Table Of ContentHigh-Dimensional Chaotic and Attractor Systems
International Series on
INTELLIGENT SYSTEMS, CONTROLAND AUTOMATION:
SCIENCE AND ENGINEERING
VOLUME 32
Editor
Professor S. G. Tzafestas, National Technical University of Athens, Greece
Editorial Advisory Board
Professor P. Antsaklis, University of Notre Dame, IN, U.S.A.
Professor P. Borne, Ecole Centrale de Lille, France
Professor D. G. Caldwell, University of Salford, U.K.
Professor C. S. Chen, University of Akron, Ohio, U.S.A.
Professor T. Fukuda, Nagoya University, Japan
Professor F. Harashima, University of Tokyo, Tokyo, Japan
Professor S. Monaco, University La Sapienza, Rome, Italy
Professor G. Schmidt, Technical University of Munich, Germany
Professor N. K. Sinha, Mc Master University, Hamilton, Ontario, Canada
Professor D. Tabak, George Mason University, Fairfax, Virginia, USA
Professor K. Valavanis, University of South Florida, USA
High-Dimensional Chaotic
and Attractor Systems
A Comprehensive Introduction
edited by
VLADIMIR G. IVANCEVIC
Defence Science and Technology Organisation, Adelaide, SA, Australia
and
TIJANAT. IVANCEVIC
The University of Adelaide, SA, Australia
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-5455-6 (HB)
ISBN-13 978-1-4020-5455-6 (HB)
ISBN-10 1-4020-5456-4 (e-book)
ISBN-13 978-1-4020-5456-3 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2007 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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and executed on a computer system, for exclusive use by the purchaser of the work.
Dedicated to Nitya, Atma and Kali
Contents
Preface ....................................................... xiii
Acknowledgments ............................................. xv
1 Introduction to Attractors and Chaos ..................... 1
1.1 Basics of Attractor and Chaotic Dynamics.................. 17
1.2 A Brief History of Chaos Theory in 5 Steps................. 29
1.2.1 Henry Poincar´e: Qualitative Dynamics, Topology and
Chaos............................................ 30
1.2.2 Stephen Smale: Topological Horseshoe and Chaos of
Stretching and Folding ............................. 38
1.2.3 Ed Lorenz: Weather Prediction and Chaos............ 48
1.2.4 Mitchell Feigenbaum: A Constant and Universality .... 51
1.2.5 Lord Robert May: Population Modelling and Chaos .... 52
1.2.6 Michel H´enon: A Special 2D Map and Its Strange
Attractor......................................... 56
1.3 Some Classical Attractor and Chaotic Systems .............. 59
1.4 Basics of Continuous Dynamical Analysis................... 72
1.4.1 A Motivating Example............................. 72
1.4.2 Systems of ODEs.................................. 75
1.4.3 Linear Autonomous Dynamics: Attractors & Repellors . 78
1.4.4 Conservative versus Dissipative Dynamics ............ 82
1.4.5 Basics of Nonlinear Dynamics....................... 89
1.4.6 Ergodic Systems ..................................108
1.5 Continuous Chaotic Dynamics ............................109
1.5.1 Dynamics and Non–Equilibrium Statistical Mechanics..111
1.5.2 Statistical Mechanics of Nonlinear Oscillator Chains ...124
1.5.3 Geometrical Modelling of Continuous Dynamics .......125
1.5.4 Lagrangian Chaos .................................128
1.6 Standard Map and Hamiltonian Chaos .....................136
1.7 Chaotic Dynamics of Binary Systems ......................142
1.7.1 Examples of Dynamical Maps.......................144
1.7.2 Correlation Dimension of an Attractor ...............148
1.8 Basic Hamiltonian Model of Biodynamics...................149
vii
viii Contents
2 Smale Horseshoes and Homoclinic Dynamics ..............153
2.1 Smale Horseshoe Orbits and Symbolic Dynamics ............153
2.1.1 Horseshoe Trellis ..................................157
2.1.2 Trellis–Forced Dynamics ...........................161
2.1.3 Homoclinic Braid Type ............................164
2.2 Homoclinic Classes for Generic Vector–Fields ...............165
2.2.1 Lyapunov Stability ................................168
2.2.2 Homoclinic Classes ................................171
2.3 Complex–Valued H´enon Maps and Horseshoes ..............174
2.3.1 Complex Henon–Like Maps.........................175
2.3.2 Complex Horseshoes ...............................178
2.4 Chaos in Functional Delay Equations ......................181
2.4.1 Poincar´e Maps and Homoclinic Solutions .............184
2.4.2 Starting Value and Targets .........................192
2.4.3 Successive Modifications of g........................199
2.4.4 Transversality.....................................215
