics of the network operat­ punov exponents, and are of importance to both the the­ ing in both modes, and in particular we will look for the oretical and the experimental understanding of dynamical type and the number of stable steady state solutions, using systems. exact mathematical methods whenever we can, or else ap­ Ernest Barreto proximate and numerical engineering approaches. We will Brian R. Hunt also present some examples of biological pattern formation Celso Grebogi similar to the ones obtained with reaction-diffusion PDEs, James A. Yorke and show how the coupling between cells is computed to University of Maryland, College Park, MD create such patterns. Patrick Thiran Swiss Federal Institute of Technology at Lausanne MS40 Switzerland Optimal Periodic Orbits of Chaotic Systems: Con­ Gianluca Setti trol and bifurcations University of Bologna Italy Invariant sets embedded in a chaotic attractor can gener­ ate time averages that differ from the average generated by typical orbits on the attractor. Motivated by two different topics (namely, controlling chaos and riddled basins of at­ MS39 traction), we consider the question of which invariant set Traveling Waves, Equilibria, and the Computation yields the largest (optimal) value of an average of a given of Spatial Entropy for Spatially Discrete Reaction smooth function of the system state. We present numerical Diffusion Equations evidence and analysis which indicate that the optimal aver­ In this talk recent work on the dynamics of lattice differen­ age is typically achieved by a low period unstable periodic tial equations, u = - feu), arranged on the sites of a spatial orbit embedded in the chaotic attractor. lattice, is surveyed. In particular, results on propagation Brian R. Hunt failure and lattice induced anisotropy for traveling wave or Edward Ott plane wave solutions in higher space dimensions spatially University of Maryland, College Park, MD discrete bistable reaction-diffusion systems are considered. In addition, analysis of and spatial chaos in the equilib­ rium states of spatially discrete reaction-diffusion systems are discussed. We study stable equilibria for such systems, MS40 from the point of view of pattern formation and spatial Periodic Shadowing chaos, where these terms mean that the spatial entropy Shadowing techniques will be used to establish the exis­ of the set of stable equilibria is zero, respectively, posi­ tence of true periodic orbits near numerically computed tive. In particular, for an idealized class of nonlinearities pseudo periodic orbits in chaotic dynamical systems. The f corresponding to a "double obstacle" at u = ±l with computational techniques will be illustrated to exhibit long feu) = ,U in between, it is natural to consider "mosaic solutions," namely equilibria which assume only the values Abstracts 101

unstable periodic orbits in specific equations. Gloria Sanchez Huseyin Kocak University of Merida, Venezuela University of Miami, Coral Gables, Fl Naomi Leonard Princeton University, Princeton, NJ MS40 Unstable Dimension Variability: A Source of Non­ hyperbolicity in Chaotic Systems MS41 The (non)hyperbolicity of a chaotic set has profound im­ The Phase for the Spatial Three-body Problem The plications for the dynamics on the set. A familiar mecha­ three-body problem concerns a dynamical system whose nism causing nonhyperbolicity is the tangency of the stable configuration space is the space of triangles. If the triangle and unstable manifolds at points on the chaotic set. We formed at time T is similar to the triangle at time 0, what is investigate a different situation, first considered by Abra­ the rigid rotation which takes one triangle to the other,up ham and Smale in 1970, in which the dimensions of the to scale? The answer is well-known for the planar problem. unstable (and stable) tangent spaces are not constant over We derive the answer for the spatial problem and fit it into the chaotic set, thus causing nonhyperbolicity. A simple the current Berry phase formalism. two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed. Richard Montgomery UC Santa Cruz, Santa Cruz, CA Eric J. Kostelich Arizona State University, Tempe, AZ

Ittai Kan MS41 George Mason University Vortex Dynamics and the Geometric Phase C. Grebogi, E. Ott and J. A. Yorke, We will focus on applications of the Hannay-Berry (H-B) University of Maryland, College Park, MD phase to vortex dynamics problems in 2-D incompress­ ible ideal fluid flows. These flows include the three vor­ tex problem, a vortex and particle in a circular domain, MS41 and a point vortex model for shear layer roll-up and pair­ Coriolis Forces as an SO(3) Gauge Field: Applica­ ing. Asymptotic methods which can be used to derive long tions to Molecular Dynamics time stretching rates for passive interfaces in these and other flowfields will be described and related to the geo­ Coriolis forces in the n-body problem are described by a non-Abelian SO(3) gauge field on the shape space of the metric phase. system. This talk focuses on the new, geometrical perspec­ Paul K. Newton tive which this point of view brings to traditional problems University of Southern California, Los Angeles, CA in molecular dynamics. In the problem of small vibra­ tions about an equilibrium, a natural choice of coordinates and gauge are Riemann normal coordinates and Poincare MS42 gauge, which turn out to be identical with the traditional DNA Sequence Matching and Nonequilibrium Dy­ Eckart conventions. Other problems, such as the Wat­ namics with Multiplicative Noise sonian term in the quantum Hamiltonian, dynamics near collinear configurations, Yang-Mills field equations for the Alignment algorithms used to search distant homologies Coriolis field, and the 4-body kinetic energy operator, will among DNA (and protein) sequences are shown to belong be discussed. to the same universality class as diffusive processes with correlated multiplicative noise. Analysis based on the dy­ Robert G. Littlejohn namics analogy enables one to relate optimal choices of UC Berkeley, Berkeley, CA 94720 penalty parameters to the underlying evolutionary process responsible for sequence divergence. The popular "local" alignment method of Smith and Waterman can be cast in MS41 the same formalism, by introducing a constant source to Geometric Phases and the Stabilization of Balance the diffusive process. Systems Terence Hwa and Miguel A. Munoz We discuss stabilization of mechanical systems with sym­ Physics Department metry (rigid body with an internal rotor, the inverted University of California at San Diego pendulum and underwater vehicles). Starting from a La­ La Jolla, CA grangian with an unstable relative equilibrium, we intro­ duce a modified Lagrangian whose Euler-Lagrange equa­ Dirk Drasdo and Michael Lassig tions differ from the given ones by terms that can be iden­ Max-Planck Institute tified with control forces. Equilibria of the modified La­ Teltow, Germany grangian are then analyzed using the energy-momentum method. When the method is used for tracking, it is nec­ essary to take the geometric phase into account. MS42 Jerrold E. Marsden RNA Virus Evolution: Fluctuation-Driven Motion Caltech, Pasadena, CA on a Smooth Landscape Recent experiments on the evolution of RNA viruses have Anthony Bloch shown several interesting dynamical features, all of which University of Michigan, Ann Arbor, MI can be simply interpreted as being due to the process tak- 102 Abstracts ing place on a smooth fitness landscape. Population dy­ to families of periodic orbits. This short-path property of namics on such a landscape are surprising; independent a regular system results in a parametrically larger suscep­ of the population size, one cannot use mean-field theory tibility than for a chaotic system. (the Eigen-Schuster equations) since the leading edge of Harold U. Baranger the population (the most fit variants) always come in small Bell Labs, Lucent Technologies numbers. We present here an exact asymptotic (in time) 700 Mountain Ave., room 1D-230 solution of a simple model of such a process and compare Murray Hill, NJ 07974-0636 these results to simpler heurisitic methods. [email protected] Herbert Levine, Lev Tsimring and Douglas Ridgway Institute for Nonlinear Science, Univ. of Calif., San Diego La Jolla, CA 92093 MS43 Chaotic Dynamics and Quantum Level Statistics David Kessler Dept. of Physics Gutzwiller's trace formula relates quantum energy levels Bar-Han Univ. to classical periodic orbits. It can therefore be used to Ramat-Gan, Israel describe the correlations between quantum levels in terms of the statistical properties of the classical dynamics in a given system. The talk will be a review of some aspects of this theory, and the links with the corresponding cor­ MS42 relations between the zeros of the Riemann zeta function. Why Do Proteins Look Like Proteins?

Protein structures in nature often exhibit a high degree of Jonathan P. Keating regularity (secondary structures, tertiary symmetries, etc.) School of Mathematics, University of Bristol, Bristol BS8 absent in random compact conformations. We demonstrate ITW and BRIMS, Hewlett-Packard Labs, Bristol BS12 in a simple lattice model of protein folding that structural 6QZ regularities are related to high designability, evolutionary Bristol, UK stability, and thermodynamic stability. Hao Li, Chao Tang, Ned Win green J [email protected] NEC Research Institute 4 Independence Way Princeton, N J 08540 MS43 Experimental Studies of Quantum Chaos in Semi­ Robert Helling conductor Microstructures University of Hamburg Hamburg, Germany The experimental aspects of mesoscopic physics will be discussed, including some details of fabricated and mea­ surement at low temperature, as well as the strenghts and limitations of this experimental system. Recent measure­ MS42 ments of electron transport through both open and clas­ Dynamics of Flocking: How Birds Fly Together sically isolated chaotic quantum dots will be compared to We propose a non-equilibrium continuum dynamical model semiclassical and random matrix theory results. for the collective motion of large groups of biological or­ Charles M. Marcus ganisms (e.g., flocks of birds, slime molds, etc.) Our model Department of Physics becomes highly non-trivial, and different from the equilib­ Stanford University rium model, for dimension d < de = 4; nonetheless, we are Stanford, CA 94305-4060 able to determine its scaling exponents exactly in d = 2, [email protected] and show that, unlike equilibrium systems, our model ex­ hibits a broken continuous symmetry even in d = 2.

Yuhai Tu MS44 IBM Watson Research, P. O. Box 218, Yorktown Heights, Lyapunov Spectra of Various Model Systems in NY10598 Nonequilibrium Steady States John Toner We discuss our recent numerical results for the Lyapunov Department of Physics, University of Oregon, Eugene, Ore­ spectra of the driven Lorentz gas, and for perturbed hard­ gon 97403 disk and hard-sphere systems in two and three dimensions, respectively. To achieve a steady state various computer thermostats and time-reversible motion equations are used. Both mechanical (color conductivity) and thermal (shear MS43 viscosity) perturbations are considered. The ergodicity and Quantum Interference in Microstructures: Signa­ macroscopic irreversibility of these systems are discussed. tures of Chaos Harald A. Posch The nature of the classical dynamics of electrons in a mi­ University of Vienna, Vienna, Austria crostructure can have a large effect on its quantum prop­ erties: in two examples we use semiclassical techniques Christoph Dellago to show that regular dynamics gives qualitatively differ­ University of Vienna, Vienna, Austria ent results from chaotic dynamics. First, the shape of the low-field magnetoresistance differs because of the long-time dynamics, an effect observed experimentally. Second, the magnetic susceptibility of interacting electrons is sensitive 103 Abstracts

MS44 TMorique Transport and Entropy Production in Low­ Campus Plaine, C.P. 231,1050 Bruxelles, Belgium Dimensional Chaotic Systems Phone: +32 2 650-5819 Fax: +32 2 650-5824 We invoke concepts from the theory of dynamical systems e-mail: [email protected] to discuss the production of thermodynamic entropy from a microscopic point of view. For closed systems subjected A. Hohl, A. Gavrielides and V. Kovanis to an external field, the necessity for thermostating leads to Nonlinear Optics Center, Phillips Laboratory PL/LIDN the well-known result that the entropy production is given 3350 Aberdeen Ave. SE, Kirtland AFB, NM 87117-5776 by the sum of all averaged Lyapunov exponents governing the chaotic motion. When opening the system to allow for coupling to a particle reservoir, one has to deal with open systems exhibiting transient chaos. In this case the MS45 concept of the conditional probability density is used to Entrainment of Coupled Solid State Lasers define an entropy. The production of entropy turns out to A discussion of the dynamics and control of coupled solid be proportional to the difference of the escape rate from state lasers with an injected field will be presented. This the open system and the sum of all averaged Lyapunov will focus on the basic equations and entrainment mecha­ exponents on the chaotic saddle governing the dynamics. nism, and the use of phase models to describe the dynamics The relation between the closed and open system approach in certain regimes. Predictions, including the strongly non­ will be illustrated by the example of a multibaker map. monotonic dependence of the output power on the drive Tamas Tel amplitude below the entrainment threshold, and the sup­ W. Breymann and J. Vollmer pression of amplitude instabilities by the injected field, will Eotvos University, Budapest, Hungary be discussed. T.A.Brian Kennedy,Y. Braiman, A. Khibnik and K. Wiesenfeld MS44 School of Physics Lyapunov Exponents and Irreversible Entropy Pro­ Georgia Tech duction in Low Density Many-Particle Systems Atlanta GA 30332-0430

For dilute many-particle systems kinetic theory methods Phone: 404-894-5221 can be used to calculate characteristic dynamical proper­ Fax: 404-894-9958 ties, such as Lyapunov exponents and Kolmogorov-Sinai email: [email protected] entropies. The change in these quantities when the sys­ tem is brought out of equilibrium into a stationary or quasi-stationary state can be related to transport prop­ erties (e.g. diffusion coefficients) of the system. Especially, MS45 for a system subjected to a constant driving field and a Design of Laser Resonators for Enhanced Spatial Gaussian thermostat keeping the kinetic energy constant, Coherence and Mode Control the sum of all Lyapunov exponents is directly proportional Most laser resonators employ conventional lenses and mir­ to the macroscopic irreversible entropy production. A cor­ rors to establish a fundamental laser mode. We explore the rect identification requires a coarse-graining of the phase larger design space made available by utilizing diffractive space density as it keeps concentrating on a fractal attrac­ . optical elements as end mirrors and intracavity elements tor. Similarly, for systems with open boundaries one finds with complex reflectivities. Coherence is established across a positive irreversible entropy production related to coarse semiconductor laser arrays by designing external cavity res­ graining of the phase space density on the fractal repeller, onators that establish a common array mode. Mode con­ i.e. the set of trajectories that remain trapped within the trol of high Fresnel number solid-state lasers is enhanced system forever. by intra-cavity phase plates. The mathematical techniques Henk van Beijeren used to design laser resonators for specific mode shapes and Universiteit Utrecht optimum modal discrimination are reviewed. Utrecht, The Netherlands James R. Leger Department of Electrical Engineering J. Robert Dorfman University of Minnesota University of Maryland Minneapolis, Minnesota 55455 College Park, MD Phone: (612) 625-0838 Fax: (612) 625-4583 MS45 e-mail: [email protected] Two Coupled Lasers Steady and oscillatory forms of synchronization of two coupled lasers are analyzed in detail. The mathematical MS45 problem is motivated by recent experiments on coupled Nonlinear Dynamics of Semiconductor Laser Ar- solid state lasers and coupled semiconductor lasers. We rays investigate the bifurcation diagram of the steady and pe­ Semiconductor laser arrays are of great interest as high riodic states analytically and numerically. We control the power coherent sources. They are also of fundamental strength of the mutual coupling, the pump level of the in­ interest because they exhibit complex spatiotemporal dy­ dividuallasers, and the frequency detuning between them. namics that can be understood in terms of simple coupled­ T. Erneux mode theories or more general partial differential equation Universite Libre de Bruxelles, Optique Nonlineaire models. We review the theoretical approaches to laser ar- 104 Abstracts ray dynamics and present a survey of relevant experimental MS46 results. Resonances and Radiation Damping in Conser­ Herbert G. Winful vative Nonlinear Waves Consider a linear wave EECS Department or Schrodinger equation which supports (stable) time­ University of Michigan periodic and spatially localized solutions (bound states). Ann Arbor, MI 48109 This talk concerns recent work on the character of such bound states under nonlinear and conservative perturba­ Phone: (313) 747-1804 tions in the dynamics. Some bound states may not persist Fax: (313) 763-1503 under perturbations and may decay due to nonlinear res­ e-mail: [email protected] onant interactions with radiation modes. Such states may decay very slowly as t --t 00 and are therefore metastable states. A system is studied in which the excited states de­ MS46 cay and a ground state is selected by the evolution. This On Oscillations of Solutions of Randomly Forced­ is joint work with Avy Soffer. Damped NLS Equations We consider the small-dispersion and small-diffusion non­ linear Schrodinger equation Michael 1. Weinstein ~ De,rartment of Mathematics 1 2: {j := y {jr + {j~ 'trMversity of Michigan, Ann Arbor where the space-variable x belongs to the unit n-cube (n :s:: 3) and u satisfies Dirichlet boundary conditions. Assuming that the force (is a zero-meanvalue random field, smooth in MS47 x and stationary in t with decaying correlations, we prove Targeting in Soft Chaotic Hamiltonian Systems that the -norms in x with m 3 of solutions u, averaged em 2: The problem of directing a trajectory of a chaotic dynami­ in ensemble and locally averaged in time, are larger than cal system to a target has been previously considered, and {j-Km, K, ~ 1/5. This means that the length-scale of a it has been shown that chaos allows targeting using only solution u decays with {j as its positive degree (at least, as small controls. In this paper we consider targeting in a (jK) and - in a sense - proves existence of turbulence for Hamiltonian system which has a mixed phase space con­ this equation. sisting of chaotic regions and KAM surfaces. The orbit Sergei B. Kuksin can be guided through regions of strong chaos relatively Steklov Mathematical Institute, Moscow quickly. Controlled motion through KAM regions may be much slower. We discuss targeting strategies under these circumstances. MS46 Christian G. Schroer and Edward Ott Self-Focusing of the Nonlinear Schrodinger Equa­ University of Maryland tion and itsBbehavior Under Small Perturbation College Park, MD The nonlinear Schrodinger equation has complicated self­ focusing behavior in the critical case (cubic nonlinearity and two space dimensions) and the modification of this be­ MS47 havior when external perturbations are present is of great Controllable Targets for use with Chaotic Con­ interest in many applications. We will examine in detail trollers the mathematical problems that arise and present some ap­ A nonlinear system with a chaotic attractor produces ran­ plications, such as the effect of small time dispersion. This dom like motion on the attractor. This observation leads is joint work with Gadi Fibich. to a very simple "Chaotic control" algorithm for bring­ George Papanicolaou ing nonlinear systems to a fixed point contained within a Department of Mathematics "controllable target". A novel approach for determining Stanford University controllable targets for discrete systems is presented. Two discrete control problems are examined. The first example is the well known Henon map and the second example is MS46 the bouncing ball. The method is also shown to be ap­ Invariant Manifolds for a Class of Dispersive, plicable to controlling a continuous double link pendulum Hamiltonian, Partial Differential Equations system. A video of the ball system and the double pendu­ lum system under control will be shown. We construct an invariant manifold of periodic orbits for a class of nonlinear Schrodinger equations. Using standard Thomas L. Vincent ideas of the theory of center manifolds, we rederive the re­ University of Arizona sults of Soffer and Weinstein on the large-time asymptotics Tucson Arizona 85721 of small solutions (scattering theory). Claude-Alain Pillet Departement de Physique Theorique MS47 Universite de Gen(we Geneva, Switzerland Limits to Deterministic Modeling Some systems have an attractor with two periodic orbits Clarence Eugene Wayne having different numbers of unstable directions. Such or­ Department of Mathematics bits typically do not have the shadowing property so nu­ The Pennsylvania State University merical orbits do not represent true orbits. The talk is University Park, PA based on joint work with Leon Poon, Celso Grebogi, and Abstracts 105

