Contents
Preface
Chapter 1 Basic geometrical point of view of dynamical systems 1
1.1 Self-excited and hidden attractors 1
1.2 Hidden oscillations in Hilbert's 16th problem and applied models 3
1.3 The main contents of this book 6
Reference 9
Chapter 2 Hidden attractors without equilibria 12
2.1 Hidden chaos without equilibria in three-dimensional autonomous system 12
2.1.1 The proposed system 13
2.1.2 Forming mechanism of the new chaotic attractors 17
2.1.3 Conclusion 22
2.2 Hidden hyperchaos without equilibria in four-dimensional autonomous system 23
2.2.1 The hyperchaotic system from generalized di.usionless Lorenz equations 25
2.2.2 Dynamical structure of the proposed hyperchaotic system 29
2.3 Conclusion 35
Reference 35
Chapter 3 Hidden hyperchaotic attractors in a modi-ed Lorenz-Stenflo system 39
3.1 Introduction 39
3.2 The hyperchaotic system from Lorenz-Stenflo system 40
3.2.1 Formulation of the system 40
3.2.2 Hidden hyperchaotic attractors with only one stable equilibrium 42
3.2.3 Non-equivalence with existing hyperchaotic systems 45
3.3 Some basic properties and bifurcation analysis 45
3.3.1 Symmetry and invariance and dissipativity 45
3.3.2 Equilibria and stability 46
3.3.3 Bifurcation analysis 48
3.4 The ultimate bound and positively invariant set 52
3.4.1 Four dimensional hyperelliptic estimate of the ultimate bound and positively invariant set 52
3.4.2 Two dimensional cylindrical estimate of the ultimate bound and positively invariant set 55
3.5 Conclusion 57
Reference 60
Chapter 4 Hidden attractors, multiple limit cycles and boundedness in the generalized 4D Rabinovich system 63
4.1 Introduction 63
4.2 The proposed system and hidden hyperchaos 65
4.2.1 Formulation of the system 65
4.2.2 Hidden hyperchaotic attractors with a unique stable equilibrium 66
4.2.3 Initial conditions and coexisting attractors 69
4.3 Generation of hidden attractors 70
4.4 Local bifurcation in the generalized hyperchaotic Rabinovich system 71
4.4.1 Equilibrium and stability 71
4.4.2 Hopf bifurcation 72
4.5 Boundedness of motion for the hyperchaotic system 76
4.6 Conclusion 79
Reference 80
Chapter 5 On the periodic orbit bifurcating from one single non-hyperbolic equibrium 84
5.1 Introduction 84
5.2 The proposed system and chaotic attractors 85
5.3 The averaging theory for periodic orbits 88
5.4 Statements of the main results 89
5.5 Conclusion 95
Reference 95
Chapter 6 Hidden attractors and dynamical behaviors in an extended Rikitake system 99
6.1 Introduction 99
6.2 Existence of equilibria 100
6.3 Hidden attractors that arise from stable equilibria 102
6.3.1 Coexistence of stable equilibria and hidden attractor 103
6.3.2 Finding hidden attractors by a simple linear transformation 104
6.4 Hopf bifurcation analysis 106
6.5 Dynamics analysis at in-nity 110
6.6 Conclusion 114
Reference 115
Chapter 7 Hidden chaotic regions and complex dynamics in 3D homopolar disc dynamo 118
7.1 Introduction 118
7.2 Description of the self-exciting homopolar disc dynamo and related problems 120
7.3 Study of hidden attractors from a simple linear transformation 125
7.4 Study of hidden attractors from Hopf bifurcation 127
7.4.1 An outline of the Hopf bifurcation methods 127
7.4.2 Hopf bifurcation analysis 129
7.4.3 Hidden attractors and numerical simulations 131
7.4.4 Unstable periodic orbits 131
7.5 Existence of homoclinic orbits 134
7.6 In-nity dynamics by Poincar.e compacti-cation 137
7.7 Conclusion 143
Reference 144
Chapter 8 Hidden hyperchaos and circuit application in 5D homopolar disc dynamo 147
8.1 Introduction 147
8.2 5D hyperchaotic self-exciting homopolar disc dynamo 148
8.3 Hidden attractors and multistability 152
8.3.1 Hidden attractors with two stable equilibria 153
8.3.2 Coexistence of point, periodic, quasi-periodic and hidden chaotic attractors 156
8.4 Electronic circuit implementation of the 5D hyperchaotic system 161
8.5 Conclusion 163
Reference 164
Chapter 9 Bifurcation and circuit realization for delayed system with hidden attractors 168
9.1 Introduction 168
9.2 Hopf bifurcation analysis with multiple delays 170
9.2.1 Stability of equilibrium 171
9.2.2 Existence of Hopf bifurcation with 72
9.2.3 Existence of Hopf bifurcation with 75
9.3 Direction, stability and numerical results of Hopf bifurcation with 77
9.3.1 Direction of Hopf bifurcations and stability of the bifurcating periodic orbits 177
9.3.2 Numerical simulations 186
9.4 Circuit implementation of the multiple time-delay system 190
9.5 Conclusion 192
Reference 193
Index 196