DOCSLIB.ORG

Nonlinear Dynamics RESOURCE LETTER

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nonlinear Dynamics RESOURCE LETTER RESOURCE LETTER Roger H. Stuewer, Editor School of Physics and Astronomy, 116 Church Street University of Minnesota, Minneapolis, Minnesota 55455 This is one of a series of Resource Letters on different topics intended to guide college physicists, astronomers, and other scientists to some of the literature and other teaching aids that may help improve course content in specified fields. @The letter E after an item indicates elementary level or material of general interest to persons becoming informed in the field. The letter I, for intermediate level, indicates material of somewhat more specialized nature; and the letter A, indicates rather specialized or advanced material.# No Resource letter is meant to be exhaustive and complete; in time there may be more than one letter on some of the main subjects of interest. Comments on these materials as well as suggestions for future topics will be welcomed. Please send such communications to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School of Physics and Astronomy, 116 Church Street SE, University of Minnesota, Minneapolis, MN 55455. Resource Letter: ND-1: Nonlinear Dynamics Robert C. Hilborn Department of Physics, Amherst College, Amherst, Massachusetts 01002-5000 Nicholas B. Tufillaro Measurement Systems Department, Hewlett–Packard Company, 1501 Page Mill Road, MS4AD, Palo Alto, California 94304-1126 ~Received 10 July 1996; accepted 13 March 1997! This Resource Letter provides an introductory guide to the literature on nonlinear dynamics. Journal articles and books are cited for the following topics: general aspects of nonlinear dynamics and applications of nonlinear dynamics to various fields of physics, other sciences, and a few areas outside the sciences. Software and Internet resources are given also. © 1997 American Association of Physics Teachers. I. INTRODUCTION There are two important themes in nonlinear dynamics. Nonlinear dynamics and its subdiscipline ‘‘chaos theory’’ One emphasizes the temporal behavior of systems, most with have swept over the landscape of science, mathematics, and no significant spatial variation. The other emphasizes spatial engineering in the past two decades. The subject has ben- structures and the formation of spatial patterns. Most begin- efited from some good public relations ~Ref. 75!, but its stay- ning courses in nonlinear dynamics focus on the temporal ing power is rooted in the universality of its methods and behavior of ‘‘low-dimensional’’ systems, that is, those gov- results: Although each nonlinear system may be quite dis- erned by a few degrees of freedom, for which there now tinct in its detailed behavior, most nonlinear systems fall into exists a fairly complete body of theoretical results and ex- broad classes with common qualitative and quantitative fea- perimental techniques ~Sec. IV B and Ref. 90!. Methods of tures. Many of the theoretical and experimental advances in nonlinear time-series analysis ~Sec. V D and Ref. 53!,in nonlinear science, moreover, can be explored with only a which a sequence of values of a single dynamical variable is rudimentary background in mathematics and a pocket calcu- used to determine the qualitative and quantitative measures lator or with simple computer programs ~Refs. 42, 67, and of the temporal dynamical behavior of the entire system, Sec. VII!. Experiments, such as the forced vibrations of a have dominated both the theoretical and experimental meth- simple pendulum ~Ref. 12!, a bouncing ball ~Ref. 27!,ora odologies and are already finding their way into many prac- semiconductor diode circuit ~Sec. VI B!, are straightforward tical applications. and inexpensive. Because nonlinear effects are observed in The extension of these notions to higher-dimensional sys- everyday phenomena—such as a dripping faucet ~Sec. VI I! tems, to problems of ‘‘spatiotemporal’’ chaos, and to the or avalanches in sandpiles ~Sec. VI F!—they capture the question of turbulence is still a subject of active debate and imagination of students at all levels. inquiry ~Sec. VI F!. So too is the question of the correspon- Nonlinear dynamics can appear at several points in the dence between chaos in classical and quantum-mechanical undergraduate or graduate curriculum. It can be included as a systems ~Sec. VI J!. Both issues, though, have motivated special topic in a standard course in classical mechanics some exciting new experimental and theoretical work. ~Ref. 79! or differential equations ~Ref. 48!. Courses focus- The crucial theoretical construct in nonlinear dynamics is ing on nonlinear dynamics at the junior, senior, or beginning phase space ~also called state space!. Each of the ~indepen- graduate level are also becoming common ~Sec. IV B!. Sev- dent! dynamical variables is used as a coordinate to construct eral books aim to introduce nonlinear dynamics to students the state space for the system. For a deterministic system, the who are not necessarily science majors ~Sec. IV A and Ref. future behavior is determined by the current state of the sys- 10!. tem, represented as a point in state space. As a system 822 Am. J. Phys. 65 ~9!, September 1997 © 1997 American Association of Physics Teachers 822 evolves in time, its state-space representation maps out a Europhysics Letters trajectory in that space. Sets of trajectories form a phase Physics Letters A portrait. These phase portraits often have interesting geomet- Physical Review Letters ric properties. For example, chaotic systems can have phase Other journals that often have articles on nonlinear dy- portraits with a fractal geometric structure. namics: The important and distinctive features of nonlinear behav- American Journal of Physics ior are as follows: Computers in Physics European Journal of Physics I Symmetry-breaking, either temporal or spatial. The tem- International Journal of Nonlinear Mechanics poral response of a system need not be the same as that of Journal of Statistical Physics the ‘‘driving force.’’ Even autonomous systems ~those with no explicit time-dependent forcing! can spontane- ously develop complex temporal behavior. The most dra- III. CONFERENCE PROCEEDINGS matic of these broken symmetries is chaotic behavior, Several conferences on nonlinear dynamics are held regu- which is bounded, completely aperiodic behavior. For larly. nonlinear systems with significant spatial variation, the 1. SIAM Conference on Applications of Dynamical Systems ~every other spatial patterns may be independent of the boundary con- year!. ditions. 2. Proceedings of the First Experimental Chaos Conference, edited by I Dramatic changes in behavior, called bifurcations, which S. Vohra et al. ~World Scientific, Singapore, 1992!. occur over extremely small parameter ranges. There are 3. Proceedings of the Second Experimental Chaos Conference, edited several common classes of bifurcation events—including by W. Ditto et al. ~World Scientific, Singapore, 1995!. period doubling, intermittency, and crises—that organize 4. Measures of Spatio-Temporal Dynamics, edited by A. M. Albano, P. E. Rapp, N. B. Abraham, and A. Passamante ~North-Holland, Amster- the transitions between regular ~periodic! behavior and dam, 1996!. chaotic behavior. I Sensitive dependence on initial conditions ~the so-called Butterfly Effect!~Refs. 5 and 9!: Small changes in initial IV. TEXTBOOKS AND EXPOSITIONS conditions may lead to qualitatively and quantitatively dif- ferent long-term behavior. Such sensitivity leads to the A. Popularizations loss of long-term predictability even if the system is com- 5. Chaos, Making a New Science, J. Gleick ~Viking, New York, 1987!. pletely deterministic. A best-seller that chronicles the development of the scientific study of chaos. Includes good biographical sketches of many of the major play- I Universal scaling laws for the transitions between chaotic ers in the development of chaos. ~E! and regular behavior. 6. Does God Play Dice? The Mathematics of Chaos, I. Stewart ~Black- well, New York, 1989!. For the scientifically literate reader. Empha- Much of the analysis of the temporal behavior of nonlinear sizes the mathematical approach to chaos, but does pay some attention systems is carried out using time-series data from a single to experiments. ~E! dynamical variable. The series may be generated by using a 7. Order Out of Chaos, I. Prigogine and I. Stengers ~Bantam, New York, ‘‘stroboscopic’’ technique: Every time the state-space trajec- 1984!. A wide-ranging look at the new paradigm: how order can de- tory intersects a plane in the state space, forming a so-called velop from randomness in nonequilibrium systems. ~E! Poincare´ section, a data point is recorded. From such a se- 8. Newton’s Clock: Chaos in the Solar System, I. Peterson ~Freeman, ries, the topology and dynamics of the entire attractor can be New York, 1993!. An excellent introduction to the history of nonlinear dynamics. ~E! reconstructed. See the references in Sec. V C and the texts in 9. The Essence of Chaos, E. N. Lorenz ~University of Washington Press, Sec. IV B. Seattle, 1993!. The text of popular lectures given by Lorenz, one of the We have selected references that emphasize the breadth pioneers of chaos. ~E! and connectedness of nonlinear dynamics at a level appro- priate for someone new to the field, although we have in- cluded advanced-level citations for historically important pa- B. Introductory texts for science students pers or those that have played a significant role in the The following are listed more or less in order of increasing modern development of nonlinear dynamics. demands
Load more

Recommended publications
  • The Swinging Spring: Regular and Chaotic Motion
    References The Swinging Spring: Regular and Chaotic Motion Leah Ganis May 30th, 2013 Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Outline of Talk I Introduction to Problem I The Basics: Hamiltonian, Equations of Motion, Fixed Points, Stability I Linear Modes I The Progressing Ellipse and Other Regular Motions I Chaotic Motion I References Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Introduction The swinging spring, or elastic pendulum, is a simple mechanical system in which many different types of motion can occur. The system is comprised of a heavy mass, attached to an essentially massless spring which does not deform. The system moves under the force of gravity and in accordance with Hooke's Law. z y r φ x k m Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics We can write down the equations of motion by finding the Lagrangian of the system and using the Euler-Lagrange equations. The Lagrangian, L is given by L = T − V where T is the kinetic energy of the system and V is the potential energy. Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics In Cartesian coordinates, the kinetic energy is given by the following: 1 T = m(_x2 +y _ 2 +z _2) 2 and the potential is given by the sum of gravitational potential and the spring potential: 1 V = mgz + k(r − l )2 2 0 where m is the mass, g is the gravitational constant, k the spring constant, r the stretched length of the spring (px2 + y 2 + z2), and l0 the unstretched length of the spring.