Tim Sauer. Department of Mathematics, Arizona State University, Tempe, AZ James A. Yorke l.P.S.T. University of Maryland College Park, MD 20742-2431 MS48 Noise-Controlled Dynamics in Normal Forms Additive noise of very small amplitude qualitatively MS47 changes the behaviour found in truncated codimension-2 A Chaotically Forced Pendulum: Ship Cranes at normal form equations. Here we analyse the equations de­ Sea scribing the interaction of a Hopf bifurcation with another simple bifurcation. The effect of the noise is to postpone We present a control strategy for mitigating the pendu­ a global (heteroclinic) bifurcation; instead of going to in­ lation of the crane load during open sea transfer opera­ finity, trajectories follow noisily periodic orbits with well­ tions. This is realized by adding a pulley-brake system to defined mean amplitude and distribution about the mean. the current crane configurations. Numerical simulations show that the braking is very effective in controlling load Grant Lythe pendulations induced by representative ship roll motions. Center for Nonlinear Studies, MS-B258 Los Alamos Na­ Furthermore, the amount of braking is a desirable control tional Laboratory, NM variable for various control algorithms. Steve Tobias Guohui Yuan, Brian Hunt, Celso Grebogi, Edward Ott, JILA, University of Colorado at Boulder, CO James Yorke University of Maryland College Park, MD MS48 Eric Kostelich Noisy Cycles, the Sequel Arizona State University In an earlier paper (Stone and Holmes 1990) we reported Tempe, AZ on the effect of small amplitude noise on the cycle time of a trajectory near an attracting hetero/homoclinic orbit. In this talk I will generalize these results t.o systems to with MS48 heteroclinic connections between more complicated limit A Beteroclinic Network with Noise sets, such as limit cycles or chaotic limit sets. Ways to implement these results in an experimental setting will be In a 1994 paper, we described a heteroclinic network in discussed. which delicate types of attracting behaviour could be seen. For example, it was shown that two cycles in the net­ Emily F. Stone work could attract significant sets of initial conditions with­ Utah State University, Logan, UT out either cycle being asymptotically stable or essentially asymptotically stable. This talk describes recent work on Dieter Armbruster the effect of additive noise on the attractivity properties of Arizona State University, Tempe, AZ the heteroclinic network. Vivien Kirk The University of Auckland, Auckland, New Zealand MS49 Smale Mary Silber Horseshoes in Perturbed Nonlinear Schroedinger Northwestern University, Evanston, IL Equations The existence of Smale horseshoes and symbolic dynam­ ics is established in the neighborhood of a symmetric pair MS48 of homoclinic orbits in the perturbed NLS equation. More Probability Densities for Qualitatively Modified specifically, a list of compact invariant Cantor sets are con­ Dynamics structed through a study on the Conley-Moser conditions. The Poincare map restricted to each of the Cantor sets, is The effect of small noise in systems with slow-fast dynam­ topologically conjugate to the shift automorphism on four ics can be quite dramatic. For example, chaotic behavior of the deterministic system is replaced by noisily periodic symbols. This gives rise to deterministic chaos which offers behavior, and delay and memory effects can be eliminated. an interpretation of the numerical observation on the per­ A new global description of these dynamics can be given in turbed NLS equation: The chaotic center-wing jumping. terms of the probability density function. We give both an­ Charles Li alytical approximations and numerical solutions for these Department of Mathematics probability densities in several areas of application, includ­ Massachusettes Institut.e of Technology ing resonant interactions, nerve membrane excitability, and Cambridge, MA, 02139 laser dynamics. Rachel Kuske Tufts University, Medford, MA 02155 MS49 The Fate of Traveling Waves of the Nonlin­ George Papanicolaou ear Schrodinger equation under Ginzburg-Landau Department of Mathematics, Stanford University, CA Type Perturbations The nonlinear Schrodinger (NLS) equation is a famous in­ Steve Baer tegrable pde with many traveling wave solutions. The com- 106 Abstracts

plex Ginzburg-Landau (CGL) equation can be viewed as MS50 a dissipatively perturbed NLS equation, and is a generic Standing Waves, Heteroclinic Connections, and envelope equation derived in many applications. I will give Parity Breaking in Bifurcations from Ordered a nearly complete account of what happens to all the pe­ States of Cellular Flames riodic and solitary traveling waves of NLS type equations Cellular flames form ordered patterns of concentric rings when they arc perturbed with CGL type perturbations. which bifurcate to dynamical states. Three examples of dy­ C. David Levermore, Gustavo Cruz-Pacheco namical states will be discussed: rotating states exhibiting and Ben Luce asymmetric cells characteristic of parity-breaking bifurca­ MSB258 Los Alamos National Laboratory, tions; standing-wave states corresponding to an oscillation Los Alamos, NM 87545 of an N-fold pattern which rotates by 21'i/N on each half cycle; and intermittently ordered states in which a pattern of cells appears and disappears at irregular intervals corre­ sponding to a network of heteroclinic connections among MS49 Homoclinic Orbits in the Maxwell-Bloch Equations unstable equilibria. of Nonlinear Laser Optics Michael Gorman This talk will describe some of the spatio-temporal dy­ University of Houston namics of a pumped ring-cavity laser, which is modeled by Houston, TX the Maxwell-Bloch partial differential equations with spa­ tially periodic boundary conditions. Numerical simulations Kay Robbins of this spatially dependent system show both regular and University of Texas at San Antonio chaotic behavior of the solutions, depending on the values San Antonio, TX of the parameters, which are the cavity length, pumping strength, material relaxation rates, and cavity losses. The stable regular structures which have been observed in the MS50 simulations include traveling waves and stable spatially in­ Spatiotemporal Pattern Formation in Combustion dependent solutions. The chaotic dynamics are manifested In recent experiments with high Lewis number flames prop­ as unidirectional waves whose speed and height vary in a agating in a tube, Pearlman (cf. this minisymposium) ob­ seemingly random fashion, but whose other features, such served flames exhibiting pacemaker target patterns, spi­ as the number of "crests" per spatial period remain con­ rals and other interesting spatiotemporal patterns. In this stant in time, indicating the low-dimensional nature of the talk we will pose a simple model of flames, and exhibit so­ process. Spatially-dependent orbits homo clinic to an un­ lutions which describe these flames, well as other, as yet stable saddle-focus are proven to exist by a combination of unobserved flames. the "dressing" method from the inverse-scattering theory and the homo clinic Melnikov method. These orbits provide Alvin Bayliss a posible mechanism for the chaotic dynamics. Northwestern University, Evanston, IL

Gregor KovaCic, Jiyue J. Li, and Victor Roytburd Bernard J. Matkowsky Mathematical Sciences Department Northwestern University, Evanston, IL Rensselaer Polytechnic Institute Troy, NY, 12180

Thomas A. Wettergren MS50 Naval Undersea Warfare Center Excitability in Premixed Gas Combustion Newport, Rhode Island, 02841-5047 Target and spiral waves patterns have been observed in freely-propagating and burner-stabilized premixed gas flames when the mixture Lewis number (ratio of mixture MS49 thermal diffusivity to mass diffusivity of the deficient com­ The Nonlinear Schrodinger Equation: Asymmetric ponent) is larger than one. The occurrence of such pat­ Perturbations, Traveling Waves and Chaotic Struc­ terns appears to be extremely sensitive to the mixture tures composition as well as local heat loss. Experimental mea­ surements of the temperature and composition profiles as Hamiltonian perturbations of the nonlinear Schrodinger well as flame-induced hydrodynamic flow will be presented equation (NLS) can produce a novel type of chaotic evo­ along with details of the flame dynamics as a function of lution. The chaotic solution is characterized by random parameter space. bifurcations across standing wave states into left and right going traveling waves. In this class of problems where the Howard Pearlman solutions are not subject to even constraints, the tradi­ NASA Lewis Research Center tional mechanism of crossings of the unperturbed homo­ Cleveland, OH clinic manifolds is not observed. I will present an analyt­ ical description of chaos in Hamiltonian perturbations of the NLS that will include this new phenomena of chaotic MS50 traveling waves. Nonlinear Dynamics in Combustion Synthesis of Constance Schober Materials Department of Mathematics & Statistics, In this talk we describe a variety of nonlinear dynami­ Old Dominion University, Norfolk, VA 23529-0077 cal modes of propagation of combustion synthesis waves. We discuss both experimental observations and theoretical studies, as well as comparisons between them. Vladimir A. Volpert Abstracts 107

Northwestern University, Evanston, IL that thc bifurcation is mediated by changes in the trans­ verse stability of an infinite number of unstable periodic or­ bits. There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The MS51 bifurcation occurs when some properly weighted transverse Blowout-Type Bifurcations in Symmetric Systems eigenvalues of these two groups are balanced. Basin riddling and other associated phenomena of blowout Ying-Cheng Lai bifurcation occur naturally in systems with symmetry; this Departments of Physics and Astronomy and of Mathemat­ is due to the invariant subspace structure forced by the ics symmetry. I will discuss some connections between this The University of Kansas and related phenomena such as structurally stable hetero­ Lawrence, KS 66045 clinic cycles. Peter Ashwin Department of Mathematical and Computing Sciences PSOI University of Surrey A Hydrodynamic Theory of Mass Transport Guildford GU2 5XH, United Kingdom We present here a two dimensional mathematical picture that describes the motion of dust particles in a continuous fluid such as water. In our description of hydrodynamic MS51 forces on the particles we show how the Lagrangian for­ Synchronized Chaos, Intermingled Basins and on­ malism can be used to find a general expression for these off Intermittency in Coupled Dynamical Systems forces. This model can serve as an initial step toward for­ Recent work has shown that chaotic systems possessing mulating the hydrodynamics of sediment transport. This certain symmetries can exhibit a novel class of phenom­ is an important problem in the field of oceanography and ena including riddled basins, intermingled basins and on­ other disciplines as well. off intermittency. In this talk, we first show that intermin­ Abdulmuhsen H. Ali gled basins can be easily realized in the context of cou­ Kuwait University pled oscillators exhibiting synchronized chaos. This opens P.O. Box 5969 the possibility of investigating these phenomena in labora­ Safat 13060, Kuwait tory experiments. Then, we derive explicit conditions for E-mail: [email protected] the stability of synchronized chaos in globally coupled map lattices where the individual map can be arbitrary dimen­ sional. We further show that after the synchronized chaos loses stability one generally observes on-off intermittency. PSOl Dynamical Features Simulated by Recurrent Neu­ Mingzhou Ding ral Networks Center for Complex Systems Florida Atlantic University The evolution of simple two dimensional recurrent neural Boca Raton, FL 33431 networks with saturated linear or sigmoidal transfer func­ tions presents a whole spectrum of dynamical behaviors, ranging from highly predictable dynamics, where almost every orbit accumulates to an attracting fixed point, to MS51 the existence of chaotic regions with cycles of arbitrarily Riddling Bifurcation in Chaotic Dynamical Sys­ large period. Moreover, recurrent analog neural networks tems with rank one connecting matrices are good approximators When a chaotic attractor lies in an invariant subspace, as in of compact supported real-valued continuous maps under systems with symmetry, riddling can occur. Riddling refers a convenient choice of encoding and deconding procedures. to the situation where the basin of a chaotic attractor is Fernanda Botelho riddled with holes that belong to the basin of another at­ Department of Mathematical Sciences tractor. We establish properties of the riddling bifurcation University of Memphis that occurs when an unstable periodic orbit embedded in Memphis, TN 38152 the chaotic attractor, usually oflow period, becomes trans­ [email protected] versely unstable. An immediate physical consequence of the riddling bifurcation is that an extraordinarily low frac­ tion of the trajectories in the invariant subspace diverge when there is a symmetry-breaking. PSOI Using Spatial Disorder to Synchronize Dynamical Celso Grebogi Systems Institute for Plasma Research University of Maryland, College Park, MD 20742 We study the effect of quenched disorder on the synchro­ nization of dynamical systems. In particular, we study ar­ rays of coupled Josephson Junctions, coupled Duffing os­ cillators, and coupled maps. We find that disorder can MS51 synchronize chaotic arrays of otherwise identical elements Characterization of Blowout Bifurcation by Unsta­ and thereby tame spatiotemporal chaos. We exhibit arrays ble Periodic Orbits that evolve chaotically when homogeneous but evolve pe­ Blowout bifurcation in chaotic dynamical systems occurs riodically when heterogeneous. We find that there exists when a chaotic attractor, lying in some invariant subspace, optimal amounts of disorder which minimize the leading becomes transversely unstable. We establish quantitative Liapunov exponents of these arrays. characterization of the blowout bifurcation by unstable pe­ Yuri Braiman riodic orbits embedded in the chaotic attract or. We argue 108 Abstracts

