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]

Reprinted in 2014

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Errata [All corrected in the 2014 edition]

 

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