Control of homoclinic chaos by weak periodic perturbations
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-...
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| Hlavní autor: | |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
New Jersey
World Scientific Publishing Co. Pte. Ltd
2005
World Scientific World Scientific Publishing Company WORLD SCIENTIFIC World Scientific Publishing |
| Vydání: | 1 |
| Edice: | World Scientific series on nonlinear science. Series A, Monographs and treatises |
| Témata: | |
| ISBN: | 9789812380425, 9812380426, 9789812703514, 9812703519 |
| On-line přístup: | Získat plný text |
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- Control of homoclinic chaos by weak periodic perturbations -- Preface -- Contents -- Chapter 1: Introduction -- Chapter 2: Theoretical approach -- Chapter 3: Physical mechanisms -- Chapter 4: Applications: low-dimensional systems -- Chapter 5: Applications: high-dimensional systems -- Chapter 6: Further remarks and open problems -- Index
- 3.2.5 Geometrical resonance in spatio-temporal systems -- 3.3 Notes and references -- 4 Applications: Low-dimensional systems -- 4.1 Control of chaotic escape from a potential well -- 4.1.1 Model equations -- 4.1.2 Escape suppression theorems -- 4.1.3 Inhibition of the erosion of non-escaping basins -- 4.1.4 Role of nonlinear dissipation -- 4.1.5 Robustness of chaotic escape control -- 4.1.6 Case of incommensurate escapesuppressing excitations -- 4.2 Taming chaos in a driven Josephson junction -- 4.2.1 Model equation -- 4.2.2 Suppression of homoclinic bifurcations -- 4.2.3 Comparison withLyapunovexponent calculations -- 4.3 Suppression of chaos of charged particles in an electrostatic wave packet159 4.3 -- 4.3.1 The three wave case -- 4.3.2 Case of a general electrostatic wave packet -- 4.4 Notes and references -- 5 Applications: High-dimensional systems -- 5.1 Controlling chaos in chaotic coupled oscillators -- 5.1.1 Localized control of spatio-temporal chaos -- 5.1.2 Application to chaotic solitons in Frenkel-Kontorova chains -- 5.2 Controlling chaos in partial differential equations -- 5.2.1 Damped sineGordon equation additively driven by two spatiG temporal periodic fields -- 5.2.2 Damped sineGordon equation additively and parametrically driven by two spatio-temporal periodic fields -- 5.2.3 Damped sineGordon equation additively driven by two tem- poral periodic excitations -- 5.2.4 Nonlinear Schrodinger equation subjected to dissipative and spatially periodic perturbations -- 5.2.5 4 model additively driven by two spatic-temporal periodic fields -- 5.2.6 4 model additively and parametrically driven by two spatio-temporal periodic fields -- 5.3 Notes and references -- 6 Further Remarks and Open Problems -- 6.1 Openproblems -- 6.1.1 Beyond the main resonance -- 6.1.2 Reshaping-induced control -- 6.1.3 Amplitude modulation control
- Case of symmetric pulses -- 6.2 Further applications -- 6.2.1 Ratchet systems -- 6.2.2 Coupled Bose-Einstein condensates -- 6.3 Notes and references -- INDEX
- Intro -- CONTENTS -- Preface -- 1 Introduction -- 1.1 Control of chaotic dynamical system -- 1.2 Non-feedback control methods -- 1.3 Controlling chaos by weak periodic excitations -- 1.3.1 Robustness and flexibility -- 1.3.2 Applicability and scope -- 1.4 Harmonic versus non-harmonic excitations: the waveform effect -- 1.4.1 Reshaping-induced strange non-chaotic attractors -- 1.4.2 Reshaping-induced crisis phenomena -- 1.4.3 Reshaping-induced basin boundary fractality -- 1.4.4 Reshaping-induced escape from a potential well -- 1.4.5 Reshaping-induced control of directed transport -- 1.4.6 Reshaping-induced control of synchronization of coupled limit-cycle oscillators -- 1.5 Notes and references -- 2 Theoretical Approach -- 2.1 Dissipative systems versus Hamiltonian system -- 2.2 Stability of perturbed limit cycles -- 2.3 Non-autonomous second-order differential systems -- 2.4 Basics of Melnikov's method -- 2.4.1 Illustration: A damped driven pendulum -- 2.5 The generic Melnikov function: Deterministic case -- 2.5.1 Suppression of chaos -- 2.5.2 Enhancement of chaos -- 2.5.3 Case of non-subharmonic resonances -- 2.5.4 The special case of the main resonance -- 2.6 The generic Melnikov function: The noise effect -- 2.6.1 Additive noise -- 2.6.2 Multiplicative noise -- 2.7 Notes and references -- 3 Physical Mechanisms -- 3.1 Energy-based approach -- 3.1.1 Motivation -- 3.1.2 Geometrical resonance -- 3.1.3 Autoresonance -- 3.1.4 Stochastic resonance -- 3.2 Geometrical resonance analysis: Chaos, stability and control -- 3.2.1 Geometrical resonance in a damped pendulum subjected to p e riodic pulses -- 3.2.2 Geometrical resonance in an overdamped bistable system -- 3.2.3 Geometrical resonance approach to control of chaos by weak periodic perturbations -- 3.2.4 Geometrical resonance and globally stable limit cycle in the van der Pol oscillator