Emory University put noise intensity with either the input signal or the unit's Department of Physics output rate signal. We demonstrate that SR enhancement Atlanta, GA 30322 will occur even if the noise modulation is time-delayed with respect to the signal. We analyze SR enhancement theo­ John Lindner retically for generic nonlinear systems and demonstrate it The College of Wooster numerically in the FitzHugh-Nagumo model. Wooster Ohio 44691 Carson C. Chow Department of Mathematics 111 Cummington St. PSO! Boston University, Boston MA 02215 Information Content of Cellular Automaton Traffic [email protected] Simulations http://zion.bu.edu/ccc.html Using a symbol dynamics approach, we develop a method for measuring the information content in cellular­ Thomas T. Imhoff automaton simulations of vehicular traffic. We present ap­ NeuroMuscular Research Center plications of the method to TRANSIMS (a TRansporta­ Boston University tion ANalysis and SIMulation System being developed for Boston MA 02215 the U.S. Department of Transportation), showing how the Renyi entropy varies and scales as a function of sampling James J. Collins procedure and simulation parameters. Department of Biomedical Engineering Boston University Brian W. Bush Boston MA 02215 Los Alamos National Laboratory Los Alamos, NM 87545 PSOl Control of Uncertain Mechanical Systems Using PSO! Reduction and Adaptation Inertial Functions: Definition, Existence and Sta­ bility This paper considers the motion control problem for a broad class of uncertain mechanical systems; systems be­ We introduce a generalisation of the inertial manifold, longing to this class include holonomically constrained sys­ which we call an inertial function. This is a function whose tems, nonholonomic systems, and systems for which the graph is invariant under the dynamics and is exponentially dynamics admits a symmetry. It is proposed that a simple attracting for almost all initial conditions. The construc­ and effective solution to this problem can be obtained by tion and smoothness of this inertial function is based on first using a reduction procedure to obtain a lower dimen­ the Lyapunov exponents of the dynamical system and on sional system which retains the mechanical system struc­ properties of an absorbing set. We discuss conditions for ture of the original system, and then adaptively controlling the existence of such a function, along with its stability to the reduced system in such a way that the complete system deterministic perturbations and noise. is driven to the goal configuration. This approach is shown Kevin M. Campbell, Michael E. Davies, and Jaroslav Stark to ensure accurate motion control even in the presence of Centre for Nonlinear Dynamics and its Applications considerable uncertainty concerning the system model and University College London state. The efficacy of the proposed control strategy is illus­ Gower Street, London, WC1E 6BT, United Kingdom trated through extensive computer simulations and hard­ Email: [email protected] ware experiments. Richard Colbaugh New Mexico State University PSO! Department of Mechanical Engineering Spectra of Frequency-Modulated Waves Box 30001, Dept. 3450 Las Cruces, NM 88003 The energy spectra of a sinusoid, frequency modulated by a stationary random process of various distributions, are Ernest Barany investigated. Wide band noise as well as random noise fil­ New Mexico State University tered through Butterworth filters of even and odd order are Department of Mathematical Sciences considered. A general expression for the Butterworth filter Box 30001, Dept. 3MB case is provided The closed form solutions take the form of Las Cruces, NM 88003 complementary error functions of complex arguments and are compared to classical solutions obtained using statisti­ Gil Gallegos and Mauro Trabatti cal communication theory. New Mexico State University George D. Catalano Department of Mechanical Engineering Department of Civil and Mechanical Engineering Box 30001, Dept. 3450 U.S. Military Academy, West Point, NY 10996 Las Cruces, NM 88003 E-mail: [email protected]

PSO! PSOl The Magnetic Pendulum Enhancing Stochastic Resonance in a Single Unit In this work we study the dynamical behavior of a fer­ We show that the stochastic resonance (SR) effect in a sin­ romagnetic pendulum, which is subject to gravitational, gle nonlinear unit can be enhanced by modulating the in- magnetic and frictional forces. The magnetic force is pro- Abstracts 109

duced by three strong magnet, which are attractor or repul­ Bulgarian Academy of Sciences sor points positioned a short distance below the pendulum Institute of Mechanics and Biomechanics plane at the vertices of any triangle. The friction force " Acad. G.Bonchev" Str., BI. 4 depends on the velocity. 1113 Sofia, Bulgaria E-mail: [email protected] Francisco Cervantes de la Torre*, J. L. Fernandez Chapou UAM-AZCAPOTZALCO, CBI. Depto. de Sistemas, ** Depto. de Ciencias Basicas E-mail: [email protected] PSOI E-mail: [email protected] Newton-Picard Methods for Robust Continuation and Bifurcation Analysis of Periodic Solutions of Infinite-Dimensional Dynamical Systems with Low­ Dimensional Dynamics PSOI Symmetry-Breaking Parametric Instabilities in Newton-Picard methods combine direct solvers and Picard Circular Josephson Junction Arrays iteration schemes to solve the linearised systems that arise in the shooting method and provide good estimates for the We use equivariant bifurcation theory to investigate para­ dominant Floquet multipliers during this process. In this metric instabilities of the in-phase periodic state of a circu­ poster, we will show how this information can be used to lar array of Josephson junctions. Watanabe, et al. showed make continuation more robust and to detect bifurcation that these instabilities are associated with steps in the I-V points. We will also show how to apply the Newton-Picard curve of the array. Our analysis extends their linear results method to extended systems to accurately compute these to the weakly nonlinear regime. We determine existence bifurcation points. and stability of the period-doubled solution branches as a function of various physical parameters in the governing Koen Engelborghs, Kurt Lust, and Dirk Roose equations. Department of Computer Science K.U. Leuven, Celestijnenlaan 200A, B-300l Heverlee, Bel­ John J. Derwent and Mary Silber gium Northwestern University, Evanston, IL 60208 [email protected] E-mail: [email protected], [email protected]

PSOI Continuum Limits of Convective-Diffusive Lattice PSOl Boltzmann Methods Numerical Evidence of Standing and Breathing So­ lutions for the Oregonator with Equal Diffusivities The connection between a diffusive lattice gas and the corresponding macroscopic convective-diffusive equation We consider a simple model of the Belousov-Zhabotinskii is described. Lattice Boltzmann methods are numerical reaction a gel reactor coupled to a reservoir. Using the schemes derived as a kinetic approximation of an underly­ Oregonator reaction kinetics in the reactor we give numer­ ing lattice gas. The continuum limit of a lattice Boltzmann ical evidence of stable (or Meta-stable) large amplitude method is formally shown to converge to the macroscopic standing patterns in the case that all of the diffusivities equation. The result is based on a formal derivation in are equal. We also give numerical evidence that these pat­ which limitting moments are balanced rather than on a terns destabilize as one varies the domain size via a Hopf classical expansion. bifurcation giving rise to Breather solutions. Cecilia Fosser and C. David Levermore Jack Dockery Program in Applied Mathematics Department of Mathematical Sciences University of Arizona, Tucson 85721 USA Montana State University e-mail:[email protected] Bozeman, MT 59717 e-mail:[email protected] Richard J. Field Department of Chemistry University of Montana PSOI Missoula, Montana 59812 Reconstruction and Prediction of Stochastic Dy­ namical Systems using Topology Preserving Net­ works PSOl Starting with a scalar time series, we present the use of Modelling and Phase Analysis of Immunological Topology Preserving Networks, as for example the Koho­ Process at Primary Immune Response nen Network and the Neural Gas Algorithm, to define a Voronoi partition and its dual, the Delaunay Triangula­ A mathematical model of a specific immunological process tion. The partition is then used, to compute the transition for venous immunization of rabbits by the haemorrhagic probabilities defining the Markov chain of the system. This disease virus is developed. On the basis of the qualitative probability set can then be used to predict the time series, theory of ordinary differential equations a phase analysis of and to reconstruct the flow and the attract or of the sys­ this immunological process is carried out in the kinetic vari­ tem. The method can be used for deterministic, as well as ables plane - antigens concentration and antibodies con­ chaotic and stochastic dynamical systems. centration. The phase portait of the discussed nonlinear model in two-dimensional domain is constructed. The mu­ Markus Anderle and Sabino Gadaleta tual relationship between the antigens reproduction rate Colorado State University and phase portrait is investigated in a qualitative way. Ft. Collins, CO 80523 [email protected] Ivan Edissonov [email protected] 110 Abstracts

PSOI PSOI Complex Dynamics of Coupled Quantum Wells Effects of Anisotropy on Nonlinear Evolution of with Local Mean Field Interaction; Bifurcations of Morphological Instability in Directional Solidifica­ Periodic Orbits in a Hamiltonian System with Sym­ tion metry Effects of anisotropy of surface tension and kinetic coeffi­ We analyze a discrete semiclassical model of coupled quan­ cients on nonlinear evolution of morphological instability tum wells with short-range mean-field interaction in one in the course ofrapid directional solidification is studied. It site. The system evolves according to the time depen­ is shown that, due to anisotropy, morphological instability dent Schrodinger equation with a nonlinear electrostatic generates deformational waves propagating in a preferred term. The simplest vector field that accounts for the com­ direction governed by anisotropy. Nonlinear evolution of plex dynamical behavior present in the continuum case these waves is governed by anisotropic dissipation-modified turns out to be eight dimensional and can be written con­ Korteweg de Vries equation and anisotropic Kawahara veniently with <1:;4 variables in a hamiltonian formalism. equation. Solutions of these equations in the form of The system is invariant under rotations in <1:;4, reversible cnoidal and solitary waves and their stability are studied. (H(z, z) = H(z, z)) and autonomous. The conserved quan­ Alexander Golovin and Stephen Davis tities are the energy and the total charge. The organizing Dept of Engineering Science & Applied Mathematics centers for the complex dynamical behavior are bifurca­ Northwestern University, Evanston, IL 60201 tions of rotating periodic solutions. Its simple structure allows a thorough analytical investigation of the bifurca­ tions as the conserved quantities are varied. PSOI J. Gahin-Viogue and E. Freire Application of Centre Manifold Theory to an Departamento de Matematica Aplicada II Adaptive Control Problem Escuela Superior de Ingenieros de Sevilla Avda. Reina Mercedes sin, Sevilla 41012, Spain Certain partially known discrete-time nonlinear control E-mail: [email protected] systems are dealt with. The control objective is the asymp­ totic regulation of the output. To this purpose, an adaptive control scheme is implemented. The resulting closed loop is a discrete-time dynamical system with non-hyperbolic PSOl fixed points. By means of the Centre Manifold Theory, Heteroclinic Networks in Coupled Cell Systems a result on the stabilization of the system about its fixed We consider a system of n identical cells, with identical, points is stated. From this, a local result on the firstly all-to-all coupling. The system exhibits both internal (cell­ proposed control objective is obtained. wise) and global (permutational) symmetry. We show by Graciela A. Gonzalez construction that the system allows a heteroclinic network, Departamento de Matematica which consists of a set of equilibria connected by a series Universidad de Buenos Aires, 1428 Argentina of trajectories. We then show how this network leads to E-mail: [email protected] interesting intermittent behavior in the system, which has a natural interpretation in terms of coupled cells

David Gillis PSOI Department of Mathematics Symmetry Breaking Perturbations and Strange At­ University of Houston tractors Houston, TX 77204-3476 E-mail: [email protected] The asymmetrically forced, damped Duffing oscillator is introduced as a prototype model for analyzing the ho­ moclinic tangle of symmetric dissipative systems with symmetrybreaking disturbances. Even a slight asymme­ PSOl try in the perturbation may cause a substantial change in Control of Differentially Flat Robotic Systems In the asymptotic behavior of the system, e.g. transitions the Presence of Uncertainty from two sided to one sided strange attractors. Numeri­ This paper considers the problem of controlling uncertain cal evidence indicates that strangeattractors appear near robotic systems which are differentially fiat, that is, which curves which are found analytically and correspond to spe­ admit a set of outputs with the property that there is a one cific secondary homo clinic bifurcations. to one correspondence between output curves and system Anna Litvak Hinenzon and Vered Rom-Kedar trajectories. It is shown that a simple and effective solution The Faculty of Mathematical Sciences to this problem can be obtained by combining a trajectory The Weizmann Institute of Science generation algorithm for the flat outputs with an adaptive P.O.B. 26, Rehovot 76100, Israel tracking controller for a set of "reducing" outputs. The [email protected], utility of this approach is illustrated through both com­ [email protected] puter simulations and hardware experiments with several important robotic systems, including a mobile robot, an underwater vehicle, a space robot, and a ducted fan. PSOI Kristin Glass, Richard Colbaugh, and Tuna Saracoglu Computing the Measure of Nonattracting Chaotic New Mexico State University Sets Department of Mechanical Engineering Box 30001, Dept. 3450 Recently, a numerical method, called the PIM-triple algo­ Las Cruces, NM 88003 rithm, has been formulated for generating long orbits on nonattracting chaotic sets. We investigate under which conditions the measure for a nonattracting invariant set Abstracts 111 generated by the PIM-triple algorithm agrees with the nat­ 1996]. Excited neuron is known to demonstrate two kinds ural measure. of behavior: fast spiking and slow bursting. Action poten­ tial of such a neuron possesses two different time scales - Joeri Jacobs, Edward Ott and Celso Grebogi the average period of a bursts and the average length of Institute for Plasma Research spikes. We calculate two phases of coupled neurons (more University of Maryland exact, the phase difference) and study their behavior for College Park, MD 20742 different values of coupling strength. E-mail: [email protected] Alexander K. Kozlov Institute of Applied Physics, Russian Academy of Sciences, 46 Ulyanov St., Nizhny Novgorod 603600, Russia PSO! E-mail: [email protected] Scaling of Travelling Waves Produced from Elec­ trostatic Instabilities Recent studies of electrostatic instabilities in Vlasov PSOl plasma have shown that the amplitude equations have a Forced Symmetry-breaking in Coupled Oscillator non-trivial scaling behavior arising from singularities in the Networks nonlinear coefficients. The analysis of the amplitude equa­ tions does not establish the time-asymptotic state of the We present a numerical and theoretical investigation into instability, but other research suggests that these instabil­ the effects of small symmetry-breaking terms on the bi­ ities often result in the appearance of nonlinear travelling furcations of a ring of nominally identical oscillators. The waves. In addition, rigorous investigations of the bifurca­ fully symmetric system has Dn --symmetry and the theory tion of travelling waves (by Holloway and Dorning) prove of Hopf bifurcation with symmetry can be used to deter­ the existence of solution branches, but do not establish any mine the types of oscillation possible. When symmetry­ relation between the nonlinear travelling waves and the ini­ breaking terms are included, the behaviour near the Hopf tial value problem. Thus the scaling of the waves with -y bifurcations is significantly changed, leading to, for exam­ (the linear growth rate) near the onset of a linear insta­ ple, stabilisation of previously unstable patterns. bility is not determined. We consider a family of Vlasov Carlo Laing equilibria Fo( v, -y) that cross the threshold of linear stabil­ University of Cambridge ity at -y = 0 and formulate the equations for branches of Dept of Applied Maths & Theoretical Physics travelling waves Ftw(x,v,t,-y) that bifurcate from Fo(v,O) Silver Street and belong to the same symplectic leaves as Fo(v,-y). The Cambridge CB3 9EW, United Kingdom analysis of singularity structure of the equations provides Email: [email protected] information about the asymptotic scaling of Ftw(x, v, t,-y) as-y-tO+. Anandhan Jayaraman and John David Crawford PSO! Department of Physics and Astronomy Taking the Temperature of an Epileptic Seizure University of Pittsburgh A surprisingly accurate and easily computed indicator of Pittsburgh, PA 15260 the onset and duration of an epileptic seizure is obtained by interpreting the windowed re-embedded trajectory as a cloud of particles in phase space and monitoring the aver­ PSOl age kinetic energy of the particles in the cloud as a function Bifurcations in Systems of Coupled Bistable Units of time. Systems of coupled bistable units have been employed to in­ David E. Lerner vestigate phenomena such as phase-separation, multistabil­ Department of Mathematics ity, propagation failure and neural dynamics. Here we show University of Kansas, Lawrence, KS 66045 how to analytically determine the existence and stability properties of fixed points of piecewise-linear coupled map lattices, then use this technique to investigate the bifurca­ tions undergone by systems of diffusively-coupled bistable PSOl maps. We show how the bifurcations lead to many interest­ Global and Local in Time Solutions of Laplacian ing phenomena, such as the coexistence of travelling waves Growth Equation: Criteria for Cusps and localised stationary solutions. A geometric (and group-theoretic) description of the con­ figuration space for the Laplacian growth equation is pre­ Mark E. Johnston Department of Applied Mathematics and Theoretical sented. The differences between global in time solutions and local in time solutions are discussed. Global solutions Physics are free of cusps. The local solutions necessarily develop University of Cambridge cusps. The criteria that the initial data must satisfy in or­ Silver Street, Cambridge, CB3 9EW, U.K. der that the solutions be free of cusps (global in time) are Email: [email protected] formulated. Hanna E.Makaruk MS B 213, Los Alamos National Laboratory PSO! Numerical Analisys of Phase Synchronization of Los Alamos NM 87545 Coupled Model Neurons [email protected] We suggest the way of numerical analysis of collective be­ Mark B. Mineev-Weinstein havior of coupled chaotic model neurons. The method is MS B 213, Los Alamos National Laboratory based on evaluation of "phase" of chaotic dynamical sys­ Los Alamos NM 87545 tem according to the approach suggested by [Kurths, et al., 112 Abstracts

[email protected] PSOI Two Meaningful Alternative Representations of Dynamic Behaviour PSOI In teaching dynamics to undergraduates via time­ Using Linear Integro-Difference Equations to Em­ dependent differential equations, the computer's use rests ulate Solutions to a Temporally Stiff System of on the algorithm. However, we note that the strictly PDE's mathematical representation of dynamics (via differen­ We examine a system of evolution PDE's modelling the tial/difference equations) is computationally implemented movement of Mountain Pine Beetles in western forests. via algorithms which usually rigidly advance time uni­ Several problems are involved. The first is that the evo­ formly and re-compute the same formula(-ae) recursively. lution of different variables takes place on different time Yet, strictly algorithmic models permit one to represent scales. Moreover, a variety of spatial scales couple forest the dynamics of biological and/or social systems much and beetle interaction. Multiple scale methods allow us to more scientifically [AN EPISTLE TO DR BENJAMIN decouple the problem, creating an emulation strategy for FRANKLIN, Exposition-University Press, 1975(1974)]. the PDE solution in terms of iterations of an IDE. Whereas G. Arthur Mihram, Ph. D. the system of PDE's is impossible to solve except through P.O. Box No. 1188 computationally lengthy numerical procedures, the IDE Princeton, NJ 08542-1188 is relatively simple, saving several orders of magnitude of computational time. Danielle Mihram, Ph. D. Tyler K. McMillen University of Southern California Department of Mathematics LVL-124 [MC #0182] Utah State University Los Angeles, Calif 90089-0182 Logan, UT 84322 [email protected] E-mail: [email protected]

