High-dimensional chaotic and attractor systems

✍ Scribed by Ivancevic V.G., Ivancevic T.T.

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 711
• Series: Intelligent Systems, Control and Automation: Science and Engineering
• Category: Library

✦ Synopsis

This - is a graduatelevel monographic textbook devoted to understanding, prediction and control of highdimensional chaotic and attractor systems of real life. The objective of the book is to provide - the serious reader with a serious scientific tool that will enable the actual performance of competitive research in highdimensional chaotic and attractor dynamics. The book has nine Chapters. The first Chapter gives a textbook-like introduction into the low-dimensional attractors and chaos. This Chapter has an inspirational character, similar to other books on nonlinear dynamics and deterministic chaos. The second Chapter deals with Smales topological transformations of stretching, squeezing and folding (of the systems phasespace), developed for the purpose of chaos theory. The third Chapter is devoted to Poincarés 3-body problem and basic techniques of chaos control, mostly of Ott-Grebogi-Yorke type. The fourth Chapter is a review of both Landaus and topological phase transition theory, as well as Hakens synergetics. The fifth Chapter deals with phase synchronization in high-dimensional chaotic systems. The sixth Chapter presents high-tech Josephson junctions, the basic components for the future quantum computers. The seventh Chapter deals with fractals and fractional Hamiltonian dynamics. The 8th Chapter gives a review of modern techniques for dealing with turbulence, ranging from theparameterspace of the Lorenz attractor to the Lie symmetries. The last, 9th, Chapter attempts to give a brief on the cutting edge techniques of the high-dimensional nonlinear dynamics (including geometries, gauges and solitons, culminating into the chaos field theory).

✦ Table of Contents

High-Dimensional Chaoticand Attractor Systems, A Comprehensive Introduction......Page 4
Contents......Page 7
Preface......Page 12
Acknowledgments......Page 14
1 Introduction to Attractors and Chaos......Page 15
Motivating Example: A Playground Swing......Page 18
Dynamics Tradition and Chaos......Page 22
Basic Terms of Nonlinear Dynamics......Page 25
Phase Plane: Nonlinear Dynamics without Chaos......Page 27
Free vs. Forced Nonlinear Oscillators......Page 28
1.1 Basics of Attractor and Chaotic Dynamics......Page 31
Lyapunov Exponents......Page 36
Kolmogorov–Sinai Entropy......Page 37
Further Motivating Example: Pinball Game and Periodic Orbits......Page 38
1.2 A Brief History of Chaos Theory in 5 Steps......Page 43
1.2.1 Henry Poincar´e: Qualitative Dynamics, Topology and Chaos......Page 44
1.2.2 Stephen Smale: Topological Horseshoe and Chaos of Stretching and Folding......Page 52
1.2.3 Ed Lorenz: Weather Prediction and Chaos......Page 62
1.2.4 Mitchell Feigenbaum: A Constant and Universality......Page 65
1.2.5 Lord Robert May: Population Modelling and Chaos......Page 66
1.2.6 Michel H´enon: A Special 2D Map and Its Strange Attractor......Page 70
Other Famous 2D Chaotic Maps......Page 72
1.3 Some Classical Attractor and Chaotic Systems......Page 73
Simple Pendulum......Page 74
Van der Pol Oscillator......Page 75
Rossler System......Page 79
Chua’s Circuit......Page 80
Inverted Pendulum......Page 81
Elastic Pendulum......Page 82
Mandelbrot and Julia Sets......Page 83
Biomorphic Systems......Page 84
1.4.1 A Motivating Example......Page 86
1.4.2 Systems of ODEs......Page 89
Flow of a Linear ODE......Page 92
Canonical Linear Flows in R2......Page 94
