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.

Autorentext

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.

Inhalt

Preface

1 What is multistability

1. 1.1 Mathematical basis

2. 1.1.1 Main definitions

3. 1.1.2 Attractors and basins of attraction

4. 1.1.3 Smooth and fractal basins

5. 1.1.4 Wada basins

6. 1.1.5 Riddled basins

7. 1.2 Stability of invariant sets

8. 1.2.1 Lyapunov stability

9. 1.2.2 Asymptotic stability

10. 1.2.3 Exponential stability

11. 1.2.4 Orbital stability

12. 1.2.5 Structural stability

13. 1.2.6 Linear stability analysis

14. 1.2.7 Local Lyapunov exponents

15. 1.3 Basin stability

16. 1.3.1 Resilience

17. 1.3.2 Integral stability

18. 1.3.3 Final state sensitivity

19. 1.3.4 Survivability

20. 1.3.5 Basin catastrophe

21. 1.3.6 Basin integrity

22. 1.4 Complexity

23. 1.4.1 Basin entropy

24. 1.4.2 Spectral entropy

25. 1.4.3 Sample entropy

26. Emergence of multistability 40

27. 2.1 Bifurcations giving rise to multistability 40

28. 2.1.1 Pitchfork bifurcation 41

29. 2.1.2 Saddle-node bifurcation 41

30. 2.1.3 Andronov-Hopf bifurcation 44

31. 2.1.4 Neimark-Sacker bifurcation 45

32. 2.1.5 Multiple limit cycle bifurcation 46

33. 2.1.6 Infinite period bifurcation 47

34. 2.1.7 Inverse gluing bifurcation 48

35. 2.1.8 Symmetry-increasing bifurcation 49

36. 2.2 Mechanisms leading to multistability 50

37. 2.2.1 Homoclinic tangencies 51

38. 2.2.2 Weak dissipation 52

39. 2.2.3 Clustering 55

40. 2.2.4 Phase multistability 57

41. 2.2.5 Positive feedback 62

42. 2.2.6 Delayed feedback 63

43. 2.2.7 Periodic forcing 66

44. 2.2.8 Symmetry 68

45. 2.2.9 Structural multistability 70

46. 2.3 Methods to reveal multistability 73

47. 2.3.1 Varying initial conditions 73

48. 2.3.2 Continuation method 74

49. 2.3.3 External short pulse 75

50. 2.3.4 Stochastic perturbation 76

51. 2.3.5 Critical velocity surfaces 79

52. 2.3.6 Method of complete bifurcation group 82

53. 2.3.7 Quantifying basins of attraction 85

54. 2.3.8 Attractor identification using a final-state machine 86

55. 2.4 Hidden attractors 88

56. 2.4.1 Homotopy and continuation methods 89

57. 2.4.2 Amplitude control 93

58. 2.4.3 Offset boosting 95

59. 2.4.4 Nested double-scroll attractors 96

60. Manifestation of multistability in different systems 100

    3.1 Multistability in simple dynamical systems 100

61. 3.1.1 Hénon map 100

    2. 3.1.2 Dissipative standard nontwist map 101

62. 3.1.3 Duffing oscillator 103

63. 3.1.4 Rössler-like oscillator 106

64. 3.1.5 Lorenz-like system 107

65. 3.1.6 Chua system 109

66. 3.1.7 Jerk systems 110

    3.2 Multistability in coupled systems 111

67. 3.2.1 Coupled quadratic maps 112

68. 3.2.2 Coupled Hénon maps 112

69. 3.2.3 Coupled Duffing oscillators 115

70. 3.2.4 Coupled Rössler oscillators 119

    5. 3.2.5 Coupled Lorenz oscillators 120

71. 3.3 Multistability in neuronal systems 121

72. 3.3.1 Single neuron 122

73. 3.3.2 Coupled neurons 126

74. 3.4 Other examples of multistable systems 128

75. 3.4.1 Mechanical systems 129

76. 3.4.2 Micro- and nanosystems 130

77. 3.4.3 Thermochemical systems 134

78. 3.4.4 Climate 135

79. 3.4.5 Ecology 140

80. 3.4.6 Biosystems 144

81. 3.4.7 Astrosystems 147

82. 3.4.8 Fractional-order systems 150

    4 Multistability in lasers 152

83. 4.1 Laser as a nonlinear dynamical system 152

84. 4.1.1 History of laser dynamics 152

85. 4.1.2 Dynamical classification of lasers 155

86. 4.2 Types of multistability in optical systems 157

87. 4.2.1 Optical bistability 157

88. 4.2.2 Spatial multistability 160

89. 4.2.3 Polarization multistability 161

90. 4.3 Multistability in CO2 lasers 164

91. 4.3.1 Loss-modulated CO2 laser 164

92. 4.3.2 Targeting attractors by short pulses 166

93. 4.3.3 Bistability induced by resonant perturbations 168

94. 4.3.4 Bistability induced by a delayed feedback 169

95. 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

96. 4.5.1 Loss-modulated fiber laser 177

97. 4.5.2 Pump-modulated fiber laser 178

    5 Multistate intermittency 182

98. 5.1 Noise-induced escapes from equilibria 184

99. 5.1.1 Multiple quasipotential 184

100. 5.1.2 Escape from a fixed point with smooth basin boundaries 186

101. 5.1.3 Escape from a chaotic attractor with a fractal basin boundary 188

102. 5.2 Coherence resonance in multistable systems 192

103. 5.2.1 Stochastic resonance in multistable systems 192

104. 5.2.2 Deterministic coherence resonance in a chaotic bistable system 195

105. 5.2.3 Logical stochastic resonance 196

106. 5.3 Characterization of noise-induced multistate intermittency 201

107. 5.3.1 Mean residence times 203

108. 5.3.2 Structural properties of noise-induced multistate intermittency 206

109. 5.3.3 Entropy measures of multistate intermittency 207

110. 5.3.4 Wavelet transform method for detection of coexisting regimes 209

     4. 5.4 Manifestation of noise-induced multistate intermittency 212

111. 5.4.1 Stochastic bistable Chua system 212

112. 5.4.2 Anti-coherence resonance in a bistable neural network 213

113. 5.4.3 Multistate intermittency in semiconductor lasers 215

114. 5.5 Noise-induced preference of attractors 219

115. 5.5.1 Attractor probability density and optimal basin size 220

116. 5.5.2 Preference of attractors in laser experiments 223

117. 5.5.3 Extreme events in a multistable system 227

118. 5.5.4 Preference of attractors in a network of coupled oscillators 229

119. 5.6 Multistate intermittency in deterministic systems 230

120. 5.6.1 Multistate intermittency induced by periodic forcing 231

121. 5.6.2 Modulational intermittency in a semiconductor laser 232

122. Multistability in complex networks 235

     6.1 Stability of multistable complex networks 235

     6.1.1…