1 Overview ..................................................... 1
1.1 The Origin and Development of Chaos Theory .............. 1
1.2 Control of Chaos ........................................ 3
1.2.1 Suppression of Chaos ............................. 4
1.2.2 Synchronization of Chaos ......................... 6
1.2.3 Control and Synchronization of Spatiotemporal
Chaos ............................................ 9
1.3 Anticontrol of Chaos ................................... 10
1.4 Summary ................................................ 12
References .................................................. 13
2 Preliminaries of Nonlinear Dynamics and Chaos ............... 17
2.1 Introduction ........................................... 17
2.2 Background ............................................. 18
2.3 Existence, Uniqueness, Flow, and Dynamical Systems ..... 20
2.3.1 Existence and Uniqueness ........................ 20
2.3.2 Row and Dynamical Systems ....................... 22
2.4 Equilibrium, Periodic Orbit, Quasiperiodic Orbit, and
Poincare Map ........................................... 24
2.4.1 Equilibrium of Continuous-Time Systems .......... 24
2.4.2 Periodic and Quasiperiodic Orbits ............... 25
2.4.3 Poincare Map .................................... 28
2.5 Invariant and Attracting Sets .......................... 29
2.6 Continuous-Time Systems in the Plane ................... 31
2.7 General Solutions of Discrete-Time Linear Systems ...... 37
2.8 Discrete-Time Systems in the Plane ..................... 38
2.9 Stabilities of Trajectories I: The Lyapunov First
Method ................................................. 43
2.9.1 The Definition of Lyapunov Stability ............ 43
2.9.2 Floquet Theory .................................. 49
2.10 Stabilities of Trajectories II: The Lyapunov Second
Method ................................................. 50
2.11 Chaotic Sets and Chaotic Attractors .................... 53
2.12 Symbolic Dynamics and the Shift Map .................... 54
2.13 Lyapunov Exponent ...................................... 57
2.14 Examples ............................................... 59
2.14.1 Tent Map and Logistic Map ....................... 59
2.14.2 Smale Horseshoe ................................. 62
2.14.3 The Lorenz System ............................... 66
2.15 Basics of Functional Differential Equations Theory ..... 71
2.16 Summary ................................................ 75
References .................................................. 76
3 Entrainment and Migration Control of Chaotic Systems ........ 77
3.1 Introduction ........................................... 77
3.2 Basics on Entrainment and Migration .................... 78
3.2.1 Entrainment Control ............................. 78
3.2.2 Attractor Selection by Migration Control ........ 80
3.3 OPCL Control Scheme .................................... 84
3.4 Global Control of a Class of Continuous-Time
Polynomial Chaotic Systems ............................. 85
3.4.1 OPNCL Control Scheme ............................ 85
3.4.2 Simulations ..................................... 89
3.5 Global Control of a Class of Discrete-Time Systems ..... 95
3.5.1 OPNCL Control for Discrete-Time Systems ......... 95
3.5.2 Simulations ..................................... 97
3.6 Summary ............................................... 101
References ................................................. 102
4 Feedback Control of Chaotic Systems ........................ 103
4.1 Introduction .......................................... 103
4.2 Model-Reference Adaptive Control of a Class of
Discrete-Time Chaotic Systems ......................... 104
4.2.1 Basic Formulation .............................. 104
4.2.2 Main Results ................................... 105
4.2.3 Simulations .................................... 108
4.3 Model-Reference Adaptive Control of a Class of
Continuous-Time Chaotic Systems ....................... 111
4.3.1 Basic Formulation .............................. 111
4.3.2 Main Results ................................... 112
4.3.3 Simulations .................................... 116
4.4 Control of a Class of Chaotic Systems Based on
Inverse Optimal Control ............................... 125
4.4.1 Problem Description ............................ 125
4.4.2 Inverse Optimal Controller Design .............. 126
4.4.3 Simulations .................................... 130
4.5 Summary ............................................... 130
References ................................................. 131
5 Synchronizing Chaotic Systems Based on Feedback Control .... 133
5.1 Introduction .......................................... 133
5.2 Synchronization of Continuous-Time Chaotic Systems
with a Single Input ................................... 134
5.2.1 Basic Formulation .............................. 134
5.2.2 Main Results ................................... 136
5.2.3 Simulations .................................... 137
5.3 Synchronization of Multi-Signals in Continuous-Time
Chaotic Systems ....................................... 142
5.3.1 Basic Formulation .............................. 142
5.3.2 Main Results ................................... 145
5.3.3 Simulations .................................... 146
5.4 Synchronization of Different Continuous-Time Chaotic
Systems ............................................... 150
5.4.1 Synchronization of Different Chaotic Systems
with Parameter Perturbations ................... 150