PSOI PSOI Integro-Differential Model for Orientational Distri­ Dynamics in the Undergraduate Mathematics Pro­ bution of F-Actin in Cells gram We model angular self-organization of actin cytoskeleton as The concepts of dynamics are introduced in two courses a process of instant changing of filament orientation in the Mathematical Modeling and Differential Equations with course of peculiar actin-actin interactions. These interac­ Linear Algebra, both having a prerequisite of Calculus II. tions are modified by cross-linking actin-binding proteins. Intermediate concepts are studied in Advanced Differential The mathematical model consists of a single Boltzmann­ Equations. The opportunity to further develop these con­ like integro-differential equation for the two-dimensional cepts is an option in Senior Project, the capstone course angular distribution. The linear stability analysis, asymp­ required of all math majors. It in turn is given in two parts: totic analysis and numerical results reveal that at certain 1 cr in the Fall where the topic is chosen and outlined, and parameters of actin-actin interactions spontaneous align­ 2 cr in the Spring where it is developed in Hypercard or ment of filaments in the form of unipolar or bipolar bundles the Mathematica Notebook. Examples of projects will be or orthogonal networks can be expected. illustrated. Alex Mogilner Richard F. Melka Department of Mathematics Mathematics Department University of California at Davis University of Pittsburgh at Bradford Davis, CA 95616 Bradford, PA 16701 [email protected] E-mail: [email protected] Edith Geigant Abt. Theor. Biologie, Universitiit Bonn, Kirschallee 1, D-53115, B.onn, Germany, PSOl [email protected] Forced Vibration of a Partially Delaminated Beam The forced vibration of a partially delaminated structure Karina Ladizhansky such as an aircraft wing can result in catastrophic crack Dept. of Applied Math. and Computer Sci., growth. A cantilever beam with a single delamination at Weizmann Institute of Science, its free end is considered. We examine, both numerically Rehovot, 76100, Israel, and analytically, the resulting motions from a mathemati­ [email protected] cal model of the system subject to a harmonic forcing of the main beam. The model displays several types of motion; periodic, chattering-periodic and chaotic motion. Experi­ PSOI mental evidence for some of these is provided. Synchronization of Structures by a Small Number Roger P. Menday and M.C. Harrison of Couplings University of Loughborough The interactions of localized coherent structures formed in Loughborough, United Kingdom two almost identical but different chains of bistable ele­ E-mail: [email protected] ments both against unexcited and chaotic backgrounds are investigated. Numerical solutions are used for analysis of E.R Green the features of mutual synchronization of such structures Leicester University in the presence of localized couplings between them. The Leciestcr, United Kingdom Abstracts 113

transition to synchronization occurs through the synchro­ is found that the average transient time preceding the con­ nization "front" propagating from the point of coupling to trol obeys a scaling law that is qualitatively different from the edges of the structures. The velocity of front propaga­ the algebraic scaling law which occurs when one controls tion is plotted versus the difference in natural frequencies chaos by stabilizing unstable periodic orbits embedded in of the structures, 8. Qualitative description of the effect is a chaotic attractor. A theoretical argument is provided for proposed. It is shown that the coupled structures in syn­ the observed scaling law. chronized and non-synchronized regimes differ, first of all, Yoshihiko Nagai, Xuan-Dong Hua, and Ying-Cheng Lai by the distribution of oscillatory phases of their elements. Department of Physics and Astronomy Possible use of this circumstance for experimental verifica­ University of Kansas tion of the synchronization between this and some other Lawrence, KS 66045 structures is discussed for the situations when the charac­ E-mail: [email protected] teristics of the latters cannot be measured to a sufficient extent.

Ya.I. Molkov, M.M. Sushchik PSOl Institute of Applied Physics, Russian Academy of Science Blownout Attractors 46 Uljanov Str., 603600 Nizhny Novgorod, Russia E-mail: [email protected] Suppose a chaotic attractor A in an invariant subspace loses stability on varying a parameter. At the point of loss A.V. Yulin of stability, the most positive Lyapunov exponent of the Institute for Physics of Microstructures natural measure on A crosses zero at what has been called Russian Academy of Science a 'blowout' bifurcation. There are two types of blowout: 46 Uljanov Str., 603600 Nizhny Novgorod, Russia subcritical (or hard) and supercritical (or soft). We inves­ E-mail: [email protected] tigate the dynamical and ergodic properties of nearby in­ variant sets in both scenarios. Our analytical results center on model piecewise linear planar maps and drift-diffusion systems, but we expect our results to apply to more general PSOl systems. Hamiltonian Moment Reduction for Describing Vortices in Shear Flow Peter Ashwin and Philip Aston Department of Mathematical & Computing Sciences A general method for modeling long-lived quasigeostrophic University of Surrey vortices is given. The method is based upon the noncanon­ Guildford GU2 5XH, United Kingdom ical Hamiltonian structure of the ideal fluid and uses mo­ ments of the vorticity as dynamical variables. The method Matthew Nicol provides a means of extracting, either exact or approximate Department of Mathematics finite degree-of-freedom Hamiltonian systems from the par­ UMIST tial differential equations that describe quasigeostrophic PO Box 88, Sackville St vortex dynamics. The method is applied to the Kida prob­ Manchester M60 lQD, United Kingdom lem and the dynamics of an ellipsoidal region of uniform [email protected] potential vorticity. S. Meacham Florida State University PSOl Tallahassee, FL 32306 Enhancement of Stochastic Resonance of FitzHugh-Nagumo Neuronal Model P. J. Morrison Driven by 1/f Noise University of Texas Austin, TX 78712 We investigated the stochastic resonance driven by 1/f E-mail: [email protected] noise in FitzHugh-Nagumo neuronal model by numerical simulation. The system exhibited typical characteritics of G. Flierl stochastic resonance that there is an optimal intensity of Massachusetts Institute of Technology noise maximizing the response to subthreshold signals. The Cambridge, MA 02139 optimal intensity of 1/f noise was lower than that of white noise by a factor of 1/5. We also discuss the mechanism by which 1/f noise could enhance the performance of stochas­ tic resonance in FitzHugh-Nagumo neurons. PSOl Controlling On-Off Intermittent Dynamics Daichi Nozaki and Yoshiharu Yamamoto Graduate School of Education On-off intermittent chaotic behavior occurs in physical sys­ The University of Tokyo tems with symmetry. The phenomenon refers to the sit­ Hongo 7-3-1, Bunkyo-ku uation where one or more physical variables exhibit two Tokyo, Japan 113 distinct states in their time evolution. One is the "off" E-mail: [email protected] state where the physical variables remain constant, and [email protected] the other is the "on" state where the variables temporarily burst out of the "off" state. We demonstrate that by using arbitrarily small feedback control to an accessible parame­ ter or state of the system, the "on" state can be eliminated PSOl completely. This could be practically advantageous where Nonlinear Pattern Dynamics in Two-dimensional the desirable operational state of the system is the "off" Josephson-Junction Networks state. Relevant issues such as the influence of noise and The dynamics of a two-dimensional array of Josepson­ the time required to achieve the control are addressed. It junctions is shown to be equivalent to that of a one di- 114 Abstracts

mensional chain of nonlinear, globally, and nonuniformly erate any complex pattern. We introduce the new idea of coupled oscillators. We describe spatiotemporal patterns, the virtual object, namely, the minimal geometrical struc­ coupled oscillatory compartments and successive bifurca­ ture that generates a pattern. It is also introduced the new tions the array exhibits in response to varying input pat­ method of random Markov field (RMF) to generate virtual terns and bias current. Since boundary instabilities impede pattern. Once upon a virtual object is generated the mod­ coherent microwave radiation emission also in the presence ified functional self organization (MFSO) stochastic model of external loads, a new symmetry-breaking array architec­ is used to form clusters (of virtual objects). ture is designed to achieve coherence. Sorincl Adrian Oprisan Werner Guettinger and Joerg Oppenlaender Department of Theoretical Physics University of Tuebingen, "AI. 1. Cuza" University Institute of Theoretical Physics, Bd. Copou, No. 11,6600 Iasi, Romania D-72074 Tuebingen, Koestlinstr.6, Germany E-mail: [email protected]

PSOI PSOI The Construction of the Analytic Periodic Solu­ Scale-sensitive Linear Analysis of Granivorous Ro­ tion in the Framework of the Multiple Time Scale dent Population Dynamics Method Traditional linear analysis of ecological systems fails to con­ Our main objective is to develop an analytical method for sider temporal scale-linear regularities may only be visi­ the problem of the domain wall motion. The starting point ble within a definite time "window." Community matri­ is the Landau-Lifshitz and Gilbert equations of the mag­ ces at different temporal scales were used to predict the netic spin system dynamics. Following Sloczewski we study effect of removal of larger granivorous desert rodents on the behavior of the anisotropic magnetics with uniaxial smaller ones. The accuracy of prediction was found to be magnetic anisotropy in the z axis direction. The Bloch very scale-dependent; predictions were highly variable, es­ wall plane is infinity and parallel to the zy plane. An ex­ pecially at larger time scales. This indicates that linear ternal magnetic acts in the direction of the z axis and the analysis is an inadequate tool for prediction of perturba­ domain wall moves in the direction of the x axis. Having tion effects in this system. in mind all of this the equation of the Bloch wall motion is Nikkala A. Pack Department of Statistics + 87r/Lo Is 0: + 16dI;) North Carolina State University ( hi b. 7r3 p Raleigh, NC 27695

I 28HJs . 27rX . E-mail: [email protected] X --l-sln-- = 2IsHsmwt + l Brian A. Maurer where d is the thickness of the magnetic material, p is the Department of Zoology electrical resistivity of the magnetic material. Using the Brigham Young University multiple time scales method [?] one finds the analytic pe­ Provo, UT 84602 riodic solution of the domain-wall motion. The limit cycle solution was theoretically obtained. The stability of the periodic solution was investigated. The periodic windows theoretically found are in good agreement with the exper­ PSOI imental and numerical results previously reported. Ionic Diffusion Within the Confined Extracellular Space of Glomeruli Sorinel Adrian Oprisan Department of Theoretical Physics, "Al.1.Cuza" University Diffusion in complex media occurs in many important Bd. Copou No. 11 6600 Iasi Romania physical and biological contexts. We present a model of E-mail: [email protected] this process relevant to the olfactory system. For insects and vertebrates, these systems are similar in the organi­ zation of their synaptic neuropil into glomeruli, structures surrounded by an incomplete layer of glial processes. Why PSOI are glial cells in such an organized structural formation? Markov Random Fields and Dynamical Systems Is the organization for developmental reasons or does it with Memory in Pattern Generation and Recog­ serve functional purposes? One possible function is the nition Processes regulation of potassium. A biologically realistic model is We present a stochastic algorithm based on a modified to­ derived to examine the diffusion of potassium through the talistic rule of the Chate-Maneville type. It is used the confined extracellular volume of glomeruli. This study will random Markov field model [?, ?, ?,?] to generate vir­ aid in the understanding of the role of glial cells in any tual object. The virtual objects are the elementary cells general synaptic neuropil. that generate the desired pattern under proper geometri­ Anita Rado, Alwyn Scott, and Leslie Tolbert cal operations. The main application of the model concerns University of Arizona, Tucson, AZ 85721 pattern recognition. The model is also a realistic base of E-mail: [email protected] understanding the emergence of synchronous behavior in physics (chemistry or biology) related to some kind of self organization processes. Our main goal is to understand the process of complex pattern generation and theirs underly­ PSOI ing local mechanisms. We previously reported successful Characterization of noise in electrotelluric time se­ theoretical and numerical results on simple pattern genera­ ries tion based on stochastic algorithm [?, ?]. The present stud­ In a resent paper[1] it was suggested that the behavior ies go further and define explicit algorithm useful to gen- of the so-called electrotelluric field is corrrelated with the Abstracts ll5

preparation mechanism of eathquakes. That report was PS01 concerned with seismic activity in the Pacific coast of Mex­ A Nonlinear Model to Classify Dynamic Time Se­ ico. By means of some of the same electrotelluric files used ries in the mentionet article, we made a noise analysis. This We present a nonlinear time series model based on a com­ new study is made by using spectral techniques and also bination of a SETAR(k;l(l), ... ,I(k)) model and a structure by means of correlation coeficients. Our preliminary results of neural networks. SETAR-models consist of different lin­ suggest that white noise is predominant in electrotelluric ear parts linked by a threshold structure. We replace each fluctuations. Nevertheless, it seems that there exist an­ of them with a neural network, predict following points of other kind of fluctuations which are not of the white noise a time series by such a model and analyze the learning rate type. [lJ E. Yepez, F.Angulo-Brown, J.A. Peralta, C.G. regarding to changes in determinism and stochasticity. Ap­ Pavia and G. Gonzalez-Santos, Geophys. Res. Lett. 22 plication to the analysis of fetal breathing pattern will be (1995) 3087. presented Alejandro Ramirez-Rojas, Francisco Cervantes-de la Torre Karin Schmidt and Carlos G. Pavia-Miller Institute of Pathophysiology Universidad Autonoma Metropolitana Azcapotzalco, Mex­ Friedrich Schiller University Jena ico D.F. 02200 Loebderstr. 3, D-07740 Jena, Germany E-mail: [email protected] Fernando Angulo-Brown Escuela Superior de Fisica y Matematicas, 1.P.N., Mexico Hazel H. Szeto D.F.,07738 Department of Pharmacology Cornell University Medical College 1300 York Avenue, LC 423 PSOl New York, NY, 10021 Bifurcations and Edge Oscillations in an Inhomo­ geneous Reaction-Diffusion System We examine the influence of inhomogeneity on bifurcations PS01 affecting the stability of pulse solutions to certain reaction­ Dynamical Behavior of a Model for Hormonal Reg­ diffusion systems. Using the Evans function and a topolog­ ulation during the Menstrual Cycle in Women ical stability index, we prove analytically that two bifur­ A four dimensional system of ordinary differential equa­ cations can destabilize standing pulses, without the inho­ tions with delay was derived for the concentrations of four mogeneity. These bifurcations are analyzed in terms of the hormones important for the menstrual cycle in women. eigenvalues involved and the solutions produced. When in­ The primary components of the model are the regulation homogeneity is restored, we compute conditions for these of the synthesis and release of luteinizing hormone (LH) bifurcations to become Hopf bifurcations, yielding in-phase and follicle stimulating hormone (FSH) by estrogen (E2) and out-of-phase edge oscillations. and progesterone (P4) and the effects of LH and FSH on Jonathan E. Rubin the production of E2 and P4. The dynamical behavior of The Ohio State University, Columbus, OH this system is investigated numerically and analytically. E-mail: [email protected] Paul M. Schlosser Chemical Industry Institute of Toxicology Research Triangle Park, NC PS01 Breaking the Symmetry of an Attractor Merging James F. Selgrade Crisis North Carolina State University Raleigh, NC 27695 A dynamical system with a symmetry has an attractor E-mail: [email protected] merging crisis when two or more chaotic attractors merge, when varying a parameter, and suddenly a chaotic attrac­ tor of higher size appears. In the present work, a parameter set for which the symmetric double-well Duffing oscillator PS01 possesses an attractor merging crisis is considered. An ex­ The Separatrix Map Approximation of Critical ternal periodic non-harmonic perturbation is used to drive Motion in the 3/1 Jovian Resonance this dynamical system instead of a harmonic function . Chaotic asteroidal motion in the 3/1 Jovian resonance is This driver has a symmetry-breaking effect on the chaotic studied in the planar-elliptic restricted three-body prob­ attractors. Before the crisis, only one of the two coexisting lem averaged on the orbital time scale. The regime of chaotic attractors remains, while the other one becomes motion taking place near chaos border is considered. In periodic. After the crisis, the only chaotic attractor suffers low-eccentric modes of long duration, when motion sticks a transition in which a periodic attractor appears on the to the chaos border, the motion is shown to be naturally left and a chaotic attractor appears on the right. approximated by the separatrix map in Chirikov's form. Miguel A. F. Sanjuan Parameters of the approximating separatrix map are esti­ Departamento de Fisica e Instalaciones mated both analytically and by means of numeric simu­ E.T.S. de Arquitectura lation. The analytical and numeric estimates are in close Universidad Politecnica de Madrid agreement. 28040 Madrid-Spain Ivan 1. Shevchenko Email: [email protected] Institute of Theoretical Astronomy Russian Academy of Sciences Nab. Kutuzova 10, St. Petersburg 191187, Russia E-mail: [email protected] 116 Abstracts

