FROM
CHAOS TO ORDER:
METHODOLOGIES, PERSPECTIVES
AND APPLICATIONS
Guanrong Chen and
Xiaoning Dong
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Table of Contents
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Foreword (by A.I.Mees)
Preface
Acknowledgments
1
Introduction 1
1.1 Controlling
Chaos 2
1.1.1 Chaos Control
- in a Broader Sense 2
1.1.2 Why Chaos
Control? 3
1.1.3 Early Skepticism
and Recent Effort 6
1.1.4 Chaos Control
- Two Examples 8
1.2 Some Distinct
Features of Chaos Control 14
1.3 Organization
of the Monograph 16
2
Nonlinear Dynamical Systems
21
2.1 Nonlinear Dynamical
Systems 22
2.1.1 Nonlinear
Dynamical System Preliminaries 23
2.1.2 Periodic Orbits
and Limit Cycles 27
2.1.3 Limit Sets
and Attractors 29
2.1.4 Poincare Maps
31
2.1.5 Homoclinic
and Heteroclinic Trajectories 32
2.2 Some Analytical
Tools 34
2.2.1 Center Manifold
Theory 35
2.2.2 Poincare's
Normal Forms 36
2.2.3 Delay-Coordinates
and Embedology 37
2.3 Chaos in Nonlinear
Systems 41
2.3.1 What Is Chaos
42
2.3.2 Features of
Chaos 43
2.3.3 Bifurcations
59
2.4 A World of Chaos
66
2.4.1 Chaos is Ubiquitous
67
2.4.2 Paradigms
of Chaos 67
2.5 Symmetry, Self-Similarity,
and Stabilities 84
2.5.1 Symmetry and
Self-Similarity 84
2.5.2 Stabilities
85
3
Parameter-Dependent Approaches to Chaos Control
89
3.1 Periodic Parametric
Forcing 90
3.1.1 Parametrically
Forced Oscillators 91
3.1.2 Parametrically
Forced Convective Flows 94
3.2 Microscopic
Parametric Variation 96
3.2.1 The Original
Approach 97
3.2.2 Some Experimental
Studies 104
3.2.3 Convergence
Analysis 109
3.2.4 Targeting
111
3.2.5 Using the
Delay-Coordinates Technique 114
3.2.6 Controlling
Chaos to Higher-Periodic Orbits 117
3.2.7 A Modification
of the Original Method 119
3.2.8 Some Applications
of Parametric Variation Control 122
3.2.9 Controlling
Transient Chaos 123
3.3 Parameter Tuning
and Chaos Control 126
3.3.1 Controlling
Chaos Onset via Parameter Tuning 127
3.3.2 Controlling
the Size of a Chaotic Attractor 129
3.3.3 Parameter
Tuning Based on Bifurcation Analysis 130
3.3.4 Some Applications
of Parameter Control 131
3.4 Summary
132
4
Open-Loop Strategies for Chaos Control 135
4.1 Chaos Control
via External Forcing 136
4.1.1 Using External
Weak Periodic Forces 136
4.1.2 Local Stability
Analysis 139
4.1.3 Phase Effect
in External Weak Forcing Control 141
4.1.4 Using Random
Noise as Control Force 146
4.1.5 Using Periodic
Impulse Input as Control Force 147
4.2 Entrainment
and Migration Controls 148
4.2.1 Entrainment-Goal
Controls 149
4.2.2 Migration-Goal
Control 160
4.2.3 Entrainment
and Migration with Feedback 165
4.3 Summary
171
5
Engineering Feedback Control (I) 173
5.1 Chaos in Feedback
Control Systems 175
5.1.1 Chaos in Continuous-Time
Feedback Systems 176
5.1.2 Chaos in Discrete-Time
Feedback Systems 183
5.2 Automatic Systems
Control 190
5.2.1 Engineering
Control Using Feedback 190
5.2.2 Feedback Versus
Open-Loop Controls 194
5.2.3 Some Practical
Perspectives 195
5.2.4 Feedback Control
of Chaos 198
5.3 Chaos Control
via Lyapunov Methods 199
5.3.1 Lyapunov Theorems
201
5.3.2 Controlling
Chaos via Lyapunov Methods 206
3.3.3 Controlling
Chaos to Higher-Periodic Orbits 228
5.4 Some General
Controllability Conditions 231
5.5 Summary
234
6
Engineering Feedback Control (II) 237
6.1 Optimal Control
of Chaos 237
6.1.1 Optimal Controls
238
6.1.2 A Case of
Optimal Control of Chaos 241
