DOCSLIB.ORG

Final State Sensitivity: an Obstruction to Predictability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Final State Sensitivity: an Obstruction to Predictability Volume 99A, number 9 PHYSICS LETTERS 26 December 1983 FINAL STATE SENSITIVITY: AN OBSTRUCTION TO PREDICTABILITY Celso GREBOGI 1 Steven W. McDONALD 1 , Edward OTT 1'2 and James A. YORKE 3 University of Maryland, College Park, MD 20742, USA Received 21 October 1983 It is shown that nonlinear systems with multiple attractors commonly require very accurate initial conditions for the reliable prediction of final states. A scaling exponent for the final-state-uncertain phase space volume dependence on un- certainty in initial conditions is defined and related to the fractai dimension of basin boundaries. Typical nonlinear dynamical systems may have more is the "basin bounctary". It the initial conaitions are than one possible time-asymptotic final state. In such uncertain by an amount e, then (cf. fig. 1), for those cases the final state that is eventually reached depends initial conditions within e of the boundary, we cannot on the initial state of the system. In this letter we say a priori to which attractor the orbit eventually wish to consider the extent to which uncertainty in tends. For example, in fig. 1, 1 and 2 represent two initial conditions leads to uncertainty in the final state. initial conditions with an uncertainty e. The orbit To orient the discussion, consider the simple two- generated by initial condition 1 is attracted to attrac- dimensional phase space diagram schematically de- tor B. Initial condition 2, however, is uncertain in the picted in fig. 1. There are two possible final states sense that the orbit generated by 2 may be attracted ("attractors") denoted A and B. Initial conditions on either to A or to B. In particular, consider the fraction one side of the boundary, Z, eventually asymptote to of the uncertain phase space volume within the rec- B, while those on the other side of Z eventually go to tangle shown and denote this fraction f. For the case A. The region to the left (right) of Z is the "basin of shown in fig. 1, we clearly have f ~ e. It is one of the attraction" for attractor A (or B, respectively) and main points of this letter that, from the point of view of prediction, much worse scalings of f with e fre- quently occur in nonlinear dynamics. In particular, if 1 Laboratory for Plasma and Fusion Energy Studies, and De- partment of Physics and Astronomy. f ~e ~ , (1) 2 Also, Department of Electrical Engineering. 3 Institute for Physical Science and Technology, and Depart- with c~ < 1, we shall say that there is final state sensi- ment of Mathematics. tivity. In fact, a substantially less than one is, we be- lieve, fairly common. In such a case, a substantial im- provement in the initial condition uncertainty, e, yields only a relatively small decrease in the uncertainty of the final state as measured by f. While ¢x is equal to one for simple basin boundaries, such as that depicted in fig. 1, highly convoluted boundaries with noninteger (fractal) dimension also occur. We use here the capacity definition of dimen- sion [ 1 ], Fig. 1. A region of phase space divided by the basin boundary Y. into basins of attraction for the two attractors A and B. 1 In N(8) and 2 represent two initial conditions with uncertainty e. d = lim (2) 6--,0 ln(1/8) ' 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 415 Volume 99A, number 9 PHYSICS LETTERS 26 December 1983 where, if the dimensionality of the relevant phase space is D, then N(6) is the minimum number of D ( dimensional cubes of side 6 needed to cover the basin boundary. In general, since the basin boundary di- c vides the phase space, its dimension d must satisfy d ~> D-1. It can be proven ,1 [2] that the following c relation between the index c~ and the basin boundary dimension holds -( c~ = O - d. (3) -( For a simple boundary, such as that depicted in fig. 1, we have d = D-l, and eq. (3) then gives c~ = 1, as ex- O rr/2 Jr pected. For a fractal basin boundary, d > D-l, and Fig. 2. Basins of attraction for eqs. (4). Arrows denote the eq. (3) gives a < 1 (i.e., there is final state sensitivity). two attractors. To see heuristically how (3) comes about from (2), set the cube edge 6 equal to the initial condition un- certainty e. The volume of the uncertain region of these parameters we find numerically that the only phase space will be of the order of the total