PSOI Department of Computer Science Intermingled Basins of Attraction: Uncomputabil­ Duke University, Box 90129, Durham, NC 27708-0129 ity in a Simple Physical System Reccntly it has been shown that a symmetrical dynamical system with multiple attractors can have basins of attrac­ PSOI tion for these attractors which are intermingled in the sense Simulation of the Immune System on Parallel Com­ that every initial condition can be perturbed, with positive puters probability, into a different attractor's basin by an arbi­ The impressive advances of computer technology have trarily small random change in the location of the initial opened the way to the simulation "in machina" of com­ condition. This has the disturbing consequence that, even plex biological systems. One of the most ambitious goals if the initial condition is available with infinite precision, is the simulation of the dynamics of the immune system finite computation does not allow one to determine with response to foreign agents. In this paper we present some certainty which of the possible types of long term motion preliminary results obtained by running the Celada-Seiden the system eventually settles into. We present an exam­ immunological automaton on a highly parallel computer. ple of a physical system with a single symmetry exhibiting intermingled basins. The simplicity of our example sug­ Massimo Bernaschi gests that intermingling can occur in common situations IBM DSSC, P.G. Pastore 7 encountered in practice. 00144 Roma, Italy ([email protected] bm.com) John C. Sommerer Milton S. Eisenhower Research Center Filippo Castiglione The Johns Hopkins University Applied Physics Laboratory Department of Mathematics, University of Catania Laurel, MD 20723-6099 v.le A Doria 6, 0095, Catania, Italy Email: [email protected] ([email protected]) Edward Ott Sauro Succi University of Maryland lAC CNR, V.le Policlinico 137 College Park, MD 20742 00161 Roma, Italy ([email protected])

PSOl Global Bifurcations in Periodically Driven Zero­ PSOl Dispersion Systems Chemical Mixing using a Chaotic Flow The bifurcation analysis for non-chaotic regimes of We consider a chemical mixing device using a chaotic flow the recently discovered (Phys.Rev.E 50, R44 (1994); under the usual assumption that the reaction rate is suffi­ Phys.Rev.Lett. 76, 4453 (1996)) new type of nonlinear ciently fast that the efficiency of the reactor is determined resonance is carried out by the averaging method. It is by the efficiency of the mixing. The mixing property is shown that the global bifurcations lead to drastic changes given by the time and spatial dependence of the finite time of fluctuational transition rates. At higher amplitudes of Lyapunov exponent of the flow (e.g. see Physica D 95, the periodic drive, global bifurcations of the averaged sys­ 283-305, 1996). The limitation and optimization of such tem indicate chaos in the full system which gives rise to the devices will also be discussed. characteristic narrow-sector-like shape of chaotic regions in the bifurcation diagram. Xianzhu Tang Department of Physics S.M. Soskin College of William and Mary, Williamsburg, VA 23185 Institute of Semiconductor Physics National Ukrainian Academy of Sciences Allen H. Boozer Kiev, Ukraine Department of Applied Physics E-mail: [email protected] Columbia University, New York, NY 10027

PSOI PSOI Dimension and KLD Correlation Lengths in a Two­ Signal Compression and Information Retrieval via Dimensional Excitable Medium Symbolization We present a quantitative analysis of the complexity of Converting a continuous signal into a multi-symbol stream spatiotemporal dynamics in a two-dimensional excitable is a simple method of data compression which preserves medium. The extent to which disordered states arising much of the dynamical information embedded in the origi­ from spiral-wave breakup in a simple model of cardiac elec­ nal signal. For example, correlation timescales can be eas­ trophysiology (the Karma model) are extensively chaotic is ily recovered by varying the time delay for symbolization, assessed. In the extensively chaotic regime, we calculate di­ even at high noise levels. The presence of periodicity in the mension and KLD (Karhunen-Loeve Decomposition) cor­ signal, even if it is weak and the original signal extremely relation lengths and discuss the utility of these statistics noisy, can also be reliably detected. for characterizing the behavior of excitable media. Xianzhu Tang and Eugene R. Tracy Matthew C. Strain Department of Physics Department of Physics College of William and Mary Duke University, Box 90305, Durham, NC 27708-0305 Williamsburg, VA 23185 E-mail: [email protected] [email protected] Henry Greenside Abstracts 117

PSOI contribution. Symbolic Time Series Analysis of Continuous Dy­ Christian Uhl namical Systems Max-Planck-Institue of Cognitive Neuroscience Symbolic techniques have successfully been used to carry Inselstrasse 22-26 out parameter estimation for systems such as noisy chaotic D-04103 Leipzig, Germany maps, the Ginzburg-Landau equation, and experimental E-mail: [email protected] data. While these results are encouraging, there is much to be learned about symbolization that could potentially Raul KompaB improve its usefulness as an experimental tool. We will dis­ Institut fUr Allgemeine Psychologic cuss the effect of varying the two fundamental parameters University of Leipzig, Leipzig, Germany which define the symbolization: the choice of time delay for sampling and the partition. E. R. Tracy, X.-Z. Tang, Reggie Brown, S. Burton PSOI Physics Department Long-Time Term Behaviour of Density Oscillator College of William & Mary We examine the dynamical behaviour of a simple density Williamsburg, VA 23185 oscillator of a salt + water aqueous solution. A rhytmic E-mail: [email protected] change of the electrical potential, associated with the pe­ riodic flow was generated. We have measured the flow and liquid levels under various experimental conditions such PSOI as different molarities and several salts. As a result we Invariant Measure for Dynamical Distributed Pa­ can found a empirical fitting to the decaying behaviour of rameter Systems the oscillations that depends on a single parameter. The purpose of this work is to look for the dissipative mecha­ We present the existence of a measure invariant with re­ nisms by which the density oscillations decaying to reach spect to the dynamical systems governed by evolutive par­ tial differential equations of the first order the equilibrium. Carlos A. Vargas Universidad Autonoma Metropolitana Azcapotzalco, ou + o(Ju) = Au (0.1) Apartado Postal 13-499, ot ox Delegacion Benito Juarez, 03501 Mexico City, D.F., Mexico where we denote u(x, t) as the state variable and assume the following boundary-initial conditions: u(t,O) = 0 for Alejandro Ramirez-Rojas t 2 0 and u(O,x) = v(x) for O:S x :S 1, whith v(x) as a Universidad Autonoma Metropolitana Azcapotzalco known function. The parameter A can take values in R+ Depto. de Ciencias Basicas, and have the role of Reynolds number. By motion of (1) Apartado Postal 16-314, we mean a continuously differentiable function u( t, x) for 02200 Mexico City, D.F., Mexico which (1) is satisfied for all x 2 0, t 2 O. We consider the phenomenon of the turbulence in the state space of the motion. The behavior of the trajectory of the system de­ pends upon A. For A sufficiently small (A < 1) all motions PSOI converge to the laminar motion u == O. For large value of A Canard Explosion and Bistability in the Belousov­ (A 2 2) the system admits infinitely many turbulent mo­ Zhabotinsky Reaction tions. The purpose of this paper is to show that by use of The Belousov-Zhabotinsky (BZ) reaction displays a rich the method of compact varieties it is possible to construct variety of temporal behaviour, including bistability be­ an ergodic invariant measure for the studied systems. Such tween a limit cycle and a steady state, and canard ex­ form of a measure is positive on open non-empty sets. A re­ plosion, where the amplitude of oscillation appear to jump lationship between strictly turbulent trajectories and non­ discontinuously. A review of a co dimension three degen­ trivial invariant measure is established. An application is erate Hopf bifurcation associated with dynamical effects also presented. which could explain these observations will be presented, Zdzislaw W. Trzaska and various models of the BZ reaction will be tested to de­ Warsaw University of Technology termine the parameter ranges where bistability and canard Plac Politechniki 1 explosion are expected to occur. 00-661 Warsaw, Poland Jon-Paul Voroney E-mail: [email protected] Department of Mathematics and Statistics University of Guelph Guelph, Ontario, N1G 2W1, Canada PSOI E-mail: [email protected] Dynamics of Interacting Modes in EEG-Datasets Results of a spatiotemporal signal analysis applied to two different types of EEG-data are presented: data recorded PSOI A) during petit-mal epileptic seizures and B) during a psy­ A Model for Flows with Separation and the Circu­ chological experiment (¢ - phenomena). In case A) the lar Hydraulic Jump analysis yields a mode interaction with two different types When a liquid jet hits a plate and spreads radially, liq­ of Shilnikov dynamics [Friedrich R., Uhl, C., Physica D (in uid layer thickness increases suddenly at a certain radius. press)]. In case B) oscillatory patterns are observed and This familiar phenomenon still remains unexplained due their dynamic interaction is investigated. The underlying to strong viscous effect and presence of a free boundary. method of spatiotemporal analysis is shown in a second Moreover, our experiment observes many flow patterns and 118 Abstracts bifurcations as the flow rate is varied. Since the boundary Email: [email protected], layer theory, the standard method to treat thin viscous [email protected] flows, breaks down when separations occur, we construct a model which overcomes this difficulty. Shinya Watanabe, Vachtang Putkaradze, Tomas Bohr, PSOl Adam E. Hansen, Anders Haaning, & Clive Ellegaard The Method of Melnikov for Perturbations of Niels Bohr Institute (CATS), Multi-Degree-of-Freedom Hamiltonian Systems Blegdamsvej 17, We develop a Melnikov-type global perturbation technique DK-2100, Copenhagen, Denmark for detecting the existence of transverse homoclinic or­ bits and occurrence of homo clinic bifurcations in periodic perturbations of multi-degree-of-freedom Hamiltonian sys­ PSOl tems. The unperturbed system is assumed to have a Trapping and Wiggling: Elastohydrodynamics of saddle-center whose stable and unstable manifolds do not Driven Microfilaments coincide but intersect in a lower-dimensional manifold, and We present a general theoretical analysis of semi flexible does not have to be completely integrable. Other Melnikov­ filaments subject to viscous drag or point forcing. These type methods do not apply in this situation. We also apply are the relevant forces in dynamic experiments designed our technique to a four-degree-of-freedom model of a non­ to measure biopolymer bending moduli. By analogy with planar vibrations of a forced, buckled beam. the "Stokes problems" in hydrodynamics (fluid motion in­ Kazuyuki Yagasaki duced by that of a wall bounding a viscous fluid), we con­ Department of Mechanical Engineering sider the motion of a polymer one end of which is moved Gifu University, Gifu 501-11, Japan in an impulsive or oscillatory way. Analytical solutions for E-mail: [email protected] the time-dependent shapes of such moving polymers are obtained within an analysis applicable to small-amplitude deformations. In the case of oscillatory driving, particular PSOl attention is paid to a characteristic length determined by Chaotic Scattering in Micro Lasing Cavities the frequency of oscillation, the polymer persistence length, and the viscous drag coefficient. Experiments on actin fil­ Optical processes in micro cavities have been a subject of aments manipulated with optical traps confirm the scaling intensive study. These cavities can support high-Q whis­ law predicted by the analysis and provide a new technique pering gallery (WG) modes due to total internal reflection for measuring the elastic bending modulus. Are-analysis of the trapped light. However, inevitable deformations can of several published experiments on microtubules is also lead to the Q-spoiling of WG modes. We show that such presented. modes can be studied via chaotic scattering. We also iden­ tify the dynamical invariants that can be related to Q­ Chris H. Wiggins spoiling. Department of Physics Princeton University, Princeton, NJ 08544 Tolga Yal<;!nkaya and Ying-Cheng Lai [email protected] Department of Physics and Astronomy The University of Kansas Daniel X. Riveline and Albrecht Ott Lawrence, KS 66045 Institut Curie, Section de Physique et Chimie E-mail: [email protected] 11 Rue Pierre et Marie Curie, 75231 Paris Cedex 05 France

Raymond E. Goldstein PSOI Department of Physics and Program in Applied Mathe­ Clusters in Globally Coupled Assemblies of matics Bistable Elements University of Arizona, Tucson, AZ 85721 Assemblies of oscillators the bistability of which is caused by phase effects such as the compensation of frequency mismatch {) and its variations due to nonlinear effects PSOl (G.V. Osipov, M.M. Sushchikk, Phys. Lett. A201, 1995, Chaotic Communications in the Presence of a 205-212) are investigated. In the case of "all-to-all" cou­ Wireless Channel pling, multi stability occurs in such assemblies, i.e., there This work considers the performance of a number of com­ exist the clusters having a different number of excited el­ munication methods using chaotic continuous time sys­ ements. The features inherent in the phase mechanism of tems. The communication problem is addressed in two formation of the bistability of such an oscillator result in stages, the synchronisation method and the information nontrivial effects accompanying the interaction of such as­ encoding/decoding scheme. In particular we consider the semblies. In particular, it was revealed that when a "min­ effect of a distorting and noisy channel on a variety of per­ imal" cluster (i.e, with one excited element) of one assem­ formance measures (such as statistical analysis of the error bly is coupled with a large cluster (Nc » 1) of another signal and Bit Error Rate). Orthogonality in multi-symbol assembly, one element of the large cluster passes over to an code sets is also presented. unexcited state, while the minimal cluster persists to be in the excited state. Christopher Williams and Mark A. Beach Centre for Communications Research A.V. Yulin Merchant Venturers Building Institute for Physics of Microstructures Dept. of Electrical & Electronic Engineering Russian Academy of Science University of Bristol 46 Uljanov Str., 603600 Nizhny Novgorod, Russia Bristol, ES8 lUE, United Kingdom E-mail: [email protected] Abstracts 119

M.M. Sushchik PS02 Institute of Applied Physics Pattern Formation Inside a Laser Spectrum Russian Academy of Science The multimode class-B laser can behave as a chain of lo­ 46 Uljanov Str., 603600 Nizhny Novgorod, Russia cally coupled oscillators, each associated to one longitudi­ E-mail: [email protected] nal mode. In such a laser, one can consider the set of modes as a spatiotemporal system in which waves can propagate. We report on the observation of dispersion-induced pat­ PSOI terns in a multimode laser subjected to pump modulation. Extensive Chaos and Unstable Periodic Orbits In particular, a parametric "Faraday-like" instability is ev­ idenced analytically, numerically, and checked experimen­ Unstable periodic orbits (UPOs) associated with chaotic tally on a Nd-doped fiber laser. systems may provide an important method for analyzing and controlling high- Christophe Szwaj, Serge Bielawski, and Dominique dimensional spatially extended chaos. Using new efficient Derozier algorithms to systematically calculate UPOs for partial dif­ Laboratoire de Spectroscopie Hertzienne ferential equations with general boundary conditions, we Centre d'Etudes et de Recherches Lasers et Applications report the first calculations of UP Os for an extensively Universite des Sciences et Technologies de Lille chaotic system (the 1d Kuramoto-Sivashinsky equation) F-59655 Villeneuve d'Ascq Cedex, France and discuss some implications of these orbits for control and synchronization. Thomas Erneux Universite Libre de Bruxelles, Optique Nonlineaire Scott M. Zoldi and Henry S. Greenside Theorique, Campus Plaine, Boulevard du triomphe Department of Physics C.P.231, B-1050 Bruxelles, Belgium Duke University Box 90305, Durham, NC 27708-0305 E-mail: [email protected] PS02 Knot-Theoretic Analysis of Low-Dimensional At­ tractors PS02 Control of Impact Dynamics When embedded in a three-dimensional attractor, the in­ tertwining of unstable periodic orbits (UPO) can be an­ In this paper we show a successful numerical control of pe­ alyzed using knot theory. A systematic study of their riodic motion exhibited by one- and two-degree-of-freedom topological organization is allowed by the existence of a mechanical systems with impacts. This approach includes branched manifold, the template, on which all the UPO two new aspects. 1. We demonstrate a possibility = of can be projected while preserving their knot invariants. stability improvement of the considered periodic motion, This approach can provide a clear-cut answer to whether using MATLAB-Simulink package. 2. We propose an origi­ a model is compatible with experimental data and can be nal analytical technique applied to a one-degree-of-freedom used to construct symbolic encodings. system to predict the efficient delay loop coefficients in or­ der to achieve a desired stabilization. The last item is based G. Boulant, J. Plumecoq, S. Bielawski, D. Derozier, and on the introduction of a special "impact" map, and then M. Lefranc the problem is reduced to consideration = of the difference Laboratoire de Spectroscopie Hertzienne equations instead of differential equations. Centre d'Etudes et de Recherches Lasers et Applications Universite des Sciences et Technologies de Lille Jan Awrejcewicz and Krzysztof Tomczak F-59655 Villeneuve d'Ascq Cedex, France Technical University of Lodz Division of Control and Biomechanics (1-10) 1/15 Stefanowskiego Str., 90-924 Lodz, Poland E-mail: [email protected] PS02 Pattern Formation Inside a Laser Spectrum The multimode class-B laser can behave as a chain of lo­ PS02 cally coupled oscillators, each associated to one longitudi­ Type-II intermittency in the driven Double Scroll nal mode. In such a laser, one can consider the set of modes Circuit as a spatiotemporal system in which waves can propagate. We report on the observation of dispersion-induced pat­ In this work, we show experimental evidences, confirmed by terns in a multimode laser subjected to pump modulation. numerical results, from type-II intermittency in the driven In particular, a parametric "Faraday-like" instability is ev­ Double Scroll Circuit. Numerically, we found a new scal­ idenced analytically, numerically, and checked experimen­ ing power law dependence on the critical parameter. This tally on a Nd-doped fiber laser. result is a consequence of the first identified global bifurca­ tion scenarium for the T2 torus breakdown observed in this Christophe Szwaj, Serge Bielawski, and Dominique system: a heteroclinic saddle connection is the nonlinear Derozier mechanism responsible for the reinjection of the trajectory Laboratoire de Spectroscopie Hertzienne around a repellor focus. In fact, in this global scenarium Centre d'Etudes et de Recherches Lasers et Applications the total laminar phase is the spiraling laminar period (usu­ Universite des Sciences et Technologies de Lille ally considered) plus the time the trajectory spends in the F-59655 Villeneuve d'Ascq Cedex, France vicinity of the saddle points. Thomas Erneux Murilo S. Baptista and Ibere L. Caldas Universite Libre de Bruxelles, Optique Nonlineaire University of Sao Paulo - Institute of Physics Theorique, Campus Plaine, Boulevard du triomphe C. P. 66318, CEP 05315-970 Sao Paulo, S.P., Brazil C.P.231, B-1050 Bruxelles, Belgium E-mails:[email protected]@if.usp.br 120 Abstracts