    [Show full text]
  • Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda
    Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda • Introduction to the elastic pendulum problem • Derivations of the equations of motion • Real-life examples of an elastic pendulum • Trivial cases & equilibrium states • MATLAB models The Elastic Problem (Simple Harmonic Motion) 푑2푥 푑2푥 푘 • 퐹 = 푚 = −푘푥 = − 푥 푛푒푡 푑푡2 푑푡2 푚 • Solve this differential equation to find 푥 푡 = 푐1 cos 휔푡 + 푐2 sin 휔푡 = 퐴푐표푠(휔푡 − 휑) • With velocity and acceleration 푣 푡 = −퐴휔 sin 휔푡 + 휑 푎 푡 = −퐴휔2cos(휔푡 + 휑) • Total energy of the system 퐸 = 퐾 푡 + 푈 푡 1 1 1 = 푚푣푡2 + 푘푥2 = 푘퐴2 2 2 2 The Pendulum Problem (with some assumptions) • With position vector of point mass 푥 = 푙 푠푖푛휃푖 − 푐표푠휃푗 , define 푟 such that 푥 = 푙푟 and 휃 = 푐표푠휃푖 + 푠푖푛휃푗 • Find the first and second derivatives of the position vector: 푑푥 푑휃 = 푙 휃 푑푡 푑푡 2 푑2푥 푑2휃 푑휃 = 푙 휃 − 푙 푟 푑푡2 푑푡2 푑푡 • From Newton’s Law, (neglecting frictional force) 푑2푥 푚 = 퐹 + 퐹 푑푡2 푔 푡 The Pendulum Problem (with some assumptions) Defining force of gravity as 퐹푔 = −푚푔푗 = 푚푔푐표푠휃푟 − 푚푔푠푖푛휃휃 and tension of the string as 퐹푡 = −푇푟 : 2 푑휃 −푚푙 = 푚푔푐표푠휃 − 푇 푑푡 푑2휃 푚푙 = −푚푔푠푖푛휃 푑푡2 Define 휔0 = 푔/푙 to find the solution: 푑2휃 푔 = − 푠푖푛휃 = −휔2푠푖푛휃 푑푡2 푙 0 Derivation of Equations of Motion • m = pendulum mass • mspring = spring mass • l = unstreatched spring length • k = spring constant • g = acceleration due to gravity • Ft = pre-tension of spring 푚푔−퐹 • r = static spring stretch, 푟 = 푡 s 푠 푘 • rd = dynamic spring stretch • r = total spring stretch 푟푠 + 푟푑 Derivation of Equations of Motion
    [Show full text]
  • STRONG REGULARITY 3 Ergodic Theorem Produces Lyapunov Exponents (W.R.T
    STRONG REGULARITY by Pierre Berger and Jean-Christophe Yoccoz 1. Uniformly hyperbolic dynamical systems The theory of uniformly hyperbolic dynamical systems was constructed in the 1960’s under the dual leadership of Smale in the USA and Anosov and Sinai in the Soviet Union. It is nowadays almost complete. It encompasses various examples [Sma67]: expanding maps, horseshoes, solenoid maps, Plykin attractors, Anosov maps and DA, all of which are basic pieces. We recall standard definitions. Let f be a C1-diffeomorphism f of a finite dimen- sional manifold M. A compact f-invariant subset Λ ⊂ M is uniformly hyperbolic if the restriction to Λ of the tangent bundle TM splits into two continuous invariant subbundles TM|Λ= Es ⊕ Eu, Es being uniformly contracted and Eu being uniformly expanded. Then for every z ∈ Λ, the sets W s(z)= {z′ ∈ M : lim d(f n(z),f n(z′))=0}, n→+∞ W u(z)= {z′ ∈ M : lim d(f n(z),f n(z′))=0} n→−∞ are called the stable and unstable manifolds of z. They are immersed manifolds tangent at z to respectively Es(z) and Eu(z). s arXiv:1901.09430v1 [math.DS] 27 Jan 2019 The ǫ-local stable manifold Wǫ (z) of z is the connected component of z in the intersection of W s(z) with a ǫ-neighborhood of z. The ǫ-local unstable manifold u Wǫ (z) is defined likewise. Definition 1.1. — A basic set is a compact, f-invariant, uniformly hyperbolic set Λ which is transitive and locally maximal: there exists a neighborhood N of Λ such that n Λ= ∩n∈Zf (N).