volume attractors are the fixed points (0, -0.3) and (Tr, 0.3). of all the N(e) D-dimensional cubes of side e needed Fig. 2 shows a computer generated picture of the basins to cover the basin boundary. Since the volume of one of attraction for the two fixed point attractors. This of these D-dimensional cubes is e D, the uncertain vol- figure is constructed using a 256 X 256 grid of initial ume of phase space is of the order of eDN(e). Noting conditions. Each initial condition is iterated until it is that N(e) ~ e -d satisfies (2), we use this to estimate close to one of the two attractors (100 iterates of the the uncertain phase space volume as eDN(e) ~ e D-d, map is always sufficient to accomplish this). If an orbit which is the result predicted by eqs. (1) and (3). goes to the attractor at 0 = 0, a black dot is plotted at We now illustrate the above with a concrete exam- the corresponding initial condition. If the orbit goes ple. Subsequently, we argue that the demonstrated to the other attractor, no dot is plotted. Thus the phenomena occur commonly in nonlinear dynamics black and blank regions are essentially pictures of the (e.g., in the Lorenz system [3] and in experiments on basins of attractions for the two attractors to the ac- fluids such as those of Berg6 and Dubois [4]). curacy of the grid used and of the computer plotter. We consider the two-dimensional map (Due to the symmetry of the map we have only shown 0 ~< 0 ~ rr in fig. 2). Fine scale structure in the basins On+ 1 = On + a sin 20 n - b sin 40 n - x n sin On , (4a) of attraction is evident. This is a consequence of the Cantor set nature of the basin boundary * 2. In fact, Xn+l = -Jo cos On , (4b) magnifications of the basin boundary show that, as where 0 and 0 + 27r are identified as equivalent. This we examine it on a smaller and smaller scale, it con- map has two fixed points (0, x) = (0, -J0), and (Tr, J0), tinues to have structure. which are attracting for I1 + 2a - 4bl < 1. Numerical experiments on eqs. (4) with different sets of parame- ters have been performed. Here, as an example, we re- ~:2 The Cantor set structure of the basin boundary for eqs. (4) is due to the presence of a horseshoe in the dynamics port results forJ 0 = 0.3, a = 1.32, and b = 0.90. For [5]. In fact, eqs. (4) were chosen because they are a par- ticularly straightforward example of this. (This point will *1 For the purposes of the proof of eq. (3) and for those of be extensively discussed in ref. [9]). Cantor set basin bound- the present letter, eq. (1) should be regarded as shorthand aries due to horseshoes axe not curves (i.e., they cannot be for a = lime~olnf/ln e. In addition, we note that, if the represented as x = x(u), 0 = O(u) with x(u) and O(u) con- basins are unbounded regions, then we restrict attention tinuous functions of u), and have been known for a long to a bounded region of phase space (e.g., the rectangular time (see, for example ref. [6]). Fractal basin boundaries region of fig. 1) for the purposes of calculating f in (1) which are curves (unlike that for fig. 2) have been dis- and N(6) in (2). cussed by Grebogi et al. [7]. 416 Volume 99A, number 9 PHYSICS LETTERS 26 December 1983 We now wish to explore the consequences for pre- for e = 0.125, 59% of the initial conditions are uncer- diction of this infinitely fine scaled structure. To do tain; for e = 0.002, 26%; for e = 3 × 10 -5, 12%. Thus this, consider an initial condition (00, x0). We ask, even apparently small uncertainty in the initial con- what is the effect of a small change e in the x-coordi- ditions yieMs substantial fractions of the phase space nate? Thus we iterate the initial conditions (00, x0), which are uncertain as to which final state is eventually (00, x 0 + e), and (00, x 0 - e) until they approach one attained. The implication is that extraordinarily high of the attractors. If either or both of the perturbed accuracy of initial conditions may sometimes be nec- initial conditions yield orbits which do not approach essary for the reliable prediction of the eventual final the same attractor as the unperturbed initial condi- state. tion, we say that (00, x0) is uncertain. Now say that Furthermore, the example just discussed is by no we randomly choose a large number of initial condi- means extreme or unusual. As evidence for this we tions in the rectangle shown in fig. 2, and let fdenote cite two examples: the fraction of these which we find to be uncertain. (1) The Lorenz system.