PS02 Knot-Theoretic Analysis of Low-Dimensional At­ J.J. Collins tractors Dept. of Biomedical Engineering When embedded in a three-dimensional attractor, the in­ NeuroMuscular Research Center Boston University tertwining of unstable periodic orbits (UPO) can be an­ alyzed using knot theory. A systematic study of their 44 Cummington Street, Boston MA 02215, USA [email protected] topological organization is allowed by the existence of a branched manifold, the template, on which all the UPO can be projected while preserving their knot invariants. This approach can provide a clear-cut answer to whether PS02 a model is compatible with experimental data and can be Dynamics of the Two-Frequency Torus Breakdown used to construct symbolic encodings. in the Driven Double Scroll Circuit G. Boulant, J. Plumecoq, S. Bielawski, D. Derozier, and In this work we identified two scenarios for the two­ M. Lefranc frequency torus breakdown to chaos in the driven Dou­ Laboratoire de Spectroscopie Hertzienne ble Scroll circuit, for varying driven parameters. One is Centre d'Etudes et de Recherches Lasers et Applications through the Curry-Yorke route to chaos. For this route we Universite des Sciences et Technologies de Lille identified numerically the transition to chaos through the F-59655 Villeneuve d'Ascq Cedex, France onset of a heteroclinic tangle and its heteroclinic points. The other scenario is through the appearance of type-II intermittency for wich a quasiperiodic torus grows in size, PS02 and breaks by touching external saddle points, forming Numerical Simulation of Standing Waves in a Ver­ a heteroclinic saddle connection. This two identified dy­ tically Oscillated Granular Layer namic scenarios have two distinct structure evolution for a varying driven parameter. Chaos (measured by the Lya­ Parametric standing wave patterns spontaneously form in punov exponents) in Curry-Yorke scenario appears very a numerical simulation of vertically oscillated granular lay­ softly and alternates with phase-locking while, through ers. Particles are modeled as spheres that collide inelasti­ type-II intermittency, chaos appears abrupt ally and is pre­ cally, conserving momentum. The particles interact only served for a large range of the varying driven parameter. on contact; between collisions particles move ballistically. Spatial patterns form when the container acceleration am­ Murilo S. Baptista and Ibere L. Caldas plitude exceeds a critical value. The dependence of the University of Sao Paulo - Institute of Physics wavelength (typically 20 sphere diameters) on frequency C. P. 66318, CEP 05315-970 Sao Paulo, S.P., Brazil determined in the simulations is in agreement with exper­ E-mails:[email protected]@if.usp.br iments in our laboratory on oscillated granular layers. Chris Bizon, Mark Shattuck, Jack Swift, William McCormick, and Harry Swinney PS02 Center for Nonlinear Dynamics Optimal Dynamic Nonlinear Pricing and Capacity University of Texas Planning Austin, TX 78712 Physically constrained subscription-based telephone net­ E-mail: [email protected] work services like cellular telephone can experience op­ posing market forces which affect new product adoption. In such networks, a positive demand externality due to PS02 increases in subscribership encourages more subscribers Modular Networks for Animal Gaits to sign up whereas a negative externality in the form of congestion discourages subscriber set expansion. Dy­ The gaits of quadrupeds, walk, trot, etc., can be described namic nonlinear pricing and capacity planning strategies by spatio-temporal symmetries of periodic functions. We are solved for analytically and numerically in an optimal model gaits by periodic solutions to coupled systems of control framework. identical ODE's (or cells) with leg permutation symme­ tries. We show that four cell models cannot produce walk Susan Y. Chao and trot without leading to an undesired conjugacy be­ Department of Mechanical Engineering tween trot and pace. We circumvent this difficulty with a University of California, Berkeley modular eight cell model; one real and one virtual cell for Berkeley, CA 94720 each leg. Our model scales to many-legged animals. [email protected] Martin Golubitsky Samuel Chiu Department of Mathematics Dept. of EES-OR, Terman 320 University of Houston, Houston TX 77204-3476, USA Stanford University [email protected] Stanford, CA 94305 [email protected] Ian Stewart Mathematics Institute University of Warwick, Coventry CV4 7AL, UK [email protected] PS02 Stationary Solutions of an Integro-Differential Pietro-Luciano Buono Equation Modeling Phase Transitions: Existence Department of Mathematics University of Houston, Houston TX 77204-3476, USA [email protected] Abstracts 121

and Local Stability fluctuatc about zero are basically unshadowable. This can occur when pcriodic orbits with different numbcrs of unsta­ Wc study stationary solutions of the scalar equation ble directions coexist insidc thc attractor. The presence of a Henon ·type chaotic saddlc guarantees such coexistencc Ut = J * U - U - f(u) in a persistent manner. In this talk we will describe how This equation arises in e.g., non-local phase transition the­ these sets appear naturally in maps of more than two di­ ory. Existence of various types of patterns is analyzed mensions, how thcy can be found and what crises they through phase-plane, homotopy and variational methods. producc. Local stability/instability of those solutions is studied with Silvina Ponce Dawson and Pablo Moresco the help of linearized operators and the maximum princi­ University of Buenos Aires, Buenos Aires, Argentina ple. Adam Chmaj Brigham Young University PS02 Provo, UT 84602 Discontinuous Bifurcations and Crises in a DC IDC [email protected] Buck Converter . In our talk, we will discuss the analysis of nonlinear phe­ nomena in thc voltage controlled DC/DC buck converter. PS02 While previous numerical and experimental investigations Persistence of Compact Invariant Sets and Spectral have shown that nonlinear phenomena are indeed present Properties in the real circuit, no analytical investigations has been We discuss the persistence of compact invariant sets for or­ carried out due to the discontinuous nature of the system dinary differential equations under time-dependent pertur­ involved. By using a new kind of discrete-time mapping, bations and the behavior of the Lyapunov exponents under the impact map, we will show how conditions of existence these perturbations. Using the concept of Morse spectrum for several periodic solutions can be given and how the reccntly introduced by the authors (TAMS (1996), to ap­ Jacobian of the map can be derived analytically, allowing pear), we prove semi continuity properties the calculation of several bifurcation points. Moreover, we Fritz Colon ius will give an explanation for the sudden enlargement of the attractor, exhibited by the circuit, and show that its struc­ Institute of Mathematics University of Augsburg, 86135 Augsburg, Germany ture is organised around an unstable five-periodic solution which exists over a large range of the bifurcation param­ Wolfgang Kliemann eter and which has a large stable manifold. All through Department of Mathematics the talk, we will present both experimental and simulation Iowa State University, Ames IA 50011, USA results, which confirm what stated analytically. Mario di Bernardo Department of Engineering Mathematics PS02 University of Bristol Lattice Boltzmann Equation Model for Potassium Bristol, BS8 1TR United Kingdom Dynamics in Brain Chris J. Budd In the brain, potassium dynamics is constrained by extra­ School of Mathematical Sciences and intracellular diffusions, by active and passive trans­ University of Bath ports across cell membranes, and by the spatial buffering Bath BA7 lAY, United Kingdom mechanism. In addition, the complex brain geometry im­ poses constraints on the diffusion process. It is difficult to Alan R Champneys study such a complex system using conventional methods. Department of Engineering Mathematics Therefore, we build a lattice Boltzmann equation model for University of Bristol this system. Numerical simulations on this model are per­ Bristol, BS8 1TR United Kingdom formed, and the numerical results for the brain as a porous medium reproduce qualitatively the behavior of potassium movements obtained from the experiments with brain tis­ sue. As applications of the model, we study the effects PS02 of specific mechanisms on the clearance of potassium after Resonance Zones for Henon Maps potassium is injected into the extracellular space. We find A resonance zone is a compact set in the plane which is that various characteristics of the brain, e.g., geometry, af­ bounded by alternating initial segments of stable and un­ fects the clearance of potassium. The geometrical effects stable manifolds of saddle points. The analysis of resonance suggest that age-related potassium clearance is partly due zones in terms of their exit time decompositions is still not to age-related brain geometry changes. complete, and the goal of this work is to provide examples Longxiang Dai and Robert M. Miura of different zones where a detailed combinatorial analysis Department of Mathematics is possible. A new study of transport between adjacent University of British Columbia zones is initiated. Area preserving Henon maps are used Vancouver, B.C., V6T 1Z2 Canada to provide examples. Emails:[email protected];[email protected] Robert Easton Program in Applied Mathematics University of Colorado PS02 Boulder Co. 80309-0526 Chaos and Crises in More than Two Dimensions [email protected] Noisy chaotic trajectories with Lyapunov exponents that 122 Abstracts

PS02 St. John's, Newfoundland, Canada A1C 5S7 On Some Random Generalized Functions Having a [email protected] Mean In previous work, classes of random periodic distributions and, of random tempered distributions, were introduced, PS02 using conditions on the (random variables) coefficients se­ Image Encryption Based on Chaos quence, of the Fourier and Hermite series expansions of We present an application of two dimensional chaotic maps these random distributions. The conditions can be shortly to image encryption. A two dimensional chaotic map on a stated as follows: the sequence of the coefficients, of the torus is first generalized by introducing parameters, then it Fourier or Hermite series expansion, is a sequence of ran­ is discretized and extended to three dimensions. When ap­ dom variables whose expectations form a numeric sequence plied to a digital image, it creates a complex permutation of slow growth. It was shown that these condition char­ of pixels and gray levels. The encrypted image has a uni­ acterises the random periodic (or tempered) distributions form histogram and resembles an uncorrelated static on a having a mean. We present now, an extension of previ­ TV monitor without signal. The parameters of the chaotic ous results to random distributions' defined on manifolds, map serve as the secret key. It is shown that sensitivity by using the local character of the concept of mean of a to initial conditions and mixing guarantee good encryp­ randon distribution. tion properties. The baker map, cat map, and generalized Manuel L. Esquivel standard map are studied. Besides the applications in cryp­ Department of de Mathematics, Faculdade de Ciencias e tography, the research contributes to the study of chaotic Tecnologia maps on periodic lattices as round-off approximations to Universidade Nova de Lisboa area-preserving maps on a torus. Quinta da Torre, 2825 Monte de Caparica, Portugal Jiri Fridrich C.M.A.F.-U.L. Av. Prof. Gama Pinto, 2, Center for Intelligent Systems & 1699 Lisboa Codex, Portugal Department of Systems Science and and Centro de Matematica e Aplicagoes Industrial Engineering Fundamentais da Universidade de Lisboa, Portugal State University of New York Email: [email protected] New York, NY 13790

PS02 PS02 A Feedback Scheme for Controlling Dispersive Steps Toward the Long-Term Simulation of Turbu­ Chaos in the Complex Ginzburg-Landau Equation lent Flows through Higher-Order Incremental U n­ Experiments on convection in binary fluids in a narrow knownsjHierarchical Basis geometry exhibit irregular bursting behavior termed "dis­ Higher-order incremental unknowns/hierarchical basis are persive chaos". This system can be modeled by a complex introduced, in the context of finite-difference methods, Ginzburg-Landau equation with a large negative nonlinear- to construct effective numerical algorithms by using ap­ dispersion coefficient C2: 2 proximate inertial manifolds in order to study the long­ (8t +s8x )A = E(1+ico)AH5(1+icl)8;A+g(1+ic2) IAI A. term dynamic behavior of nonlinear dissipative evolution­ We have suppressed dispersive bursting in numerical inte­ ary equations; we consider the one-dimensional Kuramoto­ grations of this equation by applying a nonhomogeneous Sivashinsky equation (periodic hierarchical basis) and the stress parameter E' which is computed from the phase cI> of two-dimensional incompressible Navier-Stokes equations the complex variable A: E' = E+ f(cI». We also investigate (truncated hierarchical basis). The underlying matri­ the stability of the resulting profile. cial techniques are discussed in detail and an up-to­ Georg FHitgen, Yannis G. Kevrekidis date overview of the higher-order incremental unknowns Department of Chemical Engineering methodology is presented. Princeton University, Princeton, NJ 08544 Salvador Garcia Universidad de Talca, Talca, Chile Paul Kolodner E-mail: [email protected] Bell Laboratories Murray Hill, NJ 07974

PS02 PS02 Influence of Measurement Desynchronization on Causal Bifurcation Sequences in Systems Contain­ Dynamics of Ship Motion Control System ing Strong Periodic Forcing We consider the ship motion control system which includes A causal bifurcation sequence is defined here to be com­ discrete-time meters and a discrete-time controller. The posed of codimension-1 local bifurcations, global bifurca­ measurements of ship phase coordinates are performed in tions and scenarios, produced by the variation of a single the instants between two consecutive reverses ofthe rudder. parameter in a dynamical system. I will consider an ob­ The control system is called desynchronized with respect served predator-prey model behavior in which chaotic sets of measurements if the instants of measurements and for­ with symmetry evolve from attracting to transients and mation of the rudder reverses are distinct. Otherwise, it back to attracting. For this example, it is possible to de­ is called synchronized. The particular value of the rudder termine a complete set of bifurcation sequences giving rise reverse is defined regardless oftime shifts between measure­ to full or partial dynamics. ment and control instants. It leads to a change in dynamics E.A.D. Foster of controlled ship motion. This change can be characterized Department of Mathematics and Statistics by shifts of the eigenvalues of the matrix which describes a Memorial University of Newfoundland closed loop discrete control system. The estimates of these Abstracts 123 shifts for a tanker are obtained. but for f > fe stick-slip motion takes place with a dramatic increase in the sliding friction coefficient ex: fe)-1/2 as Vladimir G. Borisov N(J - Institute of Control Sciences f ---> fe, where N is the number of oscillators in the array, Moscow, 117806 Russia due to the existence of a universal saddle-node bifurcation in the dynamics. Fedor N. Grigor'yev and Nikolay A. Kuznetsov H.G.E Hentschel, Yuri Braiman, and Fereydoon Family Institute for Information Transmission Problems Department of Physics Moscow, 101447 Russia Emory University, Atlanta, GA 30322 E-mail: [email protected]

