Frontiers in the Study of Chaotic Dynamical Systems With Open Problems (World Scientific Series on Nonlinear Science, Series B)

Contents......Page 8
Preface......Page 6
1. Problems with Lorenz’s Modeling and the Algorithm of Chaos Doctrine S. OuYang and Y. Lin......Page 10
1.1. Introduction......Page 11
1.2. Lorenz’s Modeling and Problems of the Model......Page 13
1.3. Computational Schemes and What Lorenz’s Chaos Is......Page 20
1.4. Discussion......Page 32
1.5. Appendix: Another Way to Show that Chaos Theory Su ers From Flaws......Page 35
References......Page 38
2.1. Introduction......Page 40
2.2. Open Problems About Nonexistence of Chaotic Solutions......Page 41
References......Page 49
3. Some Open Problems in the Dynamics of Quadratic and Higher Degree Polynomial ODE Systems F. Zhang and J. Heidel......Page 52
3.1. First Open Problem......Page 53
3.2. Second Open Problem......Page 56
3.3. Third Open Problem......Page 57
3.4. Fourth Open Problem......Page 59
3.6. Sixth Open Problem......Page 60
References......Page 65
4. On Chaotic and Hyperchaotic Complex Nonlinear Dynamical Systems G. M. Mahmoud......Page 68
4.1. Introduction......Page 69
4.2.1. Dynamical Properties of Chaotic Complex Chen Sys- tem......Page 75
4.2.2. Hyperchaotic Complex Lorenz Systems......Page 80
4.2.2.1. Autonomous Hyperchaotic Complex Lorenz Systems......Page 81
4.2.2.2. Non-Autonomous Hyperchaotic Complex Lorenz System......Page 85
4.3. Open Problems......Page 87
4.4. Conclusions......Page 88
References......Page 89
5. On the Study of Chaotic Systems with Non-Horseshoe Template A. Ray, A. Roy Chowdhury and S. Basak......Page 94
5.1 Introduction......Page 95
5.2. Formulation......Page 96
5.3. Topological Analysis and Its Invariants......Page 99
5.4.1. Search for Close Return......Page 101
5.4.2. Topological Constant......Page 104
5.4.3. Template Identification......Page 107
5.5. Conclusion and Discussion......Page 109
References......Page 111
6.1. Introduction......Page 114
6.2. Open Problems......Page 117
References......Page 124
7. Some Conjectures About the Synchronizability and the Topology of Networks A. Caneco, J. Leonel Rocha, C. Gr´acio and S. Fernandes......Page 126
7.1. Introduction......Page 127
7.2. Related and Historical Problems About Network Syn- chronizability......Page 130
7.3. Some Physical Examples About the Real Applications of Network Synchronizability......Page 133
7.4. Preliminaries......Page 135
7.5. Complete Clustered Networks......Page 137
7.5.1. Clustering Point on Complete Clustered Networks......Page 138
7.5.2. Classification of the Clustering and the Amplitude of the Synchronization Interval......Page 142
7.5.3. Discussion......Page 145
7.6. Symbolic Dynamics and Networks Synchronization......Page 147
References......Page 153
8.1. Definitions and State of Art......Page 158
8.2. Open Problems in the Continuous Wavelet Transform and a Topology of Bounding Tori......Page 160
8.3. The Evaluation of the Continuous Wavelet Transform Using Partial Di erential Equations in Non-Cartesian Co-ordinates and Multidimensional Case......Page 161
References......Page 162
9.1. Introduction......Page 164
9.2. Basic Definitions and Notations......Page 167
9.3. Statement of the Problems......Page 170
9.3.1. Dynamic Parrondo’s Paradox and Commuting Functions......Page 172
9.3.2. Dynamics Shared by Commuting Functions......Page 175
9.3.3. Computing Problems for Large Periods T......Page 177
9.3.4. Commutativity Problems......Page 178
9.3.5. Generalization to Continuous Triangular Maps on the Square......Page 180
References......Page 181
10.1. Introduction......Page 184
10.2. Iterated Function System......Page 185
10.3. Attractor of Pair of Linear Maps......Page 187
10.4. Code Structure of Pair of Linear Maps......Page 188
10.5. Sufficient Conditions for Computing the Code Structure......Page 194
References......Page 202
11.1. Introduction......Page 204
11.2. Closed Dynamical Systems......Page 205
11.3. Open Dynamical Systems......Page 208
11.4. Open Billiards......Page 213
11.5. Physical Applications......Page 219
11.6. Discussion......Page 222
Acknowledgements......Page 223
References......Page 224
12.1. Introduction......Page 228
12.2. Attractors for Flows and Di eomorphisms......Page 229
12.3.1. A First Attempt......Page 231
12.3.2. Examples......Page 233
12.4. Experimental Information......Page 236
12.6. Conclusions......Page 237
References......Page 238
13. How to Transform a Type of Chaos in Dynamical Systems? E. Zeraoulia and J. C. Sprott......Page 240
13.2. Hyperbolification of Dynamical Systems......Page 241
13.3. Transforming Dynamical Systems to Lorenz-Type Chaos......Page 250
13.4. Transforming Dynamical Systems to Quasi-Attractor Systems......Page 253
13.5. A Common Classification of Strange Attractors of Dynamical Systems......Page 255
References......Page 256
Author Index......Page 262
Subject Index......Page 264