This book starts with an introduction to the basic concepts of multistability, then illustrates how multistability arises in different systems and explains the main mechanisms of multistability emergence. A special attention is given to noise which can convert a multistable deterministic system to a monostable stochastic one. Furthermore, the most important applications of multistability in different areas of science, engineering and technology are given attention throughout the book, including electronic circuits, lasers, secure communication, and human perception. The book aims to provide a first approach to multistability for readers, who are interested in understanding its fundamental concepts and applications in several fields. This book will be useful not only to researchers and engineers focusing on interdisciplinary studies, but also to graduate students and technicians. Both theoreticians and experimentalists will rely on it, in fields ranging from mathematics and laser physics to neuroscience and astronomy. The book is intended to fill a gap in the literature, to stimulate new discussions and bring some fundamental issues to a deeper level of understanding of the mechanisms underlying self-organization of matter and world complexity.

Auteur

Alexander N. Pisarchik is Distinguished Researcher at the Center for Biomedical Technologies of the Universidad Politécnica de Madrid, Spain, and Senior Researcher at the Innopolis University, Russia. His research interests include nonlinear and stochastic dynamics, chaos, synchronization, complex networks, chaotic cryptography, cognitive neuroscience and brain-computer interfaces. Alexander E. Hramov is Head of the Laboratory of Neuroscience and Cognitive Technologies and Full Professor at the Innopolis University, Russia, Leading Researcher at the Center for Neurotechnology and Machine Learning of the Immanuel Kant Baltic Federal University, Saratov State Medical University and Lobachevsky University of Niznij Novgorod. Scientific activity of Prof. Hramov is devoted to nonlinear dynamics, theory of complex systems, artificial intelligence and its application to the analysis and modeling of neuronal brain activity.

Contenu

Preface

1 What is multistability

1.1 Mathematical basis
1.1.1 Main definitions
1.1.2 Attractors and basins of attraction
1.1.3 Smooth and fractal basins
1.1.4 Wada basins
1.1.5 Riddled basins
1.2 Stability of invariant sets
1.2.1 Lyapunov stability
1.2.2 Asymptotic stability
1.2.3 Exponential stability
1.2.4 Orbital stability
1.2.5 Structural stability
1.2.6 Linear stability analysis
1.2.7 Local Lyapunov exponents
1.3 Basin stability
1.3.1 Resilience
1.3.2 Integral stability
1.3.3 Final state sensitivity
1.3.4 Survivability
1.3.5 Basin catastrophe
1.3.6 Basin integrity
1.4 Complexity
1.4.1 Basin entropy
1.4.2 Spectral entropy
1.4.3 Sample entropy
Emergence of multistability 40
2.1 Bifurcations giving rise to multistability 40
2.1.1 Pitchfork bifurcation 41
2.1.2 Saddle-node bifurcation 41
2.1.3 Andronov-Hopf bifurcation 44
2.1.4 Neimark-Sacker bifurcation 45
2.1.5 Multiple limit cycle bifurcation 46
2.1.6 Infinite period bifurcation 47
2.1.7 Inverse gluing bifurcation 48
2.1.8 Symmetry-increasing bifurcation 49
2.2 Mechanisms leading to multistability 50
2.2.1 Homoclinic tangencies 51
2.2.2 Weak dissipation 52
2.2.3 Clustering 55
2.2.4 Phase multistability 57
2.2.5 Positive feedback 62
2.2.6 Delayed feedback 63
2.2.7 Periodic forcing 66
2.2.8 Symmetry 68
2.2.9 Structural multistability 70
2.3 Methods to reveal multistability 73
2.3.1 Varying initial conditions 73
2.3.2 Continuation method 74
2.3.3 External short pulse 75
2.3.4 Stochastic perturbation 76
2.3.5 Critical velocity surfaces 79
2.3.6 Method of complete bifurcation group 82
2.3.7 Quantifying basins of attraction 85
2.3.8 Attractor identification using a final-state machine 86
2.4 Hidden attractors 88
2.4.1 Homotopy and continuation methods 89
2.4.2 Amplitude control 93
2.4.3 Offset boosting 95
2.4.4 Nested double-scroll attractors 96
Manifestation of multistability in different systems 100