Pavel 1. Kitsul PS02 Department of Mathematics and Statistics Using Manifolds to Generate Trajectories for Li­ Mankato State University, Mankato, MN 56002, USA bration Point Missions E-mail: [email protected] Recently, there has been accelerated interest in missions utilizing trajectories near libration points. The trajectory design issues involved in missions of such complexity go PS02 beyond the lack of preliminary baseline trajectories. Suc­ Theoretical Analysis of Pattern Formation in Cir­ cessful and efficient design of mission options requires new cular Domains perspectives and a more complete understanding of the so­ Experiments on a circular flame front exhibit a variety of lution space. Dynamical systems theory has been applied stationary and non-stationary cellular states. Numerical to better understand the geometry of the phase space in integration of a model system with circular boundaries is the three-body problem via stable and unstable manifolds. shown to exhibit similar states. It is shown how the states Then, the manifolds are used to generate various solution result from the mode coupling between "rings of cells" with arcs and establish trajectory options that have been uti­ different symmetries. In particular, mode-couplings that lized in preliminary design for various proposed missions. are not naturally present in one dimensional systems, are shown to playa part in "hopping" and "ratcheting" states Kathleen C. Howell of cells. School of Aeronautics and Astronautics, GRIS 1282 Gemunu H. Gunaratne Purdue University, West Lafayette, IN 47907-1282 Department of Physics Email: [email protected] edu University of Houston Houston, TX 77204 E-mail: [email protected] PS02 Manifold Reduction and Flow Reconstruction Us­ ing Karhunen-Loeve Transformations and Neural PS02 Reconstruction Multi-Parameter Bifurcation Analysis in Nonlinear In this talk, I discuss the concept of neural charls. Given Control Systems a flow in high-dimensional space from which there ex­ This paper addresses the problem of probing the spectrum ists an approximating manifold, the neural charts algo­ of behaviour of a nonlinear control system when analysed rithm is designed to produce a local, low-dimensional, in the context of a multi- parameter bifurcation problem. semi-analytic (nonlinear) approximation to the manifold. Using a combination of local and global numerical methods From this description, we can then develop local low­ a hybrid framework of analysis is developed, which proves dimensional models of the flow. Examples will come from to be useful in uncovering the characteristics of a wide va­ the Kuramoto-Sivashiski Equation and the Mackey-Glass riety of nonlinear continuous and discrete or sampled data differential-delay equation. control systems. A number of examples are included to Douglas R. Hundley illustrate the framework of analysis. Colorado State University, Fort Collins, CO 80523 Barry R. Haynes and Mark P. Davison Department of Electronic and Electrical Engineering The University of Leeds, Leeds, LS2 9JT, United Kingdom PS02 B [email protected] Applications of the KAM Theory to http://www.elec-eng.leeds.ac.uk/ Commensurate-Incomensurate Phase Transitions This paper gives a brief review of some of our recent works in the the applications of the KAM thory to commensurate­ PS02 incommensurate (CI) phase tranistions. We use a simple Stick-Slip Dynamics and Friction in an Array of one-dimensional model called the Frenkel-Kontorova (FK) Coupled Nonlinear Oscillators model to describe CI transitions and vaious forms of the We present the results of our study of stick-slip motion in a interparticle and external potentials are used. Many inter­ 1D array of nearest-neighbor coupled nonlinear oscillators, esting and new results have been obtained. all subject to the same external force. Our motivation is Bambi Hu to study friction and the underlying mechanism leading to Department of Physics stick-slip motion in the strongly nonlinear regime. We have and Centre for Nonlinear and Complex Systems developed simple formalism to reduce the complexity of Hong Kong Baptist University the equations and to calculate the velocity and the friction Kowloon Tong, Hong Kong coefficient of the elements in the array as a function of the E-mail: [email protected] driving force f. For f < fe only stick behaviour occurs, 124 Abstracts

and with the omega-limit set of the dynamical process. Department of Physics Hans G. Kaper University of Houston Mathematics and Computer Science Division Houston TX 77204 Argonne National Laboratory Argonne, IL 60439-4844, USA

PS02 Peter Takac, Computation of Invariant Tori Near Ll in the Fachbereich Mathematik, Earth-Sun System Universitiit Rostock, The final goal of this work is to describe the dynamics Universitiitsplatz 1, near the Lagrangian point Ll of the real Earth-Sun sys­ D-18055 Rostock, Germany tem. The first step is to build a suitable model, based on the Earth-Sun Restricted Three Body Problem plus quasiperiodic time-dependent perturbations coming from PS02 the real motion of the bodies of the Solar system. Then, Chaos Synchronization and Riddled Basins in Two using normal form techniques, we will compute several in­ Coupled One-Dimensional Maps variant tori as well as integrable approximations to the The connection between chaos synchronization and phe­ dynamics, that allow to give a complete description of the nomena of riddled basins is disscused for a family of two-di­ solutions nearby. mensional piecewise linear endomorphisms which consist of Gerard Gomez two linearly coupled one-dimensional maps. Under analyt­ Dept. Matematica Aplicada i Analisi ically given conditions chaotic behavior in both maps can Universitat de Barcelona synchronized but synchronized state is characterized by dif­ Gran Via 585 ferent types of stability. The mechanism of occurrence of 08071 Barcelona, Spain riddled basins is described in details. T. Kapitaniak Angel Jorba and Josep Masdemont Division of Dynamics Dept. Matematica Aplicada I - ETSEIB Technical University of Lodz Universitat Politecnica de Catalunya Stefanowskiego 1/15, 90-924 Lodz, Poland Diagonal 647 E-mail: [email protected] 08028 Barcelona, Spain E-mail: [email protected] Y. Maistrenko Institute of Mathematics Academy of Sciences of Ukraine PS02 3 Tereshchenkivska st, Kiev 252601 Ukraine Finitely Representable Set-Valued Maps and Com­ putation of the Conley Index The class of representable usc multivalued maps is dis­ PS02 cussed. Such maps first appeared in the Mrozek­ Dichotomies and Numerics of Stiff Problems Mischaikow proof of chaos in the Lorenz equations. Given Stiff problems are a classical problem in numerical anal­ a continuous map f in an euclidean space, its cubic-valued ysis of ordinary differential equations. In a quantitative representation F can be computed. Iheritable properties of way the problem is stiff, if the linearization of the vector F are used to provide algorithmic constuctions of isolating field along a solution has an exponential dichotomy with neighbourhoods, index pairs, the homomorphisms induced widely separated constants. We show that the existence in homology by f and, finally, the Conley Index of f. of a similar discrete dichotomy for the linearization of a Tomasz Kaczynski given discretization method is an efficient and unifying cri­ Departement de Mathematique et d'Informatique terion for the suitability of the discretization method for Universite de Sherbrooke stiff problems. Sherbrooke, Quebec, Canada JIK 2Rl B. Katzengruber and P. Szmolyan E-mail: [email protected] Institut fUr Angewandte und Numerische Mathematik Technische Universitiit Wien Wiedner Haupstrasse 8-10 PS02 1040 Wien, Austria Dynamics of the Ginzburg-Landau Equations of E-mail: [email protected] Superconductivity In the one-parameter family of "¢ = -w(V . A)" gauges (w > 0), the time-dependent Ginzburg-Landau (TDGL) PS02 equations of superconductivity define a dynamical process Application of the Theory of Optimal Control of when the applied magnetic field varies with time and a Distributed Media to Nonlinear Optical Fiber dynamical system when the applied magnetic field is sta­ Soliton propagating in optical fiber amplifiers lose ideal tionary. When the applied magnetic field is stationary, soliton shape and nonsoliton radiation appears. We con­ every solution of the TDGL equations is attracted to a set sider the optimal control problem for distributed media of stationary solutions, which are divergence free. When with functional describing deviation of ideal soliton with the applied magnetic field is asymptotically stationary, the increasing amplitude from real solution subject to the Non­ dynamical process is asymptotically autonomous and ap­ linear Schredinger Equation with variable coefficients with proaches a dynamical system, whose attractor coincides gain -y(z) as control. Obtained analytical solution de­ scribe optimal amplification of solitons without formation Abstracts 125

of nonsoliton radiation. Method was applied also for phase­ chemical catalytic reactions, decaying, regular and relaxing sensitive amplifier and for timing jitter effect. oscillations were experimentally found. The establishment of interrelation between the reaction mechanisms and os­ Vladimir Y. Khasilev cillation forms is of some interest. It is known that for Institute of Physics, University of Rostov the describing of decaying oscillation it is quite enough to Stachki Street, 194, 344090, Rostov on Don, Russia use three and more stage linear reaction schemes which E-mail: [email protected] are characterized by a single steady inner stationary state E-mail: [email protected] (ISS). If there is a single unsteady ISS, then non-decaying, FAX: 7(863-2)285044 regular and relaxing oscillations arise. Such regimes are characteristic for reactions which proceed via three and more stages when not less than two independent inter­ PS02 mediate substances take part. They are to satisfy some When Can a Dynamical System Represent an Ir­ definite requirements on stoichiometry. The systematiza­ reversible Process tion of all possible mechanisms describing auto-oscillating Processes, V, which are both extremal and reversible regimes was done corcerning the group of three- and four­ do not exist for symplectic (non-canonical) 4D physi­ stage reactions. For the analysis of the schemes of such cal systems defined by a I-form of action, A. Evolu­ reactions we determined the common stoichiometric con­ tionary vector fields must admit anholonomic differential ditions of unstability which help to identify the reaction fluctuations in position (~x-pressure) and/or in velocity schemes describing non-decaying oscillating regimes. The (~v-temperature). If ~x = 0, a kinematic dynamical sys­ requirement of singularity of ISS is taken into account. For tem exists which represents an irreversible process when the systematization of mechanisms a computer programm the virtual work I-form W=i(V)dA is not closed. was compiled which includes the following subprograms: the generation of mechanisms which are not repeated; the R. M. Kiehn analisys of unstability and singularity of ISS. Due to this Physics Department program the classification of three- and four-stage mecha­ University of Houston, Houston, TX 77204 nisms was done. The application of generated mechanisms in reactions of CO and H2 oxidation on platinum metals was investigated. PS02 N.L Koltsov The Computer Analysis of Poincare Cross-Sections Chuvash State University in Chemical Reactions Dynamics Cheboksary 428015, Russia There are some ways to investigate chemical dynamics Fax: (+78350)428090, E-mail: [email protected] which is described by a system of ordinary differential equations. One of them is based on the analysis of coeffi­ cients of characteristic polynom. The computer realization PS02 of this way is given in papers. Another way is construc­ Scaling Behavior of Transition to Chaos in tion and analysis of Poincare cross-sections for trajecto­ Quasiperiodically Driven Dynamical Systems ries of the investigated system. We worked out the com­ puter program which constructs Poincare cross- sections A route to chaos in quasiperiodically driven dynamical according to the below-described algorithm of the system systems is investigated whereby the Lyapunov exponent of differential equations of chemical kinetics. A hyperplane passes through zero linearly near the transition. A dynam­ which contains a special point of the investigated system ical consequence is that, after the transition, the collective and which divides n-measured phase space into two regions behavior of an ensemble of trajectories on the chaotic at­ is chosen. A multitude of points forming Poincare cross­ tractor exhibits an extreme type of intermittency. The sections are obtained on the base of the data on the de­ scaling behavior of various measurable quantities near the pendence of time change of the concentration of the one transition is examined. of reacting substances and cross-section points of this de­ Ying-Cheng Lai pendence with the hyperplane. The dynamics character of Department of Physics and Astronomy an initial chemical system can be established according to The University of Kansas, Lawrence, KS 66045 the topological properties of an obtained cross-section. For E-mail: [email protected] example, if the solution of the differential equation system is periodic, then the corresponding Poincare cross-section Ulrike Feudel is either one point or some points. In this case the terminal Max-Planck Arbeitsgruppe "Nichtlineare Dynamik" cycle is twisting. For decaying periodic oscillations many Universitiit Potsdam, Potsdam, Germany points are formed which direct to the single terminal point which correspond to the focus on a phase portrait. The Celso Grebogi program was applied to the investigation of self-oscillations Institute for Plasma Research in heterogeneous catalytic reaction kinetics. University of Maryland, College Park, MD 20742 Kozhevnikov LV. and Koltsov N.L Department of Physical Chemistry Chuvash State University, Cheboksary 428015 PS02 Fax: (+78350)428090, E-mail: [email protected] Noise-Induced Riddling in Chaotic Systems Recent works have considered the situation of riddling where, when a chaotic attractor lying in an invariant sub­ PS02 space is transversely stable, the basin of the attractor can The Modellong of Self-Oscillating Regimes in Cat­ be riddled with holes that belong to the basin of another at­ alytic Reaction Dynamics tractor. We show that riddling can be induced by arbitrar­ While investigating kinetic regularities of a number of ily small random noise even if the attractor is transversely 126 Abstracts

unstable, and we obtain universal scaling laws for noise­ mics induced riddling. Our results imply that the phenomenon We present a simple model of the solar system using a series of riddling can be more prevalent than expected before, as of planar circular restricted three body systems. N umeri­ noise is practically inevitable in dynamical systems. cal computations of the invariant manifolds provide a re­ Ying-Cheng Lai markable picture of the choatic diffusion process within the Department of Physics and Astronomy solar system. This is not Arnold diffusion since the system The University of Kansas, Lawrence, KS 66045 has only two degrees of freedom and is not near-integrable. E-mail: [email protected] This diffusion process is the basis for much of the dynam­ ics of the solar system. It brings together many disparate Celso Grebogi phenomena within solar system dynamics and provides a Institute for Plasma Research coherent and simple explanation for their behavior. University of Maryland, College Park, MD 20742 Martin W. Lo Jet Propulsion Laboratory California Institute of Technology PS02 4800 Oak Grove Dr. 301/142 Stacked Lagrange Tops: Analysis of Multi- Pasadena, CA 91109 Component Systems [email protected] Techniques from geometric mechanics and numerical anal­ ysis can be combined to efficiently study large nonlinear Shane D. Ross systems. Systems of coupled rigid bodies are an important Jet Propulsion Laboratory class of such systems. A representative example consists of California Institute of Technology a collection of Lagrange tops linked along their axes of sym­ 4800 Oak Grove Dr. 301/142 metry and moving under the influence of gravity. The tops Pasadena, CA 91109 can undergo steady motions consisting of combinations of [email protected] spins about their axes of symmetry and an overall spatial rotation. The equations determining such steady motions are essentially linear and the structure of the relevant ma­ PS02 trices leads to easily evaluated criteria for the existence and Transport Properties in Disordered Ratchet Poten­ stability of steady motions. Any such matrix can always tials be expressed as the sum of a triangular matrix and a rank The role of disorder in one-dimensional periodic ratchet one matrix; those associated to steady motions such that potentials is investigated by introducing: (i) impurities - a the axes of symmetry of the tops do not all lie in a common certain fraction of the asymmetric unit cells are replaced by vertical plane, have an additional 'overlapping block' struc­ unit cells with opposite asymmetry; (ii) randomness - all ture. The stability of the relative equilibria can be studied unit cells have the same asymmetry, but random size. The using the reduced energy momentum method of Simo et relevant color induced currents are determined and com­ al.. Sharp, simple, and physically intuitive stability con­ pared with the current in the ideal periodic ratchet poten­ ditions can be found for several classes of steady motions; tial. Altogether, disorder is shown to quench the effective­ these classes exist for arbitrarily large collections of tops. ness of thermal ratchets, while novel transport properties Debra Lewis become detectable. University of California, Santa Cruz F. Marchesoni Santa Cruz, CA 95064 Department of Physics E-mail: [email protected] Loomis Laboratory of Physics University of Illinois Urbana, II 61801 PS02 E-mail: [email protected] Recurrent Propagation of a Wave Front in an Ex­ citable Medium We illustrate how a diffusable substance that is concen­ PS02 trated locally in a finite domain of space may spread out Dynamical Precursor of Black Hole Formation N u­ in the form of a spatially recurrent wave. Here, spatially merical Simulation recurrent wave refers to a wave front that repetitively re­ The formation of a massive black hole in a galactic center verses its direction and returns to the site of its origin requires a dynamical mechanism for the concentration of before reaching out further in the space. Such recurrent mass in a self gravitating system. Thermodynamics sug­ waves arise in a set of 3 partial differential equations (with gests the possibility of a density singularity. Here we in­ only 1 diffusable variable) that describes the propagation vestigate the dynamics of a system of concentric spherical of calcium waves observed oocytes. A Fitzhugh-Nagumo mass shells constrained by an inner and outer reflecting analogue of such equations is formulated and analyzed. barrier. We show that, as the inner barrier becomes small, Yue-Xian Li a transition is observed and we find coexistence between Institute of Theoretical Dynamics an extended and a condensed phase. University of California, Davis, CA 95616 Bruce N. Miller and V. Paige Youngkins E-mail: [email protected] Physics Department Texas Christian University Fort Worth, TX 76129 PS02 E-mail: [email protected] The Invariant Manifolds and Solar System Dyan- Abstracts 127