6.1.3 Optimal Parametric
Variation Control 243
6.1.4 Comparison
of Optimal Chaos Control Strategies 246
6.1.5 H-infinity
Control Approach 250
6.2 Toward Robust
Control of Chaos 254
6.2.1 Chaotic Vibration
Control 255
6.2.2 A Two-Degree-of-Freedom
Controller 259
6.3 Contraction
Mapping and Sliding Mode Controls 265
6.3.1 Contraction
Mapping Based Controls 265
6.3.2 Sliding Mode
Control 269
6.3.3 Control of
Systems with Discontinuous Vector Fields 273
6.4 Discretization
Chaos and Its Control 274
6.4.1 Chaos From
Discretization 275
6.4.2 Chaos Suppression
in Sampled-Data Systems 277
6.4.3 Digital Redesign
for Chaos Control 278
6.5 Summary
291
7
Engineering Feedback Control (III) 293
7.1 Occasional Proportional
Feedback Control 293
7.2 Delayed Feedback
Control 297
7.3 Methods from
Mechanical Engineering 301
7.3.1 Dissipative
Controller Design 301
7.3.2 Absorber as
Controller 303
7.4 Stochastic Control
Approaches 305
7.4.1 A Stochastic
Control Method 306
7.4.2 Stochastic
Modeling for Chaos Control 309
7.4.3 Probabilistic
Control of Chaos 311
7.4.4 Chaos Control
under a Statistical Criterion 313
7.5 Distortion Control
via Harmonic Balance 314
7.5.1 Harmonic Balance
Analysis 314
7.5.2 Predicting
Chaos via Harmonic Balance 319
7.5.3 Distortion
Control of Chaos 321
7.6 Filtering Applied
to Chaos Control 323
7.6.1 Kalman Filter
and Chaos 323
7.6.2 Controlling
Chaos Using a Notch Filter 324
7.7 Entropy Reduction
for Chaos Rejection 328
7.8 Summary
333
8
Adaptive Control of Chaos
335
8.1 Adaptive Control:
An Example 336
8.2 Typical Adaptive
Control Algorithms 338
8.2.1 Two Typical
Adaptive Control Methods 338
8.2.2 Projection-Estimation
and LMS Schemes 342
8.3 Chaos in Adaptive
Control 344
8.3.1 Complex Dynamics
in MRAC and STAC Systems 345
8.3.2 Chaos in Other
Adaptive Control Systems 349
8.4 An Example of
Adaptive Control of Chaos 349
8.5 Gradient Based
Adaptive Control 351
8.5.1 Examples of
Gradient Based Control of Chaos 354
8.5.2 Gradient Model-Referenced
Adaptive Control 357
8.6 Self-Tuning
Adaptive Control of Chaos 358
8.6.1 Self-Tuning
Control of Cardiac Chaos 359
8.6.2 Autoregressive
Self-Tuning Feedback Control 364
8.7 The Lyapunov
Function Approach 371
8.7.1 MRAC and Differential
Inclusion Approach 372
8.7.2 Adaptive Control
of Uncertain Chaotic Systems 376
8.8 Model Reconstruction
Based Controls 385
8.8.1 Adaptive Control
Based on Model Reconstruction 386
8.8.2 Adaptive Control
Based on ARMA Models 388
8.8.3 A General
Model Reconstruction Based Approach 393
8.9 Parametric Variation
Based Controls 403
8.10 Summary
406
9
Intelligent Control of Chaos 409
9.1 Artificial Neural
Networks 411
9.1.1 General Structure
of Neural Networks 411
9.1.2 Functional
Approximation by Neural Networks 415
9.2 Chaos in Neural
Networks 419
9.3 Controlling
Chaos in Neural Networks 425
9.4 Chaos Identification
via Neural Networks 431
9.4.1 A General
Setup for Chaos Identification 431
9.4.2 A Wiener-Type
Model for Chaos Identification 433
9.5 Chaos Control
by Neural Networks 441
9.5.1 Examples of
Neural Network Based Chaos Control 441
9.5.2 Further Discussion
447
9.6 Fuzzy Control
Systems 449
9.6.1 General Structure
of Fuzzy Control Systems 451
9.6.2 Fuzzy Logic
Control: Two Basic Approaches 456
9.7 Chaos in Fuzzy
Control Systems 462
9.8 Controlling
Chaos Using Fuzzy Logic 465
9.8.1 Controlling
Chaos by Adaptive Fuzzy Method 465
