Chaos Control: Theory and Applications

Preface......Page 6
Contents......Page 9
1 Introduction: Is “Controlling Chaos” an Oxymoron?......Page 11
2 Statement of the Targeting Problem......Page 12
3 Targeting: A Simple and Instructive One-Dimensional Example......Page 14
3.1 The higher-dimensional generalization......Page 15
3.2 A web approach to higher-dimensional targeting......Page 16
3.3 Targeting through recurrence through resonance layers and to the moon......Page 17
4 Combinatorial Targeting and Symbolic Dynamics......Page 22
4.1 One-dimensional maps with a single critical point......Page 23
4.2 Higher-dimensional systems and symbolic dynamics of di.eomorphisms......Page 25
4.3 Learning the grammar in practice......Page 26
5 Forcing the Path: Feedback Control......Page 27
5.1 Parametric feedback control......Page 28
5.2 Time-delayed feedback control......Page 29
5.3 Dynamic limiting......Page 30
1 Introduction......Page 35
2 Linear Wave Equation with a van der Pol Boundary Condition......Page 39
3 Some Open Questions......Page 56
1 Introduction......Page 61
2 Hybrid Systems......Page 63
3.2 Generalization......Page 65
3.3 Convergence analysis......Page 66
4 Switched Arrival System......Page 68
5 Chua’s Circuit......Page 73
6 Conclusions......Page 77
References......Page 78
1 Introduction......Page 80
2.1 Stabilization of periodic orbits......Page 81
2.3 Discrete-time delayed feedback control......Page 82
3 The Odd Number Limitation......Page 83
3.1 The odd number limitation via discrete-time DFC......Page 84
3.3 Overcoming the odd number limitation......Page 85
4.1 Half-periodic DFC......Page 86
5 Dynamic DFC......Page 87
5.1 Observer-based DFC......Page 88
5.2 State feedback......Page 89
5.3 Extended DFC......Page 90
6 Recursive DFC......Page 91
7.1 Second-order dynamic delayed feedback controller......Page 92
7.2 Recursive delayed feedback controller......Page 93
References......Page 94
1.2 The perturbed nonlinear Schr¨odinger equation (NSL)......Page 97
1.4 The extended Fisber-Kolmogorov (F-K) equation......Page 98
2.1 The Lyapunov exponent method......Page 99
2.2 Measurement of chaos by entropy......Page 104
2.3 Perturbation theory and the Melnikov method......Page 105
2.4 Some numerical and analytic methods......Page 106
2.5 Use decay mutual information to characterize spatiotemporal chaotic dynamics.......Page 109
3 Controlling chaos to a class of PDEs by applying invariant manifold and structure stability theory......Page 110
Appendix......Page 121
References......Page 122
1 Introduction......Page 124
2 Master-Slave Synchronization: Autonomous Case......Page 125
2.2 Master-slave synchronization using dynamic output feedback......Page 127
2.3 Robust synchronization......Page 128
2.4 Synchronization with time delay......Page 129
3 Nonlinear H Synchronization......Page 131
4.1 Impulsive synchronization: state feedback case......Page 133
4.2 Impulsive synchronization: measurement feedback case......Page 134
5 Examples......Page 135
6 Conclusions......Page 138
References......Page 139
1 Introduction......Page 143
2 Inverse Optimal Control......Page 144
3.1 Mathematical description......Page 146
3.2 Model-following as a stabilization problem......Page 147
3.3 Inverse optimal control......Page 148
3.4 Chaos reproduction......Page 151
4 Chaos Synchronization via Adaptive Recurrent Neural Control......Page 153
4.1 Recurrent high-order neural network......Page 155
4.3 Tracking analysis......Page 156
4.4 Tracking error stabilization......Page 158
4.5 Inverse optimal control......Page 159
4.6 Chaos synchronization......Page 160
5 Conclusions......Page 162
1 Introduction......Page 165
2.2 Chaos in the sense of Li-Yorke......Page 167
3 Chaotification: Problem Formulation......Page 169
