Preface
1 What is multistability
1. 1.1 Mathematical basis
1. 1.1.1 Main definitions
2. 1.1.2 Attractors and basins of attraction
3. 1.1.3 Smooth and fractal basins
4. 1.1.4 Wada basins
5. 1.1.5 Riddled basins
2. 1.2 Stability of invariant sets
1. 1.2.1 Lyapunov stability
2. 1.2.2 Asymptotic stability
3. 1.2.3 Exponential stability
4. 1.2.4 Orbital stability
5. 1.2.5 Structural stability
6. 1.2.6 Linear stability analysis
7. 1.2.7 Local Lyapunov exponents
3. 1.3 Basin stability
1. 1.3.1 Resilience
2. 1.3.2 Integral stability
3. 1.3.3 Final state sensitivity
4. 1.3.4 Survivability
5. 1.3.5 Basin catastrophe
6. 1.3.6 Basin integrity
4. 1.4 Complexity
1. 1.4.1 Basin entropy
2. 1.4.2 Spectral entropy
3. 1.4.3 Sample entropy
2. Emergence of multistability 40
1. 2.1 Bifurcations giving rise to multistability 40
1. 2.1.1 Pitchfork bifurcation 41
2. 2.1.2 Saddle-node bifurcation 41
3. 2.1.3 Andronov-Hopf bifurcation 44
4. 2.1.4 Neimark-Sacker bifurcation 45
5. 2.1.5 Multiple limit cycle bifurcation 46
6. 2.1.6 Infinite period bifurcation 47
7. 2.1.7 Inverse gluing bifurcation 48
8. 2.1.8 Symmetry-increasing bifurcation 49
2. 2.2 Mechanisms leading to multistability 50
1. 2.2.1 Homoclinic tangencies 51
2. 2.2.2 Weak dissipation 52
3. 2.2.3 Clustering 55
4. 2.2.4 Phase multistability 57
5. 2.2.5 Positive feedback 62
6. 2.2.6 Delayed feedback 63
7. 2.2.7 Periodic forcing 66
8. 2.2.8 Symmetry 68
9. 2.2.9 Structural multistability 70
3. 2.3 Methods to reveal multistability 73
1. 2.3.1 Varying initial conditions 73
2. 2.3.2 Continuation method 74
3. 2.3.3 External short pulse 75
4. 2.3.4 Stochastic perturbation 76
5. 2.3.5 Critical velocity surfaces 79
6. 2.3.6 Method of complete bifurcation group 82
7
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