1.4.4 Conservative versus Dissipative Dynamics......Page 96
Dissipative Systems......Page 97
Thermodynamic Equilibrium......Page 98
Nonlinearity......Page 99
The Second Law of Thermodynamics......Page 100
Geometry of Phase Space......Page 101
Real 1-DOF Hamiltonian Dynamics......Page 103
Complex One–DOF Hamiltonian Dynamics......Page 110
Library of Basic Hamiltonian Systems......Page 113
n-DOF Hamiltonian Dynamics......Page 120
1.4.6 Ergodic Systems......Page 122
1.5 Continuous Chaotic Dynamics......Page 123
1.5.1 Dynamics and Non–Equilibrium Statistical Mechanics......Page 125
1.5.2 Statistical Mechanics of Nonlinear Oscillator Chains......Page 138
1.5.3 Geometrical Modelling of Continuous Dynamics......Page 139
1.5.4 Lagrangian Chaos......Page 142
Lagrangian Chaos in 2D–Flows......Page 146
1.6 Standard Map and Hamiltonian Chaos......Page 150
1.7 Chaotic Dynamics of Binary Systems......Page 156
1.7.1 Examples of Dynamical Maps......Page 158
1.7.2 Correlation Dimension of an Attractor......Page 162
1.8 Basic Hamiltonian Model of Biodynamics......Page 163
2.1 Smale Horseshoe Orbits and Symbolic Dynamics......Page 166
2.1.1 Horseshoe Trellis......Page 170
2.1.2 Trellis–Forced Dynamics......Page 174
2.1.3 Homoclinic Braid Type......Page 177
2.2 Homoclinic Classes for Generic Vector–Fields......Page 178
2.2.1 Lyapunov Stability......Page 181
2.2.2 Homoclinic Classes......Page 184
2.3 Complex–Valued H´enon Maps and Horseshoes......Page 187
2.3.1 Complex Henon–Like Maps......Page 188
2.3.2 Complex Horseshoes......Page 191
2.4 Chaos in Functional Delay Equations......Page 194
2.4.1 Poincar´e Maps and Homoclinic Solutions......Page 197
2.4.2 Starting Value and Targets......Page 205
2.4.3 Successive Modi.cations of g......Page 212
2.4.4 Transversality......Page 228
2.4.5 Transversally Homoclinic Solutions......Page 234
3.1 Mechanical Origin of Chaos......Page 236
3.1.1 Restricted 3–Body Problem......Page 237
Equations of Motion for the Restricted 3–Body Problem......Page 239
Birkhoff Normalization......Page 240
3.1.2 Scaling and Reduction in the 3–Body Problem......Page 249
3.1.3 Periodic Solutions of the 3–Body Problem......Page 252
3.1.4 Bifurcating Periodic Solutions of the 3–Body Problem......Page 253
3.1.5 Bifurcations in Lagrangian Equilibria......Page 254
Hamiltonian Hopf Bifurcation......Page 257
3.1.6 Continuation of KAM–Tori......Page 260
Lyapunov Exponents......Page 262
Mathieu Equation......Page 263
Parametric Resonance Model......Page 264
3.2.1 Feedback and Non–Feedback Algorithms for Chaos Control......Page 265
Hybrid Systems and Homotopy ODEs......Page 268
3.2.2 Exploiting Critical Sensitivity......Page 269
3.2.3 Lyapunov Exponents and Kaplan–Yorke Dimension......Page 270
3.2.4 Kolmogorov–Sinai Entropy......Page 272
3.2.5 Chaos Control by Ott, Grebogi and Yorke (OGY)......Page 274
Simple Example of Chaos Control: a 1D Map
......Page 275
3.2.6 Floquet Stability Analysis and OGY Control......Page 277
3.2.7 Blind Chaos Control......Page 281
3.2.8 Jerk Functions of Simple Chaotic Flows......Page 285
3.2.9 Example: Chaos Control in Molecular Dynamics......Page 288
Hamiltonian Chaotic Dynamics......Page 289
Optimal Control Algorithm......Page 290
Following the Mean Trajectory......Page 293
Following a Fixed Trajectory......Page 295
4.1.1 Equilibrium Phase Transitions......Page 298
4.1.2 Classi.cation of Phase Transitions......Page 299
4.1.3 Basic Properties of Phase Transitions......Page 301
4.1.4 Landau’s Theory of Phase Transitions......Page 303
Classical Partition Function......Page 305
Quantum Partition Function......Page 306