3.1 Multistability in simple dynamical systems 100

3.1.1 Hénon map 100
1. 3.1.2 Dissipative standard nontwist map 101
3.1.3 Duffing oscillator 103
3.1.4 Rössler-like oscillator 106
3.1.5 Lorenz-like system 107
3.1.6 Chua system 109
3.1.7 Jerk systems 110

3.2 Multistability in coupled systems 111

3.2.1 Coupled quadratic maps 112
3.2.2 Coupled Hénon maps 112
3.2.3 Coupled Duffing oscillators 115
3.2.4 Coupled Rössler oscillators 119
1. 3.2.5 Coupled Lorenz oscillators 120
3.3 Multistability in neuronal systems 121
3.3.1 Single neuron 122
3.3.2 Coupled neurons 126
3.4 Other examples of multistable systems 128
3.4.1 Mechanical systems 129
3.4.2 Micro- and nanosystems 130
3.4.3 Thermochemical systems 134
3.4.4 Climate 135
3.4.5 Ecology 140
3.4.6 Biosystems 144
3.4.7 Astrosystems 147
3.4.8 Fractional-order systems 150

4 Multistability in lasers 152

4.1 Laser as a nonlinear dynamical system 152
4.1.1 History of laser dynamics 152
4.1.2 Dynamical classification of lasers 155
4.2 Types of multistability in optical systems 157
4.2.1 Optical bistability 157
4.2.2 Spatial multistability 160
4.2.3 Polarization multistability 161
4.3 Multistability in CO2 lasers 164
4.3.1 Loss-modulated CO2 laser 164
4.3.2 Targeting attractors by short pulses 166
4.3.3 Bistability induced by resonant perturbations 168
4.3.4 Bistability induced by a delayed feedback 169
4.4 Multistability in semiconductor lasers 171

4.4.1 Multistability in a semiconductor lasers with delayed feedback 171

4.4.2 Multistability in a directly modulated semiconductor lasers 173

4.5 Multistability in fiber lasers 175

4.5.1 Loss-modulated fiber laser 177
4.5.2 Pump-modulated fiber laser 178

5 Multistate intermittency 182

5.1 Noise-induced escapes from equilibria 184
5.1.1 Multiple quasipotential 184
5.1.2 Escape from a fixed point with smooth basin boundaries 186
5.1.3 Escape from a chaotic attractor with a fractal basin boundary 188
5.2 Coherence resonance in multistable systems 192
5.2.1 Stochastic resonance in multistable systems 192
5.2.2 Deterministic coherence resonance in a chaotic bistable system 195
5.2.3 Logical stochastic resonance 196
5.3 Characterization of noise-induced multistate intermittency 201
5.3.1 Mean residence times 203
5.3.2 Structural properties of noise-induced multistate intermittency 206
5.3.3 Entropy measures of multistate intermittency 207
5.3.4 Wavelet transform method for detection of coexisting regimes 209
1. 5.4 Manifestation of noise-induced multistate intermittency 212
5.4.1 Stochastic bistable Chua system 212
5.4.2 Anti-coherence resonance in a bistable neural network 213
5.4.3 Multistate intermittency in semiconductor lasers 215
5.5 Noise-induced preference of attractors 219
5.5.1 Attractor probability density and optimal basin size 220
5.5.2 Preference of attractors in laser experiments 223
5.5.3 Extreme events in a multistable system 227
5.5.4 Preference of attractors in a network of coupled oscillators 229
5.6 Multistate intermittency in deterministic systems 230
5.6.1 Multistate intermittency induced by periodic forcing 231
5.6.2 Modulational intermittency in a se…