Load more

Recommended publications
  • Chaotic Advection, Diffusion, and Reactions in Open Flows Tamás Tél, György Károlyi, Áron Péntek, István Scheuring, Zoltán Toroczkai, Celso Grebogi, and James Kadtke
    Chaotic advection, diffusion, and reactions in open flows Tamás Tél, György Károlyi, Áron Péntek, István Scheuring, Zoltán Toroczkai, Celso Grebogi, and James Kadtke Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 10, 89 (2000); doi: 10.1063/1.166478 View online: http://dx.doi.org/10.1063/1.166478 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/10/1?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.173.125.76 On: Mon, 24 Mar 2014 14:44:01 CHAOS VOLUME 10, NUMBER 1 MARCH 2000 Chaotic advection, diffusion, and reactions in open flows Tama´sTe´l Institute for Theoretical Physics, Eo¨tvo¨s University, P.O. Box 32, H-1518 Budapest, Hungary Gyo¨rgy Ka´rolyi Department of Civil Engineering Mechanics, Technical University of Budapest, Mu¨egyetem rpk. 3, H-1521 Budapest, Hungary A´ ron Pe´ntek Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093-0238 Istva´n Scheuring Department of Plant Taxonomy and Ecology, Research Group of Ecology and Theoretical Biology, Eo¨tvo¨s University, Ludovika te´r 2, H-1083 Budapest, Hungary Zolta´n Toroczkai Department of Physics, University of Maryland, College Park, Maryland 20742-4111 and Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 Celso Grebogi Institute for Plasma Research, University of Maryland, College Park, Maryland 20742 James Kadtke Marine Physical Laboratory, Scripps Institution of Oceanography, University of California at San Diego, La Jolla, California 92093-0238 ͑Received 30 July 1999; accepted for publication 8 November 1999͒ We review and generalize recent results on advection of particles in open time-periodic hydrodynamical flows.
    [Show full text]
  • Table of Contents (Print, Part 2)
    PERIODICALS PHYSICAL REVIEW E Postmaster send address changes to: For editorial and subscription correspondence, APS Subscription Services please see inside front cover Suite 1NO1 „ISSN: 1539-3755… 2 Huntington Quadrangle Melville, NY 11747-4502 THIRD SERIES, VOLUME 75, NUMBER 6 CONTENTS JUNE 2007 PART 2: STATISTICAL AND NONLINEAR PHYSICS, FLUID DYNAMICS, AND RELATED TOPICS RAPID COMMUNICATIONS Chaos and pattern formation Giant acceleration in slow-fast space-periodic Hamiltonian systems (4 pages) ........................... 065201͑R͒ D. V. Makarov and M. Yu. Uleysky Controlling spatiotemporal chaos using multiple delays (4 pages) ..................................... 065202͑R͒ Alexander Ahlborn and Ulrich Parlitz Signatures of fractal clustering of aerosols advected under gravity (4 pages) ............................ 065203͑R͒ Rafael D. Vilela, Tamás Tél, Alessandro P. S. de Moura, and Celso Grebogi Fluid dynamics Clustering of matter in waves and currents (4 pages) ............................................... 065301͑R͒ Marija Vucelja, Gregory Falkovich, and Itzhak Fouxon Experimental study of the bursting of inviscid bubbles (4 pages) ..................................... 065302͑R͒ Frank Müller, Ulrike Kornek, and Ralf Stannarius Surface thermal capacity and its effects on the boundary conditions at fluid-fluid interfaces (4 pages) ....... 065303͑R͒ Kausik S. Das and C. A. Ward Plasma physics Direct evidence of strongly inhomogeneous energy deposition in target heating with laser-produced ion beams (4 pages) ..................................................................................
    [Show full text]
  • Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems
    Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems Wen-Xu Wanga;b, Ying-Cheng Laic;d;e, and Celso Grebogie a School of Systems Science, Beijing Normal University, Beijing, 100875, China b Business School, University of Shanghai for Science and Technology, Shanghai 200093, China c School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA d Department of Physics, Arizona State University, Tempe, Arizona 85287, USA e Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK Wen-Xu Wang, Ying-Cheng Lai, and Celso Grebogi 1 Abstract The problem of reconstructing nonlinear and complex dynamical systems from measured data or time series is central to many scientific disciplines including physical, biological, computer, and social sciences, as well as engineering and economics. The classic approach to phase-space re- construction through the methodology of delay-coordinate embedding has been practiced for more than three decades, but the paradigm is effective mostly for low-dimensional dynamical systems. Often, the methodology yields only a topological correspondence of the original system. There are situations in various fields of science and engineering where the systems of interest are complex and high dimensional with many interacting components. A complex system typically exhibits a rich variety of collective dynamics, and it is of great interest to be able to detect, classify, under- stand, predict, and control the dynamics using data that are becoming increasingly accessible due to the advances of modern information technology. To accomplish these tasks, especially prediction and control, an accurate reconstruction of the original system is required.
