This monograph presents a reasonably rigorous theory of a highly relevant chaos control method: suppression-enhancement of chaos by weak periodic excitations in low-dimensional, dissipative and non-autonomous systems. The theory provides analytical estimates of the ranges of parameters of the chaos-controlling excitation for suppression-enhancement of the initial chaos. The important applications of the theory presented in the text include: control of chaotic escape from a potential well; suppression of chaos in a driven Josephson junction; and control of chaotic solitons in Frenkel-Kontorova chains.
Author(s): Ricardo Chacon
Series: World Scientific Series on Nonlinear Science: Series a
Publisher: World Scientific Publishing Company
Year: 2006
Language: English
Pages: 223
2.5.2 Enhancement of chaos......Page 12
Preface......Page 8
1.1 Control of chaotic dynamical system......Page 16
1.2 Non-feedback control methods......Page 17
1.3.1 Robustness and flexibility......Page 18
1.4 Harmonic versus non-harmonic excitations: the waveform effect......Page 19
1.4.1 Reshaping-induced strange non-chaotic attractors......Page 21
1.4.2 Reshaping-induced crisis phenomena......Page 29
1.4.3 Reshaping-induced basin boundary fractality......Page 30
1.4.4 Reshaping-induced escape from a potential well......Page 31
1.4.5 Reshaping-induced control of directed transport......Page 35
1.4.6 Reshaping-induced control of synchronization of coupled limit-cycle oscillators......Page 41
1.5 Notes and references......Page 42
2.1 Dissipative systems versus Hamiltonian system......Page 46
2.2 Stability of perturbed limit cycles......Page 47
2.4 Basics of Melnikov’s method......Page 49
2.4.1 Illustration: A damped driven pendulum......Page 53
2.5.1 Suppression of chaos......Page 55
2.5.3 Case of non-subharmonic resonances......Page 75
2.5.4 The special case of the main resonance......Page 83
2.6 The generic Melnikov function: The noise effect......Page 95
2.6.1 Additive noise......Page 96
2.6.2 Multiplicative noise......Page 99
2.7 Notes and references......Page 100
3.1.1 Motivation......Page 106
3.1.2 Geometrical resonance......Page 107
3.1.3 Autoresonance......Page 109
3.1.4 Stochastic resonance......Page 117
3.2.1 Geometrical resonance in a damped pendulum subjected to p e riodic pulses......Page 121
3.2.2 Geometrical resonance in an overdamped bistable system......Page 125
3.2.3 Geometrical resonance approach to control of chaos by weak periodic perturbations......Page 128
3.2.4 Geometrical resonance and globally stable limit cycle in the van der Pol oscillator......Page 131
3.2.5 Geometrical resonance in spatio-temporal systems......Page 134
3.3 Notes and references......Page 136
4.1 Control of chaotic escape from a potential well......Page 140
4.1.1 Model equations......Page 141
4.1.2 Escape suppression theorems......Page 143
4.1.3 Inhibition of the erosion of non-escaping basins......Page 147
4.1.4 Role of nonlinear dissipation......Page 148
4.1.5 Robustness of chaotic escape control......Page 151
4.1.6 Case of incommensurate escapesuppressing excitations......Page 154
4.2.1 Model equation......Page 159
4.2.2 Suppression of homoclinic bifurcations......Page 160
4.2.3 Comparison withLyapunovexponent calculations......Page 166
4.3.1 The three wave case......Page 174
4.3.2 Case of a general electrostatic wave packet......Page 182
4.4 Notes and references......Page 193
5.1.1 Localized control of spatio-temporal chaos......Page 196
5.1.2 Application to chaotic solitons in Frenkel-Kontorova chains......Page 199
5.2 Controlling chaos in partial differential equations......Page 205
5.2.1 Damped sineGordon equation additively driven by two spatiG temporal periodic fields......Page 206
5.2.2 Damped sineGordon equation additively and parametrically driven by two spatio-temporal periodic fields......Page 210
5.2.3 Damped sineGordon equation additively driven by two tem- poral periodic excitations......Page 213
5.2.4 Nonlinear Schrodinger equation subjected to dissipative and spatially periodic perturbations......Page 217
5.2.5 4 model additively driven by two spatic-temporal periodic fields......Page 219
5.2.6 4 model additively and parametrically driven by two spatio-temporal periodic fields......Page 222
5.3 Notes and references......Page 225
6.1.2 Reshaping-induced control......Page 228
Case of symmetric pulses......Page 229
6.2.1 Ratchet systems......Page 231
6.2.2 Coupled Bose-Einstein condensates......Page 233
6.3 Notes and references......Page 234
INDEX......Page 236