    [Show full text]
  • DOUBLE STANDARD MAPS 1. Introduction It Is a Usual
    DOUBLE STANDARD MAPS MICHALMISIUREWICZ AND ANA RODRIGUES Abstract. We investigate the family of double standard maps of the circle onto itself, given by fa,b(x) = 2x + a + (b/π) sin(2πx) (mod 1), where the parameters a, b are real and 0 ≤ b ≤ 1. Similarly to the well known family of (Arnold) standard maps of the circle, Aa,b(x) = x + a + (b/(2π)) sin(2πx) (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a, b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues, that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior. 1. Introduction It is a usual procedure that in order to understand the behavior of a system in higher dimension one investigates first a one-dimensional system that is somewhat similar. The classical example is the H´enonmap and similar systems, where a serious progress occurred only after unimodal interval maps have been thoroughly understood. Another example of this type was investigation by V. Arnold of the family of standard maps of the circle, given by the formula b (1.1) A (x) = x + a + sin(2πx) (mod 1) a,b 2π (when we write “mod 1,” we mean that both the arguments and the values are taken modulo 1).
    [Show full text]
  • Spatial Accessibility to Amenities in Fractal and Non Fractal Urban Patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser
    Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser To cite this version: Cécile Tannier, Gilles Vuidel, Hélène Houot, Pierre Frankhauser. Spatial accessibility to amenities in fractal and non fractal urban patterns. Environment and Planning B: Planning and Design, SAGE Publications, 2012, 39 (5), pp.801-819. 10.1068/b37132. hal-00804263 HAL Id: hal-00804263 https://hal.archives-ouvertes.fr/hal-00804263 Submitted on 14 Jun 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. TANNIER C., VUIDEL G., HOUOT H., FRANKHAUSER P. (2012), Spatial accessibility to amenities in fractal and non fractal urban patterns, Environment and Planning B: Planning and Design, vol. 39, n°5, pp. 801-819. EPB 137-132: Spatial accessibility to amenities in fractal and non fractal urban patterns Cécile TANNIER* ([email protected]) - corresponding author Gilles VUIDEL* ([email protected]) Hélène HOUOT* ([email protected]) Pierre FRANKHAUSER* ([email protected]) * ThéMA, CNRS - University of Franche-Comté 32 rue Mégevand F-25 030 Besançon Cedex, France Tel: +33 381 66 54 81 Fax: +33 381 66 53 55 1 Spatial accessibility to amenities in fractal and non fractal urban patterns Abstract One of the challenges of urban planning and design is to come up with an optimal urban form that meets all of the environmental, social and economic expectations of sustainable urban development.