Vibrations of Coupled Oscillators......Page 307
4.1.6 Noise–Induced Non–equilibrium Phase Transitions......Page 312
General Zero–Dimensional System......Page 313
General d-Dimensional System......Page 315
4.2 Elements of Haken’s Synergetics......Page 319
4.2.1 Phase Transitions......Page 321
4.2.2 Mezoscopic Derivation of Order Parameters......Page 323
4.2.3 Example: Synergetic Control of Biodynamics......Page 325
4.2.4 Example: Chaotic Psychodynamics of Perception......Page 326
4.2.5 Kick Dynamics and Dissipation–Fluctuation Theorem Deterministic Delayed Kicks......Page 330
Random Kicks and Langevin Equation......Page 331
4.3 Synergetics of Recurrent and Attractor Neural Networks......Page 333
4.3.1 Stochastic Dynamics of Neuronal Firing States......Page 335
4.3.2 Synaptic Symmetry and Lyapunov Functions......Page 340
4.3.3 Detailed Balance and Equilibrium Statistical Mechanics......Page 342
4.3.4 Simple Recurrent Networks with Binary Neurons Networks with Uniform Synapses......Page 347
Phenomenology of Hopfield Models......Page 350
Analysis of Hopfield Models Away From Saturation......Page 351
4.3.5 Simple Recurrent Networks of Coupled Oscillators Coupled Oscillators with Uniform Synapses......Page 356
Coupled Oscillator Attractor Networks......Page 358
4.3.6 Attractor Neural Networks with Binary Neurons......Page 363
Closed Macroscopic Laws for Sequential Dynamics......Page 364
Application to Separable Attractor Networks......Page 367
Closed Macroscopic Laws for Parallel Dynamics......Page 371
Application to Separable Attractor Networks......Page 374
4.3.7 Attractor Neural Networks with Continuous Neurons Closed Macroscopic Laws......Page 375
Application to Graded–Response Attractor Networks......Page 378
Fluctuation–Dissipation Theorems......Page 382
Simple Attractor Networks with Binary Neurons......Page 386
Graded–Response Neurons with Uniform Synapses......Page 390
4.3.9 Path–Integral Approach for Complex Dynamics......Page 391
Partition–Function Analysis for Binary Neurons......Page 395
Extremely Diluted Attractor Networks Near Saturation......Page 409
4.3.10 Hierarchical Self–Programming in Neural Networks......Page 412
4.4.1 Phase Transitions in Hamiltonian Systems......Page 419
4.4.2 Geometry of the Largest Lyapunov Exponent......Page 421
4.4.3 Euler Characteristics of Hamiltonian Systems......Page 425
4.4.4 Pathways to Self–Organization in Human Biodynamics......Page 429
5.1 Lyapunov Vectors and Lyapunov Exponents......Page 432
5.1.1 Forced R¨ossler Oscillator......Page 435
5.2 Phase Synchronization in Coupled Chaotic Oscillators......Page 439
5.3 Oscillatory Phase Neurodynamics......Page 443
5.3.1 Kuramoto Synchronization Model......Page 445
5.3.2 Lyapunov Chaotic Synchronization......Page 446
5.4.1 Geometry of Coupled Nonlinear Oscillators......Page 448
5.4.2 Noisy Coupled Nonlinear Oscillators......Page 452
5.4.3 Synchronization Condition......Page 461
5.5 Complex Networks and Chaotic Transients......Page 464
6 Josephson Junctions and Quantum Engineering......Page 469
6.0.1 Josephson E.ect......Page 472
6.0.2 Pendulum Analog......Page 473
6.1 Dissipative Josephson Junction......Page 475
6.1.1 Junction Hamiltonian and its Eigenstates......Page 476
6.1.2 Transition Rate......Page 478
6.2 Josephson Junction Ladder (JJL)......Page 479
6.2.1 Underdamped JJL......Page 485
6.3 Synchronization in Arrays of Josephson Junctions......Page 489
6.3.1 Phase Model for Underdamped JJL......Page 491
Multiple time scale analysis......Page 494
6.3.2 Comparison of LKM2 and RCSJ Models......Page 497
6.3.3 ‘Small–World’ Connections in JJL Arrays......Page 498
