Synchronization in Complex Networks of Nonlinear Dynamical Systems

✍ Scribed by Chai Wah Wu

• Publisher: World Scientific Publishing Company
• Year: 2007
• Tongue: English
• Leaves: 168
• Category: Library

✦ Synopsis

This book brings together two emerging research areas: synchronization in coupled nonlinear systems and complex networks, and study conditions under which a complex network of dynamical systems synchronizes. While there are many texts that study synchronization in chaotic systems or properties of complex networks, there are few texts that consider the intersection of these two very active and interdisciplinary research areas. The main theme of this book is that synchronization conditions can be related to graph theoretical properties of the underlying coupling topology. The book introduces ideas from systems theory, linear algebra and graph theory and the synergy between them that are necessary to derive synchronization conditions. Many of the results, which have been obtained fairly recently and have until now not appeared in textbook form, are presented with complete proofs. This text is suitable for graduate-level study or for researchers who would like to be better acquainted with the latest research in this area.

✦ Table of Contents

Contents......Page 10
Preface......Page 8
1. Introduction......Page 14
2.1 Graphs and digraphs......Page 18
2.2 Matrices and graphs......Page 21
2.3.1 Undirected graphs......Page 31
2.3.2 Directed graphs......Page 32
2.3.3 Basic properties of a1 and b1......Page 34
2.3.4 Basic properties of a2......Page 37
2.4 Locally connected graphs......Page 38
2.5 Examples......Page 40
2.6 Hypergraphs......Page 41
2.7 Further reading......Page 42
3.1.1 Neural networks......Page 44
3.1.2 Transportation networks......Page 45
3.1.5 Film actor network......Page 46
3.1.6 Citation network......Page 47
3.2.1 Algebraic connectivity of random graphs......Page 48
3.3 Small-world networks......Page 50
3.3.3 Algebraic connectivity of small-world networks......Page 51
3.4 Scale-free networks......Page 52
3.5 Random geometric graphs......Page 53
3.6 Graphs with a prescribed degree sequence......Page 54
3.7.1.1 Construction 1: graph with low 2 and r......Page 58
3.7.2 Graphs with prescribed degree sequence......Page 60
3.7.2.1 Construction 1: graph with low 2 and r......Page 61
3.8 Further reading......Page 62
4. Synchronization in Networks of Nonlinear Continuous-time Dynamical Systems......Page 64
4.1 Static coupling topology......Page 66
4.1.1 Properties of (G)......Page 67
4.1.2 Computing (G)......Page 72
4.1.3 Zero row sums matrices......Page 73
4.1.4 Matrices in W......Page 74
4.2 Coupling topology with a spanning directed tree......Page 75
4.3 Time-varying coupling topology......Page 79
4.4 Coupling between delayed state variables......Page 80
4.4.1 Choosing the factorization B1B2 = UG V D......Page 82
4.4.3 Choosing the matrix K......Page 83
4.4.4 The case D = 0......Page 84
4.5 Synchronization criteria based on algebraic connectivity......Page 85
4.6 Further reading......Page 87
5. Synchronization in Networks of Coupled Discrete-time Systems......Page 88
5.1 Synchronization of coupled scalar maps via contractivity of operators......Page 94
6.1 Autonomous coupling......Page 96
6.2 Nonautonomous coupling: continuous-time case......Page 97
6.2.1 Slowly varying coupling......Page 98
6.3.1 A discrete-time consensus problem......Page 99
6.4 Ergodicity of inhomogeneous Markov chains......Page 100
6.5 Contractive matrices......Page 109
6.5.1 Pseudocontractivity and scrambling stochastic matrices......Page 111
6.5.2 Set-nonexpansive and set-contractive operators......Page 112
6.5.3 Set-contractivity under the max-norm......Page 116
6.5.5 Set-contractivity under a weighted Euclidean norm......Page 117
6.5.6 Set-contractivity and coefficient of ergodicity......Page 121
6.6 Further reading......Page 122
7.1 Continuous-time models......Page 124
7.1.2 Dynamic coupling topology......Page 126
7.2 Discrete-time models......Page 127
7.2.1 Follow the leader dynamics and leadership in coordinated agents......Page 131
7.3 A nonlinear model for consensus......Page 132
7.4 Agreement in random networks......Page 133
7.4.1 Synchronization in random networks without the scrambling condition......Page 135
7.5 Further reading......Page 136
A.1 Properties of algebraic connectivity......Page 138
A.3.1 Maximum directed cut......Page 148
A.3.3 Bisection width......Page 149
A.3.4 Isoperimetric number......Page 150
A.3.6 Independence number......Page 151
A.4 Semibalanced graphs......Page 152
Bibliography......Page 156
Index......Page 166