    [Show full text]
  • Strange Attractors That Are Not Chaotic
    Physica 13D (1984) 261-268 North-Holland, Amsterdam STRANGE ATTRACTORS THAT ARE NOT CHAOTIC Celso GREBOGI, a Edward OTT, ~b Steven PELIKAN, c and James A. YORKE d Received 7. February 1984 It is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic. Here we use the word strange to refer to the geometry or shape of the attracting set, while the word chaotic refers to the dynamics of orbits on the attractor (in particular, the exponential divergence of nearby trajectories). We first give examples for which it can be demonstrated that there is a strange nonchaotic attractor. These examples apply to a class of maps which model nonlinear oscillators (continuous time) which are externally driven at two incommensurate frequencies. It is then shown that such attractors are persistent under perturbations which preserve the original system type (i.e., there are two incommensurate external driving frequencies). This suggests that, for systems of the type which we have considered, nonchaotic strange attractors may be expected to occur for a finite interval of parameter values. On the other hand, when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit. Thus we conjecture that, in general, continuous time systems (" flows") which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space. 1. Introduction and definitions ties [1] such as noninteger fractal dimension, Cantor set structure, or nowhere differentiability.
    [Show full text]
  • Crises, Sudden Changes in Chaotic Attractors, and Transient Chaos
    Physica 7D (1983) 181-200 North-Holland Publishing Company CRISES, SUDDEN CHANGES IN CHAOTIC ATTRACTORS, AND TRANSIENT CHAOS Celso GREBOGI and Edward OTT Laboratory for Plasma and Fusion Energy Studies, Unit'ersity of Maryland, College Park, Md. 20742, USA and James A. YORKE Institute for Physical Science and Technology and Department of Mathematics, Unit ersi O, of Mao'land, College Park, Md. 20742, USA The occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated. It is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor. We call such collisions crises. Phenomena associated with crises include sudden changes in the size of chaotic attractors, sudden appearances of chaotic attractors (a possible route to chaos), and sudden destructions of chaotic attractors and their basins. This paper presents examples illustrating that crisis events are prevalent in many circumstances and systems, and that, just past a crisis, certain characteristic statistical behavior (whose type depends on the type of crisis) occurs. In particular the phenomenon of chaotic transients is investigated. The examples discussed illustrate crises in progressively higher dimension and include the one-dimensional quadratic map, the (two-dimensional) H~non map, systems of ordinary differential equations in three dimensions and a three-dimensional map. In the case of our study of the three-dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four, respectively. Based on the examples presented the following conjecture is proposed: almost all sudden changes in the size of chaotic attractors and almost all sudden destructions or creations of chaotic attractors and their basins are due to crises.
    [Show full text]
  • Data Based Identification and Prediction of Nonlinear And
    Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems Wen-Xu Wanga;b, Ying-Cheng Laic;d;e, and Celso Grebogie a School of Systems Science, Beijing Normal University, Beijing, 100875, China b Business School, University of Shanghai for Science and Technology, Shanghai 200093, China c School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA d Department of Physics, Arizona State University, Tempe, Arizona 85287, USA e Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK arXiv:1704.08764v1 [physics.data-an] 27 Apr 2017 Wen-Xu Wang, Ying-Cheng Lai, and Celso Grebogi 1 Abstract The problem of reconstructing nonlinear and complex dynamical systems from measured data or time series is central to many scientific disciplines including physical, biological, computer, and social sciences, as well as engineering and economics. The classic approach to phase-space re- construction through the methodology of delay-coordinate embedding has been practiced for more than three decades, but the paradigm is effective mostly for low-dimensional dynamical systems. Often, the methodology yields only a topological correspondence of the original system. There are situations in various fields of science and engineering where the systems of interest are complex and high dimensional with many interacting components. A complex system typically exhibits a rich variety of collective dynamics, and it is of great interest to be able to detect, classify, under- stand, predict, and control the dynamics using data that are becoming increasingly accessible due to the advances of modern information technology.