7.1.1 Mandelbrot Set......Page 502
7.2.1 Quasi–Periodically Forced Maps Forced Logistic Map......Page 505
Forced Circle Map......Page 506
7.2.2 2D Map on a Torus Existence of SNA......Page 508
Rational Approximation: Origin of SNA’s......Page 512
7.2.3 High Dimensional Maps Radial Perturbations of the Torus Map......Page 513
Map on a High–Dimensional Torus......Page 514
7.3 E.ective Dynamics in Hamiltonian Systems......Page 516
7.3.1 E.ective Dynamical Invariants......Page 518
7.4 Formation of Fractal Structure in Many–Body Systems......Page 519
7.4.2 Linear Perturbation Analysis......Page 520
Collisionless Boltzmann Equation......Page 521
Dynamical Stability......Page 522
7.5.1 Fractional Calculus Fractional Derivatives......Page 523
Numerical Calculation of Fractional Derivatives......Page 524
7.5.2 Fractional–Order Chua’s Circuit......Page 525
7.5.3 Feedback Control of Chaos......Page 526
7.6 Fractional Gradient and Hamiltonian Dynamics......Page 527
7.6.1 Gradient Systems......Page 528
7.6.2 Fractional Di.erential Forms......Page 529
7.6.3 Fractional Gradient Systems......Page 530
Examples......Page 532
Lorenz System as a Fractional Gradient System......Page 533
7.6.4 Hamiltonian Systems......Page 534
7.6.5 Fractional Hamiltonian Systems......Page 535
8.1 Parameter–Space Analysis of the Lorenz Attractor......Page 539
8.1.1 Structure of the Parameter–Space......Page 541
8.1.2 Attractors and Bifurcations......Page 548
8.2 Periodically–Driven Lorenz Dynamics......Page 549
8.2.1 Toy Model Illustration......Page 552
8.3 Lorenzian Di.usion......Page 555
8.4 Turbulence......Page 559
8.4.1 Turbulent Flow......Page 560
8.4.2 The Governing Equations of Turbulence......Page 562
8.4.3 Global Well-Posedness of the Navier–Stokes Equations......Page 563
8.4.4 Spatio–Temporal Chaos and Turbulence in PDEs......Page 564
8.4.5 General Fluid Dynamics......Page 569
8.4.6 Computational Fluid Dynamics......Page 573
8.5 Turbulence Kinetics......Page 575
8.5.1 Kinetic Theory......Page 577
8.5.2 Filtered Kinetic Theory......Page 580
8.5.3 Hydrodynamic Limit......Page 582
8.5.4 Hydrodynamic Equations......Page 584
8.6 Lie Symmetries in the Models of Turbulence......Page 585
Lie Symmetry Groups......Page 586
Prolongations......Page 589
Noether Symmetries......Page 595
Lie Symmetries in Biophysics......Page 598
8.6.2 Noether Theorem and Navier–Stokes Equations......Page 599
Turbulent Viscosity Models......Page 601
Gradient–Type Models......Page 602
Similarity–Type Models......Page 603
8.6.4 Model Analysis......Page 604
Translational Invariance......Page 605
Rotational and Re.ective Invariance......Page 606
Scaling Invariance......Page 607
Material Indi.erence......Page 608
Invariance under the Symmetries......Page 610
Consequences of the Second Law......Page 611
8.6.6 Stability of Turbulence Models......Page 612
8.7.1 Advective Fluid Flow......Page 613
8.7.2 Chaotic Flows......Page 616
8.8.1 Random Walk Model......Page 617
8.8.3 Advection–Di.usion......Page 619
8.8.4 Beyond the Di.usion Coe.cient......Page 624
9.1 Chaotic Dynamics and Riemannian Geometry......Page 627
9.2 Chaos in Physical Gauge Fields......Page 630
9.3.1 History of Solitons in Brief......Page 635
9.3.2 The Fermi–Pasta–Ulam Experiments......Page 640
9.3.3 The Kruskal–Zabusky Experiments......Page 645
9.3.4 A First Look at the KdV Equation......Page 648
9.3.5 Split–Stepping KdV......Page 651
9.3.6 Solitons from a Pendulum Chain......Page 653
9.3.7 1D Crystal Soliton......Page 654
9.3.8 Solitons and Chaotic Systems......Page 655
9.4 Chaos Field Theory......Page 659
References......Page 662
Index......Page 698