2.4.5 Transversally Homoclinic Solutions ..................221
3 3–Body Problem and Chaos Control.......................223
3.1 Mechanical Origin of Chaos...............................223
3.1.1 Restricted 3–Body Problem.........................224
3.1.2 Scaling and Reduction in the 3–Body Problem ........236
3.1.3 Periodic Solutions of the 3–Body Problem ............239
3.1.4 Bifurcating Periodic Solutions of the 3–Body Problem..240
3.1.5 Bifurcations in Lagrangian Equilibria ................241
3.1.6 Continuation of KAM–Tori .........................247
3.1.7 Parametric Resonance and Chaos in Cosmology .......249
3.2 Elements of Chaos Control ...............................252
3.2.1 Feedback and Non–Feedback Algorithms for Chaos
Control ..........................................252
3.2.2 Exploiting Critical Sensitivity.......................256
3.2.3 Lyapunov Exponents and Kaplan–Yorke Dimension....257
3.2.4 Kolmogorov–Sinai Entropy .........................259
3.2.5 Chaos Control by Ott, Grebogi and Yorke (OGY) .....261
3.2.6 Floquet Stability Analysis and OGY Control..........264
3.2.7 Blind Chaos Control...............................268
3.2.8 Jerk Functions of Simple Chaotic Flows ..............272
3.2.9 Example: Chaos Control in Molecular Dynamics.......275
4 Phase Transitions and Synergetics .........................285
4.1 Phase Transitions, Partition Function and Noise.............285
4.1.1 Equilibrium Phase Transitions ......................285
4.1.2 Classification of Phase Transitions...................286
4.1.3 Basic Properties of Phase Transitions ................288
4.1.4 Landau’s Theory of Phase Transitions................290
Contents ix
4.1.5 Partition Function.................................292
4.1.6 Noise–Induced Non–equilibrium Phase Transitions.....299
4.2 Elements of Haken’s Synergetics...........................306
4.2.1 Phase Transitions .................................308
4.2.2 Mezoscopic Derivation of Order Parameters...........310
4.2.3 Example: Synergetic Control of Biodynamics..........312
4.2.4 Example: Chaotic Psychodynamics of Perception ......313
4.2.5 Kick Dynamics and Dissipation–Fluctuation Theorem..317
4.3 Synergetics of Recurrent and Attractor Neural Networks .....320
4.3.1 Stochastic Dynamics of Neuronal Firing States........322
4.3.2 Synaptic Symmetry and Lyapunov Functions .........327
4.3.3 Detailed Balance and Equilibrium Statistical Mechanics 329
4.3.4 Simple Recurrent Networks with Binary Neurons ......334
4.3.5 Simple Recurrent Networks of Coupled Oscillators.....343
4.3.6 Attractor Neural Networks with Binary Neurons ......350
4.3.7 Attractor Neural Networks with Continuous Neurons ..362
4.3.8 Correlation– and Response–Functions................369
4.3.9 Path–Integral Approach for Complex Dynamics .......378
4.3.10 Hierarchical Self–Programming in Neural Networks ....399
4.4 Topological Phase Transitions and Hamiltonian Chaos .......406
4.4.1 Phase Transitions in Hamiltonian Systems............406
4.4.2 Geometry of the Largest Lyapunov Exponent .........408
4.4.3 Euler Characteristics of Hamiltonian Systems .........412
4.4.4 Pathways to Self–Organization in Human Biodynamics.416
5 Phase Synchronization in Chaotic Systems.................419
5.1 Lyapunov Vectors and Lyapunov Exponents ................419
5.1.1 Forced R¨ossler Oscillator ...........................422
5.1.2 Second Lyapunov Exponent: Perturbative Calculation..424
5.2 Phase Synchronization in Coupled Chaotic Oscillators........426
5.3 Oscillatory Phase Neurodynamics .........................430
5.3.1 Kuramoto Synchronization Model ...................432
5.3.2 Lyapunov Chaotic Synchronization ..................433
5.4 Synchronization Geometry................................435
5.4.1 Geometry of Coupled Nonlinear Oscillators ...........435
5.4.2 Noisy Coupled Nonlinear Oscillators .................439
5.4.3 Synchronization Condition..........................448
5.5 Complex Networks and Chaotic Transients .................451
6 Josephson Junctions and Quantum Engineering............457
6.0.1 Josephson Effect ..................................460
6.0.2 Pendulum Analog .................................461
6.1 Dissipative Josephson Junction............................463
6.1.1 Junction Hamiltonian and its Eigenstates.............464
6.1.2 Transition Rate ...................................466
x Contents
6.2 Josephson Junction Ladder (JJL)..........................467
6.2.1 Underdamped JJL.................................473
6.3 Synchronization in Arrays of Josephson Junctions ...........477