4 An Illustrative Example......Page 174
5 Some Remarks on the Continuous-Time Case......Page 179
6 Concluding Remarks......Page 180
References......Page 181
1 Introduction......Page 184
2.1 Generating chaos in a linear diffential equation......Page 185
2.3 Generating chaos in feedback linearizable systems......Page 187
3.1 Normal form of chaotic systems......Page 193
3.2 Generating chaos in linear controllable systems......Page 197
3.3 Generating chaos in feedback linearizable systems......Page 199
3.4 Generating chaos in feedback unlinearizable systems......Page 201
4 Generating Chaos in Continuous-Time Systems Using Impulsive Control......Page 205
5 Conclusions......Page 207
References......Page 208
1 Introduction......Page 210
2.1 The particle-core model......Page 212
2.2 Breathing model interaction with particle motion......Page 213
2.3 Nonlinear resonances and induced chaos......Page 217
2.4 Halo formation in three-dimensional bunches......Page 220
2.5 Canonical transformation method......Page 226
2.6 The matrix transfer method......Page 230
3 Nonlinear Feedback Control of Beam Halo Chaos in PFCs......Page 232
4 Wavelet-Based Feedback Control......Page 235
4.1 Main control results via wavelet-based feedback......Page 236
5 Switching Manifold Control Method......Page 240
6 Time-Delayed Self-Control Feedback Method......Page 244
7 Conclusions and Discussions......Page 247
1 Introduction......Page 251
2 Pseudo-Random Number Generators......Page 252
3 Algorithmic Complexity and Randomness......Page 254
4 Computational Complexity and Randomness......Page 255
5 Determinism, Chaos and Randomness......Page 258
6 Shadowing......Page 261
7 Chaos-Based Pseudo-Random Number Generators......Page 262
References......Page 266
1 Introduction......Page 269
2.1 Category I: Modulation based on the chaotic signals......Page 272
2.2 Category II: Modulation based on chaotic state functions......Page 275
2.3 Category III: Modulation based on the system parameters......Page 279
3.1 An ergodic approach to chaos communication......Page 280
3.2 System model of E-DCSK......Page 282
3.3 BER of E-DCSK......Page 283
4.1 Noise performance in an AWGN channel......Page 286
4.2 BER performance of E-DCSK under special noise environment......Page 293
5 Conclusions......Page 296
1 Introduction......Page 299
2 Some Tools for Computing Expectations of Quantized Chaotic Trajectories......Page 302
3 Asynchronous DS-CDMA Systems Model......Page 308
4 System Performance Merit Figure......Page 311
5 Performance Optimization in Dispersive and Non-Dispersive Channels A......Page 313
6 Chaos-Based Generation of Optimized Spreading Sequences......Page 315
7 Chaos-Based DS-CDMA System Prototype and Measurement Results......Page 317
7.2 Performance over exponential channel......Page 320
8 Conclusions......Page 321
References......Page 322
1 Introduction......Page 325
2 Piecewise ACne Systems......Page 326
3 Piecewise Smooth Systems......Page 327
4.1 Formulation......Page 329
4.2 Repetitive calculation of UPOs embedded in a chaotic attractor......Page 331
4.3 Calculation of UPOs in a piecewise smooth system......Page 333
5.1 Stability analysis of EFC for piecewise smooth systems......Page 337
5.2 EFC for piecewise smooth systems......Page 340
6 Conclusions......Page 342
References......Page 343
1 Introduction......Page 345
2 Generation of Low-EMI Clock Signals......Page 349
2.1 Random FM clock modulation......Page 350
2.2 Statistical features of chaotic processes and chaos-based FM signals......Page 354
2.3 Experimental results......Page 361
3 Generation of Low-EMI PWM Signals......Page 362
3.1 Chaos-based PWM signals......Page 364
3.2 Experimental results......Page 368
4 Conclusions......Page 370