    [Show full text]
  • Chaos, Qualitative Reasoning, and the Predictability Proble M
    CHAOS, QUALITATIVE REASONING, AND THE PREDICTABILITY PROBLE M Bernardo A. Huberma n Xerox Palo Alto Research Center Palo Alto, CA . 94304 Peter Struss Siemens Corp. Corporate Research and Development Munich, West Germany Abstract Qualitative Physics attempts to predict the behavior of physical systems based on qualitativ e information . No one would consider using these methods to predict the dynamics of a tennis ball on th e frictionless edge of a knife . The reason: a strong instability caused by the edge which makes tw o arbitrarily small different initial conditions have very different outcomes . And yet, research on chaos suggests that sharp edges are not so unique : most nonlinear systems can be shown to be as unstable as the ball on the cutting edge. As a result, prediction is often impossible regardless of how simple the deterministic system is . We analyze the predictabilit y problem and its origins in some detail, show how it is generic, and discuss the impact of these result s on commonsense reasoning and Qualitative Physics . This done by presenting an example from the world of billiards and attempting to uncover the underlying assumptions that may lead physics , commonsense reasoning, and Qualitative Physics to generate different results . 2 1 Introduction Qualitative Physics (QP) aims at describing, explaining, and predicting the behavior of physical systems based only on qualitative information ([Bobrow 841 , [Hobbs-Moore 85]) . Predicting the future development of some part of the world give n knowledge of its current situation is also a central problem when attempting t o formalize the commonsense by some sort of non-monotonic logic (eg .
    [Show full text]
  • Counterexample to the Bohigas Conjecture for Transmission Through a One-Dimensional Lattice
    Counterexample to the Bohigas Conjecture for Transmission Through a One-Dimensional Lattice Ahmed A. Elkamshishy∗ Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907 USA Chris H. Greeney Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907 USA and Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907 USA Resonances in particle transmission through a 1D finite lattice are studied in the presence of a finite number of impurities. Although this is a one-dimensional system that is classically integrable and has no chaos, studying the statistical properties of the spectrum such as the level spacing distribution and the spectral rigidity shows quantum chaos signatures. Using a dimensionless parameter that reflects the degree of state localization, we demonstrate how the transition from regularity to chaos is affected by state localization. The resonance positions are calculated using both the Wigner-Smith time-delay and a Siegert state method, which are in good agreement. Our results give evidence for the existence of quantum chaos in one dimension which is a counter-example to the Bohigas- Giannoni-Schmit conjecture. In 1984, Bohigas, Giannoni and Schmit stated the cel- BGS conjecture, and a second case where a signature of ebrated (BGS) conjecture[1] that describes the statisti- chaos arises in the quantum spectrum, which matches the cal properties of chaotic spectra. This conjecture draws GOE statistics for both the nearest-neighbor distribution a connection between quantum systems whose classical and the spectral rigidity. The chaotic behavior is shown analog is chaotic and random matrix theory (RMT). to depend on a dimensionless parameter that represents Classical chaos is a consequence of the non-linearity of the degree of state localization, thereby suggesting a con- the Newtonian equations of motion, while Schr¨odinger's nection between quantum chaos and the phenomenon of equation is linear and strictly speaking has no chaos.
    [Show full text]
  • Some Elements for a History of the Dynamical Systems Theory Christophe Letellier,1 Ralph Abraham,2 Dima L
    Some elements for a history of the dynamical systems theory Christophe Letellier,1 Ralph Abraham,2 Dima L. Shepelyansky,3 Otto E. Rössler,4 Philip Holmes,5 René Lozi,6 Leon Glass,7 Arkady Pikovsky,8 Lars F. Olsen,9 Ichiro Tsuda,10 Celso Grebogi,11 Ulrich Parlitz,12 Robert Gilmore,13 Louis M. Pecora,14 and Thomas L. Carroll14 1)Normandie Université — CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France. 2)University of California, Santa Cruz, USA 3)Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 31062 Toulouse, Francea) 4)University of Tübingen, Germany 5)Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA 6)Université Côte d’Azur, CNRS, Laboratoire Jean Alexandre Dieudonné, Nice, France 7)Department of Physiology, McGill University, 3655 Promenade Sir William Osler, Montreal H3G 1Y6, Quebec, Canada 8)Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany 9)Institute of Biochemistry and Molecular Biology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark 10)Chubu University Academy of Emerging Sciences, Matumoto-cho 1200, Kasugai, Aichi 487-8501, Japan 11)Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen AB24 3UE, Scotland 12)Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany — Institute for the Dynamics of Complex Systems, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 13)Drexel University, Philadelphia, USA 14)Code 6392, U.S.