PS02 cal and Quantal Distributions Integrals of Motion and the Shape of the Attractor We present the first analytically derived asymptotic-time for the Lorenz Model diagnostic for chaos in the dynamics of distributions in a In this work, we consider three-dimensional dynamical sys­ Hamiltonian system. We establish that in a chaotic system tems, as for example the Lorenz model. For these sys­ a phase-space distribution acquires structure at increas­ tems, we introduce a method for obtaining families of two­ ingly smaller scales at an exponential rate given by the dimensional surfaces such that trajectories cross each sur­ largest Lyapunov exponent of the point dynamics. Our fi­ face of the family in the same direction. For obtaining these nal expression for the Lyapunov exponent is trajectory in­ surfaces, we are guided by the integrals of motion that ex­ dependent: We show how this method may be applied, for ist for particular values of the parameters of the system. instance, in the case of chaotic advection of passive scalars Nonetheless families of surfaces are obtained for arbitrary in fluid dynamics. We also apply this method to the classi­ values of these parameters. Only a bounded region of the cal and quantal Cat Map and show that the Ii -+ 0 limit of phase space is not filled by these surfaces. The global at­ the quantum map yields the classical, fully chaotic, result. tractor of the system must be contained in this region. In We also demonstrate that chaos is supressed as Ii increases this way, we obtain information on the shape and location and the growth of structure saturates at O( Ii). of the global attractor; this information can be used to Arjendu K. Pattanayak and Paul Brumer improve the calculation of the Lyapunov exponents of the Chemical Physics Theory Group attractor. University of Toronto Sebastien Neukirch and Hector Giacomini Toronto, Ontario, Canada M5S 3H6 C.N.R.S. Laboratoire de Mathematiques et de Physique Email: [email protected] Theorique Universite de Tours, France E-mail: [email protected] PS02 A Master Stability Function for Synchronization in Coupled Arrays of Oscillators PS02 It's well know from past work that it is possible to synchro­ Two-dimensional Josephson-Junction Network Ar­ nize coupled oscillators, whether chaotic or limit cycle. We chitecture for Maximum Microwave Radiation show here that it is possible to solve, once and for all, Emission the stability problem of synchronizing any array of identi­ We design a two-dimensional Josephson-junction array ar­ cal oscillators. The scheme which gives a master stability chitecture such that all array oscillators self-synchronize function will work for any array of linearly-coupled oscil­ into a coherent in-phase state with maximum microwave lators. We apply this to a system of 8 coupled Rossler-like radiation output in the entire frequency range. Flux quan­ circuits. tization implies a global current coupling scheme so that Lou Pecora, Tom Carroll, Gregg Johnson, and Doug Mar the in-phase oscillation is a global attractor ofthe nonlinear Code 6343, Naval Research Laboratory network dynamics independent of initial conditions, noise Washington, DC 20375-5000 and junction imperfections. The microwave power coupled E-mail: [email protected] to a matched load depends only on flc and the size of the http:// natasha. umsl.edu/Exp_Chaos4 array. Joerg Oppenlaender and Werner Guettinger University of Tuebingen PS02 Institute for Theoretical Physics On a Certain Class of Zero-Sum Discrete-Time D-72074 Tuebingen, Koestlinstr.6, Germany Stochastic Games We develop some aspects of a two-players zero-sum stochastic dynamic game theory for a class of discrete-time PS02 nonlinear dynamic systems which depends on upon a jump Scaling in Forced Laser Synchronization parameter. We consider the class of Markovian Strategies. We have found a scaling law for the loss of forced synchro­ The concept of ess inf (sup) in conjunction with the no­ nization in single-mode lasers. The degrading parameters tion of Markovian bifurcation devised here, and a certain are the corresponding cavity frequency and the atomic fre­ consistency property, are the cornerstone of our work. Re­ quency of the synchronizing lasers. There is a small region lying on these elements we derive, inter alia, the following in the parameter space of the frequencies where the scaling results: law breaks up. The corresponding scaling exponents are 1- The saddle point is generically characterized in two dif­ universal in the sense that they do not depend on the laser ferent ways: via a probabilistic principle of optimality operation regime or whether the laser belongs to class-A, and also from a martingale point of view; class-B or class-C. 2- A stochastic Isaacs equation. Carlos L. Pando L. Universidad Aut6noma de Puebla D. Leao Pinto, Jr. Instituto de Fisica Universidade Estadual de Campinas - UNICAMP Apdo. Postal J-48 Faculdade de Engenharia Eletrica, C. P. 6101, Puebla,Pue 72570, Mexico CEP 13081-970 - Campinas, Sao Paulo, Brazil

Marcelo Dutra Fragoso PS02 National Laboratory for Scientific Computing - LNCC - Lyapunov Exponents from the Dynamics of Classi- CNPq- Rua Lauro Muller 455, CEP 22.290-160 - RJ- Brazil 128 Abstracts

E-mail: [email protected] We sought solutions u of the equation

Joao Bosco Ribeiro do Val ,uxxxx = U xx + feu). (3) Universidade Estadual de Campinas - UNICAMP Faculdade de Engenharia Eletrica, C. P. 6101, Consistent with (2), we required that CEP 13081-970 - Campinas, Sao Paulo, Brazil (u, u', u", U'") --+ (1,0,0,0) as x --+ 00. (4)

PS02 The presentation is concerned with some qualitative prop­ Flow-Reversal Systems with Fast Switching erties of solutions u of Problem (3), (4) without making a-priori restrictions that they are even or monotone. We present a class of dynamical systems described by a composition of two conjugated flows. This class of sys­ A.I. Rotariu-Bruma and L. A. Peletier tems arose from experimental studies of catalytic reactors Department of Mathematics with periodic flow-reversal. We present an abstract frame­ University of Leiden work for such systems and some results of numerical exper­ Niels Bohrweg 1 iments revealing a rich structure of spatio-temporal chaos. 2300 RA Leiden, The Netherlands Then we focus at the" fast switching" for finite dimensional E-mail: [email protected] flow-reversal systems and use linearization and Campbell­ Baker-Hausdorff formula to obtain approximations of the W. C. Troy asymptotic behavior of such systems. University of Pittsburgh Pittsburg, P A 15260 Jan Rehacek Center for Nonlinear Studies, MS-B258 Los Alamos National Laboratory Los Alamos, NM 87544 PS02 [email protected] Instabilities of Hexagon Patterns in a Model for Rotating Convection Alois Klic, Pavel Pokorny When rotation about a vertical axis breaks the chiral sym­ Department of Mathematics metry in a convection system, the Kiippers-Lortz insta­ Institute of Chemical Technology bility leads to a switching between convection rolls of dif­ Technicka 5 ferent orientations. If the vertical reflection symmetry is 16628 Prague, Czech Republic broken hexagonal patterns arise. We investigate in an [email protected], [email protected] extended Swift-Hohenberg equation whether a Kiippers­ Lortz type instability can arise for hexagons; i.e. whether the hexagons can be unstable to hexagons rotated at an PS02 angle ¢ "# 60°,120°, .... Numerical analysis will determine Sudden Changes of Invariant Chaotic Sets in Pla­ the nonlinear behavior. nar Dynamical Systems Filip Sain and Hermann Riecke We investigate the nature of sudden discontinuous changes Dept Engineering Sciences & Applied Mathematics in invariant sets as a parameter is varied by analyzing the Northwestern University, Evanston, IL 60208 behavior of different crises and metamorphoses occuring in E-mail: [email protected] 2D invertible maps of the plane. This leads to a gener­ alization of the notions of crises of chaotic attractors and metamorphoses of basins boundaries to the occurrence of PS02 sudden changes of chaotic invariant sets in general. Classification of Nonlinear Time Series using Clus­ Carl Robert ter Analysis University of Maryland, College Park, MD 20742 We address the problem of distinguishing different dynam­ ical states of a system, based on time series data. Usu­ Kathleen T. Alligood ally, this is done by measuring one or a few characteristic George Mason University, Fairfax, VA 22030 quantities. Then these quantities are arranged into groups. We propose to avoid this severe reduction of information Edward Ott and James A. Yorke by comparing the time series directly, not via estimates University of Maryland, College Park, MD 20742 of parameters. A dissimilarity matrix, as obtained e.g. from nonlinear cross-predictions, can then be used to form groups of similar signals. PS02 Thomas Schreiber Pulse Like Spatial Patterns Described by Higher Physics Department Order Model Equations University of Wuppertal We studied the stationary solutions of the equation D-42097 Wuppertal, Germany E-mail: [email protected] Ut = -,Uxxxx + U xx + feu). (1) Of particular interest to us were equations in which f is PS02 such that the related equation Ut = feu) has two stable The Geometric Phase in a Point Vortex Flow in a states u = u± separeted by one unstable state u = Uo. A typical function f is Circle The existense of a geometric phase is shown in the adia­ 2 f(u)=(u-a)(1-u ) for aE(-l,l). (2) batic evolution of the angle variable of a fluid particle in a 129 Abstracts planar inviscid, incompressible point vortex flow in a circu­ a previous study on the limit cycle of the Van der Pol's lar domain. The velocity field is the linear superposition of equation (1990, 1984). The recurrence of this phenomenon that due to an isolated vortex in an unbounded plane and in the present problem suggests that it is very likely to that arising due to the circular boundary. The latter field affect convergence of the perturbation series for periodic makes the point vortex in any eccentric position move in a solutions of other differential equations. circular path with constant frequency and introduces time­ Mohammad Tajdari varying periodic coefficients, due to the circular motion of Department of Mathematical & Computer Science the vortex, in the equations of motion of the fluid particle. University of New England, Biddeford, ME 04005-9599 We set up a non-dimensional time period T which grows E-mail: [email protected] without bound as the fluid particle approaches the parent vortex and, thus, defines an adiabatic process. The peri­ odic coefficients can then be viewed as varying on a "slow" time. Using classical multi-scale asymptotics we then show PS02 Can Markoff Chain Theory Determine a Signals that the leading order term in the angle variable of the fluid Origin: Chaos or Random? particle decomposes into a! "fast" term and a "slow" term. The value of the "slow" term at the end of the period T We present a hypothesis test that can be used to place is shown to be independent of T. Further, it is expressible limits on the likelihood of randomness in complex time se­ as the integral of a I-form defined on R2 - {O} around the ries. This process uses time based clustering to transform circular path of the vortex and is therefore identified as a the original time series into a symbol string, that is ana­ geometric phase. lyzed for symbol combination exclusions. These exclusions are easily visualized by using an Iterated Function System B. N. Shashikanth and P. K. Newton Map. Finally, the hypothesis test is implemented using Department of Aerospace Engineering Markoff chains to calculate the probability of missing and University of Southern California existing symbol combinations. Los Angeles, CA 90089 Email: [email protected] Charles R. Tolle Utah State University Rocky Mountain NASA Space Grant Consortium Department of Electrical and Computer Engineering PS02 Nonlinear Dynamics of Discrete Ecosystem Models Logan, UT 84322 Subject to Periodic Forcing David Peak Discrete ecosystem models are, in general, nonlinear, multi­ Utah State University species, and involve difference equations containing control Department of Physics parameters that relate to such properties as the intrinsic Logan, UT 84322 growth-rate of a species. To simulate seasonal effects, we subject to periodic forcing the control parameters in well­ Robert Gunderson known one-species models, including those of May, Moran­ Utah State University Ricker, and Hassell, and also a Maynard Smith predator­ Department of Electrical and Computer Engineering prey model. Rich dynamical behavior, including chaos, Logan, UT 84322 is found in each model for certain forcing amplitudes and frequencies. Danny Summers, Justin G. Cranford, and Brian P. Healey PS02 Department of Mathematics and Statistics Deterministic chaos and analysis of singularities Memorial University of Newfoundland Of practical interest to us is the transition from chaotic to St. John's, Newfoundland nonchaotic dynamics in Hamiltonian systems. This tran­ A1C 5S7, Canada sition plays a fundamental role in a very wide variety of physical, chemical, etc. problems (turbulence in incom­ pressible fluids, scattering of atoms from metal surfaces, PS02 complex formation in atom-diatom collisions, etc.). The Moving Singularities of the Perturbation Ex­ We study this phenomenon on the example of singularly pansion of the Classical Kepler Problem 2£..u d 2 2 perturbed equation E dx4 + ~ = y +y, where E is a small The convergence properties of the perturbation expansion parameter. The subject of our interest are deformations of 2 for the periodic solution of the Kepler problem are dis­ the homo clinic (separatrix) solution y = 1.5sinh- (x/2) cussed as a function of an "amplitude" parameter a asso­ to the nonperturbed equations. Starting with the Kruskal­ ciated with the solutions. An infinite number of movable Segur method of "asymptotic beyond all orders" , we show singularities occurring the complex a-plane as conjugate that chaotic dynamics is governed by interaction of 3 reg­ pairs are found to cause the divergence of the perturba­ ular singular points in the so-called Borel plane. tion solution. The movable singularities occur at loca­ Alexander Tovbis tions where the differential equation under consideration Department of Mathematics is singular. These singularities can explain the nonuni­ University of Central Florida form convergence noted by Melvin (1977) in his study on Orlando, FL 32816-1364 constructing Poincare-Lindstedt series for a class of differ­ E-mail: [email protected] ential equations of oscillator type. Analysis near one of the complex singularities indicates that distinct branches of solution occur there. These solutions undergo transition at the singularity to develop into new solution branches PS02 with distinctively different properties. Moving singulari­ A Hilbert Space Framework for the Analysis of Bi- ties similar to those for the Kepler problem were found in 130 Abstracts furcations in Control PS02 For the bifurcation analysis in nonlinear parametric opti­ Non-Collision Singularities in a 6 Body Problem mization one uses a Banach space setting for the smooth We show the existence of noncollision singularities in a bifurcation analysis and a Hilbert space setting for the Newtonian 6-body problem. The proof uses the "collision spectral analysis. In this talk we present a framework manifold" technique of McGehee and as well as a numeri­ addressing both issues and demonstrate how bifurcations cal component in which the unstable manifold of a certain arise from a loss of controllability of the linearized dynam­ fixed point are carefully traced. The existence of non- colli­ ical system as well as a second order condition. sion singularities was recently shown in a 5-body problem. Aubrey B. Poore and Timothy J. Trenary The present work has the advantage that it does not re­ Colorado State University, Fort Collins, Co 80523 quire any small mass ratios as in the previous work. E-mail: [email protected] Jeff Xia and Todd Young Northwestern University, Evanston, IL 60208 E-mail: [email protected] PS02 Use of Parametric Excitation for Unstable Fixed Point Detection and Capture PS02 If a parameter of a deterministic system is intentionally Global Bifurcations in a Reaction-Difussion Model perturbed by small erratic variations, the system may visit Leading to Spatiotemporal Chaos unstable orbits distant from the unperturbed attractor. The traveling-wave ODE of a two-species reaction-diffusion Several mechanisms will be discussed whereby parameter PDE in one spatial dimension is investigated. It is found perturbations can lead to distant excursions from the sys­ that the break-up of a heteroclinic cycle between two fixed tem's unperturbed attractor set, allowing the system to points is at the onset of spatiotemporal chaos in the PDE enter the local area of unstable orbits. Once local to the model. This global bifurcation unfolds with two branches orbit, certain stabilizing linear control strategies can be ap­ of homo clinic orbits, emmanating from the co dimension 2 plied. point. In terms of the PDE simulations, we can argue that onset of spatiotemporal chaos corresponds to a stable pulse Peter M. Van Wirt, Maj, USAF solution "replaced" by another unstable one. Air Force Institute of Technology Martin Zimmermann, Sascha Firle, Center for Intelligent Systems Department of Quantum Chemistry, Department of Electrical and Computer Engineering Uppsala University, Sweden, Utah State University Logan, Utah 84322 Mario Natiello, Department of Mathematics, Robert Gunderson, PhD Royal Institute of Technology, Stockholm, Sweden, Center for Intelligent Systems Department of Electrical and Computer Engineering Michael Hildebrand, Markus Eiswirth, Utah State University Fritz-Haber Institut, Logan, Utah 84322 Max-Planck Gesellschaft, Berlin, Germany,

David Peak, PhD Markus Bar, Department of Physics Max-Planck-Institut fUr Physik Komplexer Systeme, Utah State University Dresden, Germany Logan, Utah 84322 A. K. Bangia and Y. Kevrekidis Dept. of Chemical Engineering, PS02 Princeton University, USA. Lyapunov Transforms and Invariant Stability Ex­ ponents We investigate the conditions under which a Lyapunov transform can yield stability exponents which are coordi­ nate frame invariant. The maximal group of transforma­ tions which permit this is shown to be general autonomous linear transformations. The standard Lyapunov exponents are invariant only under rotations, while an eigenvalue de­ composition does yield the predicted coordinate invariance. Other possible decompositions of the fundamental matrix are explored numerically. William E. Wiesel Air Force Institute of Technology/ENY Wright-Patterson AFB, OH 45434-7765 Email: [email protected]