    [Show full text]
  • An Entire Transcendental Family with a Persistent Siegel Disk
    An entire transcendental family with a persistent Siegel disk Rub´en Berenguel, N´uria Fagella March 12, 2009 Abstract We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disk. After normalization ∗ this is a one parameter family fa with a ∈ C which includes the semi- standard map λzez at a = 1, approaches the exponential map when a → 0 and a quadratic polynomial when a → ∞. We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disk in terms of the parameter. 1 Introduction Given a holomorphic endomorphism f : S → S on a Riemann surface S we consider the dynamical system generated by the iterates of f, denoted by n) n f = f◦ ··· ◦f. The orbit of an initial condition z0 ∈ S is the sequence + n O (z0)= {f (z0)}n N and we are interested in classifying the initial condi- ∈ tions in the phase space or dynamical plane S, according to the asymptotic behaviour of their orbits when n tends to infinity. There is a dynamically natural partition of the phase space S into the Fatou set F (f) (open) where the iterates of f form a normal family and the Julia set J (f)= S\F (f) which is its complement (closed). If S = C = C ∪∞ then f is a rational map. If S = C and f does not extend to the point at infinity, then f is an entire transcendental map, b that is, infinity is an essential singularity.
    [Show full text]
  • Role of Nonlinear Dynamics and Chaos in Applied Sciences
    v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings.
    [Show full text]
  • Instructional Experiments on Nonlinear Dynamics & Chaos (And
    Bibliography of instructional experiments on nonlinear dynamics and chaos Page 1 of 20 Colorado Virtual Campus of Physics Mechanics & Nonlinear Dynamics Cluster Nonlinear Dynamics & Chaos Lab Instructional Experiments on Nonlinear Dynamics & Chaos (and some related theory papers) overviews of nonlinear & chaotic dynamics prototypical nonlinear equations and their simulation analysis of data from chaotic systems control of chaos fractals solitons chaos in Hamiltonian/nondissipative systems & Lagrangian chaos in fluid flow quantum chaos nonlinear oscillators, vibrations & strings chaotic electronic circuits coupled systems, mode interaction & synchronization bouncing ball, dripping faucet, kicked rotor & other discrete interval dynamics nonlinear dynamics of the pendulum inverted pendulum swinging Atwood's machine pumping a swing parametric instability instabilities, bifurcations & catastrophes chemical and biological oscillators & reaction/diffusions systems other pattern forming systems & self-organized criticality miscellaneous nonlinear & chaotic systems -overviews of nonlinear & chaotic dynamics To top? Briggs, K. (1987), "Simple experiments in chaotic dynamics," Am. J. Phys. 55 (12), 1083-9. Hilborn, R. C. (2004), "Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics," Am. J. Phys. 72 (4), 425-7. Hilborn, R. C. and N. B. Tufillaro (1997), "Resource Letter: ND-1: nonlinear dynamics," Am. J. Phys. 65 (9), 822-34. Laws, P. W. (2004), "A unit on oscillations, determinism and chaos for introductory physics students," Am. J. Phys. 72 (4), 446-52. Sungar, N., J. P. Sharpe, M. J. Moelter, N. Fleishon, K. Morrison, J. McDill, and R. Schoonover (2001), "A laboratory-based nonlinear dynamics course for science and engineering students," Am. J. Phys. 69 (5), 591-7. http://carbon.cudenver.edu/~rtagg/CVCP/Ctr_dynamics/Lab_nonlinear_dyn/Bibex_nonline..
    [Show full text]
  • International Journal of Engineering and Advanced Technology (IJEAT)
    IInntteerrnnaattiioonnaall JJoouurrnnaall ooff EEnnggiinneeeerriinngg aanndd AAddvvaanncceedd TTeecchhnnoollooggyy ISSN : 2249 - 8958 Website: www.ijeat.org Volume-2 Issue-5, June 2013 Published by: Blue Eyes Intelligence Engineering and Sciences Publication Pvt. Ltd. ced Te van ch d no A l d o n g y a g n i r e e IJEat n i I E n g X N t P e L IO n O E T r R A I V n NG NO f IN a o t l i o a n n r a u l J o www.ijeat.org Exploring Innovation Editor In Chief Dr. Shiv K Sahu Ph.D. (CSE), M.Tech. (IT, Honors), B.Tech. (IT) Director, Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., Bhopal (M.P.), India Dr. Shachi Sahu Ph.D. (Chemistry), M.Sc. (Organic Chemistry) Additional Director, Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., Bhopal (M.P.), India Vice Editor In Chief Dr. Vahid Nourani Professor, Faculty of Civil Engineering, University of Tabriz, Iran Prof.(Dr.) Anuranjan Misra Professor & Head, Computer Science & Engineering and Information Technology & Engineering, Noida International University, Noida (U.P.), India Chief Advisory Board Prof. (Dr.) Hamid Saremi Vice Chancellor of Islamic Azad University of Iran, Quchan Branch, Quchan-Iran Dr. Uma Shanker Professor & Head, Department of Mathematics, CEC, Bilaspur(C.G.), India Dr. Rama Shanker Professor & Head, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea Dr. Vinita Kumari Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., India Dr. Kapil Kumar Bansal Head (Research and Publication), SRM University, Gaziabad (U.P.), India Dr.