9.8.2 A Combined
Modeling and Control Approach 475
9.8.3 Some Related
Developments 479
9.9 Summary
481
10
Chaos Control in Distributed Systems 483
10.1 Distributed
Parameter Control Systems 484
10.2 Chaos in Spatiotemporal
Systems 486
10.3 Controlling
Spatiotemporal Chaos 487
10.3.1 Controlling
Spatiotemporal Chaos in Plasma 489
10.3.2 Controlling
Chaos in Isotropic Systems 493
10.4 Controlling
Transport in Chaotic Lattices 498
10.5 Localized Control
of Chaos 502
10.5.1 Local Controls
of a Nonlinear Network Model 503
10.5.2 Controlling
a Chain of Coupled Nonlinear Maps 507
10.5.3 Local Control
of Coupled Logistic Networks 511
10.5.4 Parameter
Variation versus Feedback Pinning 514
10.5.5 Local versus
Global Controls 522
10.6 Decentralized
Control of Chaos 524
10.7 Controlling
Chaos in DAI Systems 528
10.7.1 Chaos in
DAI Systems 529
10.7.2 Using Reward
Policy to Control Chaos 531
10.8 Summary
534
11
Chaos Synchronization
537
11.1 What is Synchronization?
538
11.1.1 Synchronization
as Result of Coupled Oscillations 538
11.1.2 Chaos Synchronization:
A Drive-Responses Setup 543
11.2 Synchronization
Based on System Decomposition 546
11.2.1 Realizing
Drive and Response by Decomposition 546
11.2.2 Synchronizing
by Homogeneous Driving 547
11.2.3 Stability
Analysis for Chaos Synchronization 551
11.3 Chaos Synchronization
via Feedback 557
11.3.1 Synchronization
of Identical Subsystems 557
11.3.2 Adaptive
Synchronization 558
11.3.3 Chaos Synchronization
with Observer 567
11.3.4 Two Simple
Ways to Synchronize Chaos 569
11.4 Chaos Synchronization
via System Inverse 572
11.5 More on Chaos
Synchronization 575
11.5.1 Generalized
Synchronization 576
11.5.2 Phase Synchronization
577
11.5.3 Synchronizing
Higher-Dimensional and
Spatiotemporal Chaos 579
11.5.4 Synchronization
via the OLC Method 585
11.5.5 Synchronization
Using Contraction Mappings 588
11.5.6 Stochastic
Synchronization 590
11.5.7 Dead-Beat
Chaos Synchronization 591
11.6 Chaos Control
versus Chaos Synchronization 595
11.7 Synchronization
and Communication 597
11.7.1 Communication
Based on Chaos Synchronization 598
11.7.2 Robustness
of Chaos Synchronization 603
11.7.3 Dead-Beat
Synchronization for Communication 606
11.7.4 Implementation
of Chaos Synchronization Based
Communication 609
11.7.5 Chaotic Signal
Encoding: A Biological Inspiration 614
11.8 Summary
617
12 More
on Chaos Control 619
12.1 Controlling
Bifurcations 620
12.1.1 Bifurcations
in Control Systems 621
12.1.2 Some Bifurcation
Control Approaches 628
12.2 Controlling
Multiple Limit Cycles 638
12.2.1 Graphical
Hopf Bifurcation Theorem 639
12.2.2 Controlling
the Birth of Multiple Limit Cycles 642
12.2.3 Controlling
the Amplitudes of Limit Cycles 646
12.3 Chaos, Synchronization,
and Disorder 653
12.3.1 Taming Spatiotemporal
Chaos with Disorder 653
12.3.2 Disorder-Enhanced
Synchronization 657
12.4 Pattern Formation,
Self-Organization, and Chaos Control 657
12.4.1 Pattern Formation
659
12.4.2 Classification
of Patterns 663
12.4.3 Pattern Formation
and Control 666
12.5 Anticontrol
of Chaos 674
12.5.1 Why Anticontrol
of Chaos? 677
12.5.2 Toward Anticontrol
of Chaos 679
12.6 Summary
691
Epilogue
693
References
695
Notation
749
Index
753
World Scientific
Pub. Co., Singapore, May 1998. US$88.00
ISBN 981-02-2569-5
[with 760pp, 312 Figs, and 728 Refs]
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