6.3.1 Phase Model for Underdamped JJL..................479
6.3.2 Comparison of LKM2 and RCSJ Models .............485
6.3.3 ‘Small–World’ Connections in JJL Arrays ............486
7 Fractals and Fractional Dynamics..........................491
7.1 Fractals ................................................491
7.1.1 Mandelbrot Set ...................................491
7.2 Robust Strange Non–Chaotic Attractors....................494
7.2.1 Quasi–Periodically Forced Maps.....................494
7.2.2 2D Map on a Torus................................497
7.2.3 High Dimensional Maps ............................502
7.3 Effective Dynamics in Hamiltonian Systems.................505
7.3.1 Effective Dynamical Invariants ......................507
7.4 Formation of Fractal Structure in Many–Body Systems.......508
7.4.1 A Many–Body Hamiltonian.........................509
7.4.2 Linear Perturbation Analysis .......................509
7.5 Fractional Calculus and Chaos Control .....................512
7.5.1 Fractional Calculus ................................512
7.5.2 Fractional–Order Chua’s Circuit.....................514
7.5.3 Feedback Control of Chaos .........................515
7.6 Fractional Gradient and Hamiltonian Dynamics .............516
7.6.1 Gradient Systems .................................517
7.6.2 Fractional Differential Forms........................518
7.6.3 Fractional Gradient Systems ........................519
7.6.4 Hamiltonian Systems ..............................523
7.6.5 Fractional Hamiltonian Systems.....................524
8 Turbulence ................................................529
8.1 Parameter–Space Analysis of the Lorenz Attractor...........529
8.1.1 Structure of the Parameter–Space ...................531
8.1.2 Attractors and Bifurcations.........................538
8.2 Periodically–Driven Lorenz Dynamics ......................539
8.2.1 Toy Model Illustration .............................542
8.3 Lorenzian Diffusion ......................................545
8.4 Turbulence .............................................549
8.4.1 Turbulent Flow ...................................550
8.4.2 The Governing Equations of Turbulence..............552
8.4.3 Global Well-Posedness of the Navier–Stokes Equations .553
8.4.4 Spatio–Temporal Chaos and Turbulence in PDEs......554
8.4.5 General Fluid Dynamics............................559
8.4.6 Computational Fluid Dynamics .....................563
8.5 Turbulence Kinetics .....................................565
Contents xi
8.5.1 Kinetic Theory....................................567
8.5.2 Filtered Kinetic Theory ............................570
8.5.3 Hydrodynamic Limit...............................572
8.5.4 Hydrodynamic Equations...........................574
8.6 Lie Symmetries in the Models of Turbulence ................575
8.6.1 Lie Symmetries and Prolongations on Manifolds.......576
8.6.2 Noether Theorem and Navier–Stokes Equations .......589
8.6.3 Large–Eddy Simulation ............................591
8.6.4 Model Analysis ...................................594
8.6.5 Thermodynamic Consistence........................600
8.6.6 Stability of Turbulence Models......................602
8.7 Advection of Vector–Fields by Chaotic Flows ...............603
8.7.1 Advective Fluid Flow ..............................603
8.7.2 Chaotic Flows ....................................606
8.8 Brownian Motion and Diffusion ...........................607
8.8.1 Random Walk Model ..............................607
8.8.2 More Complicated Transport Processes ..............609
8.8.3 Advection–Diffusion ...............................609
8.8.4 Beyond the Diffusion Coefficient.....................614
9 Geometry, Solitons and Chaos Field Theory ...............617
9.1 Chaotic Dynamics and Riemannian Geometry...............617
9.2 Chaos in Physical Gauge Fields ...........................620
9.3 Solitions................................................625
9.3.1 History of Solitons in Brief .........................625
9.3.2 The Fermi–Pasta–Ulam Experiments.................630
9.3.3 The Kruskal–Zabusky Experiments ..................635
9.3.4 A First Look at the KdV Equation ..................638
9.3.5 Split–Stepping KdV ...............................641
9.3.6 Solitons from a Pendulum Chain ....................643
9.3.7 1D Crystal Soliton.................................644
9.3.8 Solitons and Chaotic Systems .......................645
9.4 Chaos Field Theory......................................649
References.....................................................653
Index..........................................................689