5.4.2 Simulations .................................... 151
5.5 Synchronization of Discrete-Time Chaotic Systems ...... 157
5.5.1 Basic Formulation .............................. 157
5.5.2 Main Results ................................... 159
5.5.3 Simulations .................................... 164
5.6 Summary ............................................... 167
References ................................................. 168
6 Synchronizing Chaotic Systems via Impulsive Control ........ 169
6.1 Introduction .......................................... 169
6.2 Complete Synchronization of a Class of Chaotic
Systems via Impulsive Control ......................... 170
6.2.1 System Description and the Synchronization
Problem ........................................ 170
6.2.2 Main Results ................................... 171
6.2.3 Simulations .................................... 176
6.3 Lag-Synchronization of the Unified Systems via
Impulsive Control ..................................... 180
6.3.1 Preliminaries .................................. 181
6.3.2 Main Results ................................... 183
6.3.3 Simulations .................................... 186
6.4 Impulsive Synchronization of Different Chaotic
Systems ............................................... 187
6.4.1 System Description and Synchronization
Problem ........................................ 188
6.4.2 Main Results ................................... 189
6.4.3 Simulations .................................... 194
6.5 Impulsive Synchronization of a Class of Chaotic
Delayed Neural Networks ............................... 198
6.5.1 Preliminaries .................................. 198
6.5.2 Main Results ................................... 200
6.5.3 Simulations .................................... 204
6.6 Summary ............................................... 206
References ................................................. 206
7 Synchronization of Chaotic Systems with Time Delay ......... 209
7.1 Introduction .......................................... 209
7.2 Adaptive Synchronization of a Class of Delayed
Chaotic Systems ....................................... 210
7.2.1 Adaptive Synchronization of the Delayed Chen
Chaotic Systems ................................ 210
7.2.2 Adaptive Synchronization of the Delayed
Lorenz Systems ................................. 216
7.3 Adaptive Synchronization Between Two Different
Delayed Chaotic Systems ............................... 221
7.3.1 Adaptive Synchronization Between Delayed Chen
Chaotic Systems and Delayed Lorenz Chaotic
Systems ........................................ 222
7.3.2 Adaptive Synchronization Between Two
Different Delayed Ikeda Chaotic Systems ........ 229
7.3.3 Adaptive Synchronization Between Two Different
Delayed Liao Chaotic Systems ................... 233
7.4 Synchronization of Chaotic Delayed Neural Networks .... 238
7.4.1 Synchronization of a Class of Chaotic Delayed
Neural Networks Based on Inverse Optimal
Control Theory ................................. 238
7.4.2 Adaptive Synchronization Between Two
Different Chaotic Delayed Neural Networks ...... 252
7.5 Summary ............................................... 266
References ................................................. 267
8 Synchronizing Chaotic Systems Based on Fuzzy Models ........ 269
8.1 Introduction .......................................... 269
8.2 Modeling Chaotic Systems via T-S Fuzzy Models ......... 270
8.3 Synchronization of Hyperchaotic Systems via T-S
Fuzzy Models .......................................... 277
8.3.1 Fuzzy Modeling of Hyperchaotic Systems ......... 277
8.3.2 Synchronization of Two Different Hyperchaotic
Systems ........................................ 289
8.3.3 Simulations .................................... 291
8.4 Synchronizing Fuzzy Chaotic Systems with Time-
Varying Delays ........................................ 298
8.4.1 Basic Formulation .............................. 298
8.4.2 Main Result .................................... 300
8.4.3 Simulations .................................... 302
8.5 Summary ............................................... 306
References ................................................. 306
9 Chaotification of Nonchaotic Systems ....................... 309
9.1 Introduction .......................................... 309
9.2 Chaotification of Discrete-Time Fuzzy Hyperbolic
Model with Uncertain Parameters ....................... 310
9.2.1 Basic Formulation .............................. 310
9.2.2 Main Results ................................... 312
9.2.3 Simulations .................................... 315
9.3 Chaotification of Continuous-Time Fuzzy Hyperbolic
Model Using Impulsive Control ......................... 318
9.3.1 Basic Formulation .............................. 318
9.3.2 Main Results ................................... 320
9.3.3 Simulations .................................... 323
9.4 Chaotification of Linear Systems Using Sampled Data
Control ............................................... 327
9.4.1 Basic Formulation .............................. 327
9.4.2 Main Results ................................... 328
9.4.3 Simulations .................................... 336
9.5 Summary ............................................... 339
References ................................................. 340
Index ......................................................... 343
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