    [Show full text]
  • Elliptic Bubbles in Moser's 4D Quadratic Map: the Quadfurcation
    Elliptic Bubbles in Moser's 4DQuadratic Map: the Quadfurcation ∗ Arnd B¨ackery and James D. Meissz Abstract. Moser derived a normal form for the family of four-dimensional, quadratic, symplectic maps in 1994. This six-parameter family generalizes H´enon'subiquitous 2d map and provides a local approximation for the dynamics of more general 4D maps. We show that the bounded dynamics of Moser's family is organized by a codimension-three bifurcation that creates four fixed points|a bifurcation analogous to a doubled, saddle-center|which we call a quadfurcation. In some sectors of parameter space a quadfurcation creates four fixed points from none, and in others it is the collision of a pair of fixed points that re-emerge as two or possibly four. In the simplest case the dynamics is similar to the cross product of a pair of H´enonmaps, but more typically the stability of the created fixed points does not have this simple form. Up to two of the fixed points can be doubly-elliptic and be surrounded by bubbles of invariant two-tori; these dominate the set of bounded orbits. The quadfurcation can also create one or two complex-unstable (Krein) fixed points. Special cases of the quadfurcation correspond to a pair of weakly coupled H´enonmaps near their saddle-center bifurcations. Key words. H´enonmap, symplectic maps, saddle-center bifurcation, Krein bifurcation, invariant tori AMS subject classifications. 37J10 37J20 37J25 70K43 1. Introduction. Multi-dimensional Hamiltonian systems model dynamics on scales rang- ing from zettameters, for the dynamics of stars in galaxies [1, 2], to nanometers, in atoms and molecules [3, 4].
    [Show full text]
  • Control of Chaos: Methods and Applications
    Automation and Remote Control, Vol. 64, No. 5, 2003, pp. 673{713. Translated from Avtomatika i Telemekhanika, No. 5, 2003, pp. 3{45. Original Russian Text Copyright c 2003 by Andrievskii, Fradkov. REVIEWS Control of Chaos: Methods and Applications. I. Methods1 B. R. Andrievskii and A. L. Fradkov Institute of Mechanical Engineering Problems, Russian Academy of Sciences, St. Petersburg, Russia Received October 15, 2002 Abstract|The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system ex- citation, the method of Poincar´e map linearization (OGY method), and the method of time- delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications. 1. INTRODUCTION The term control of chaos is used mostly to denote the area of studies lying at the interfaces between the control theory and the theory of dynamic systems studying the methods of control of deterministic systems with nonregular, chaotic behavior. In the ancient mythology and philosophy, the word \χαωσ" (chaos) meant the disordered state of unformed matter supposed to have existed before the ordered universe. The combination \control of chaos" assumes a paradoxical sense arousing additional interest in the subject.
    [Show full text]
  • An Extended Correlation Dimension of Complex Networks
    entropy Article An Extended Correlation Dimension of Complex Networks Sheng Zhang 1,*, Wenxiang Lan 1,*, Weikai Dai 1, Feng Wu 1 and Caisen Chen 2 1 School of Information Engineering, Nanchang Hangkong University, 696 Fenghe South Avenue, Nanchang 330063, China; [email protected] (W.D.); [email protected] (F.W.) 2 Military Exercise and Training Center, Academy of Army Armored Force, Beijing 100072, China; [email protected] * Correspondence: [email protected] (S.Z.); [email protected] (W.L.) Abstract: Fractal and self-similarity are important characteristics of complex networks. The correla- tion dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimen- sion is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects. Keywords: fractal property; correlation dimension; weighted networks; small-world network Citation: Zhang, S.; Lan, W.; Dai, W.; Wu, F.; Chen, C. An Extended 1. Introduction Correlation Dimension of Complex Recently, revealing and characterizing complex systems from a complex networks Networks.
    [Show full text]