    [Show full text]
  • Appendix a the Mathematics of Discontinuity
    Appendix A The Mathematics of Discontinuity On the plane of philosophy properly speaking, of metaphysics, catastrophe theory cannot, to be sure, supply any answer to the great problems which torment mankind. But it favors a dialectical, Heraclitean view of the universe, of a world which is the continual theatre of the battle between ‘logoi,’ between archetypes. René Thom (1975, “Catastrophe Theory: Its Present State and Future Perspectives,” p. 382) Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Benoit B. Mandelbrot (1983, The Fractal Geometry of Nature, p. 1) A.1 General Overview Somehow it is appropriate if ironic that sharply divergent opinions exist in the mathematical House of Discontinuity with respect to the appropriate method for analyzing discontinuous phenomena. Different methods include catastrophe theory, chaos theory, fractal geometry, synergetics theory, self-organizing criticality, spin glass theory, and emergent complexity. All have been applied to economics in one way or another. What these approaches have in common is more important than what divides them. They all see discontinuities as fundamental to the nature of nonlinear dynam- ical reality.1 In the broadest sense, discontinuity theory is bifurcation theory of which all of these are subsets. Ironically then we must consider the bifurcation of bifurcation theory into competing schools. After examining the historical origins of this bifurcation of bifurcation theory, we shall consider the possibility of a recon- ciliation and synthesis within the House of Discontinuity between these fractious factions. J.B. Rosser, Complex Evolutionary Dynamics in Urban-Regional 213 and Ecologic-Economic Systems, DOI 10.1007/978-1-4419-8828-7, C Springer Science+Business Media, LLC 2011 214 Appendix A A.2 The Founding Fathers The conflict over continuity versus discontinuity can be traced deep into a variety of disputes among the ancient Greek philosophers.
    [Show full text]
  • Relativistic Quantum Chaos—An Emergent Interdisciplinary Field Ying-Cheng Lai, Hong-Ya Xu, Liang Huang, and Celso Grebogi
    Relativistic quantum chaos—An emergent interdisciplinary field Ying-Cheng Lai, Hong-Ya Xu, Liang Huang, and Celso Grebogi Citation: Chaos 28, 052101 (2018); doi: 10.1063/1.5026904 View online: https://doi.org/10.1063/1.5026904 View Table of Contents: http://aip.scitation.org/toc/cha/28/5 Published by the American Institute of Physics Articles you may be interested in When chaos meets relativistic quantum mechanics Scilight 2018, 180006 (2018); 10.1063/1.5037088 Hybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 041101 (2018); 10.1063/1.5028373 Accurate detection of hierarchical communities in complex networks based on nonlinear dynamical evolution Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 043119 (2018); 10.1063/1.5025646 Using machine learning to replicate chaotic attractors and calculate Lyapunov exponents from data Chaos: An Interdisciplinary Journal of Nonlinear Science 27, 121102 (2017); 10.1063/1.5010300 Optimal control of networks in the presence of attackers and defenders Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 051103 (2018); 10.1063/1.5030899 Impacts of opinion leaders on social contagions Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 053103 (2018); 10.1063/1.5017515 CHAOS 28, 052101 (2018) Relativistic quantum chaos—An emergent interdisciplinary field Ying-Cheng Lai,1,2,a) Hong-Ya Xu,1 Liang Huang,3 and Celso Grebogi4 1School of Electrical, Computer and Energy
    [Show full text]
  • Two-Week Loan Py
    LBL-1 0903 (! at the International Conference on Plasma Physics, Nagoya, Japan, Apri 1 7-11 , 1980, and to be published in the Proceedings STATISTICAL ANALYSIS OF A DETERMINISTIC STOCHASTIC ORBIT Allan N. Kaufman, Henry O.I. Abarbanel and Celso Grebogi May 1980 TWO-WEEK LOAN PY This is a Library Copy which may be borrowed for two weeks. For a personal retention copyy call Tech. Info. Division; Ext 6782. Prepared for the U.S. Department of Energy under Contract W-7405-ENG-48 DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. Submitted to the Proceedings of the LBL-10903 International Conference on Plasma Physics Nagoya, Japan, April 1980 Statistical Analysis of a Deterministic Stochastic Orbit* Allan N.
    [Show full text]