Synchronization in complex networks of nonlinear dynamical systems

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 co...

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Bibliographic Details
Main Author Wu, Chai Wah
Format eBook
LanguageEnglish
Published Singapore World Scientific Publishing Co. Pte. Ltd 2007
World Scientific Publishing Company
World Scientific Publishing
Edition1
Subjects
Differentiable dynamical systems
Nonlinear theories
SCIENCE
Synchronization
System analysis
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Table of Contents:
  • Synchronization in complex networks of nonlinear dynamical systems -- Preface -- Contents -- Chapter 1: Introduction -- Chapter 2: Graphs, Networks, Laplacian Matrices and Algebraic Connectivity -- Chapter 3: Graph Models -- Chapter 4: Synchronization in Networks of Nonlinear Continuous-time Dynamical Systems -- Chapter 5: Synchronization in Networks of Coupled Discrete-time Systems -- Chapter 6: Synchronization in Network of Systems with Linear Dynamics -- Chapter 7: Agreement and Consensus Problems in Groups of Interacting Agents -- Appendix A: Algebraic Connectivity and Combinatorial Properties of a Graph -- Bibliography -- Index
  • Intro -- Contents -- Preface -- 1. Introduction -- 2. Graphs, Networks, Laplacian Matrices and Algebraic Connectivity -- 2.1 Graphs and digraphs -- 2.2 Matrices and graphs -- 2.3 Algebraic connectivity -- 2.3.1 Undirected graphs -- 2.3.2 Directed graphs -- 2.3.3 Basic properties of a1 and b1 -- 2.3.4 Basic properties of a2 -- 2.4 Locally connected graphs -- 2.4.1 Basic properties of a3 and a4 -- 2.5 Examples -- 2.6 Hypergraphs -- 2.7 Further reading -- 3. Graph Models -- 3.1 Examples of complex networks -- 3.1.1 Neural networks -- 3.1.2 Transportation networks -- 3.1.3 World Wide Web -- 3.1.4 Erdos number and scientific collaboration network -- 3.1.5 Film actor network -- 3.1.6 Citation network -- 3.2 Classical random graph models -- 3.2.1 Algebraic connectivity of random graphs -- 3.3 Small-world networks -- 3.3.1 Watts-Strogatz model -- 3.3.2 Newman-Watts model -- 3.3.3 Algebraic connectivity of small-world networks -- 3.4 Scale-free networks -- 3.5 Random geometric graphs -- 3.6 Graphs with a prescribed degree sequence -- 3.7 Algebraic connectivity and degree sequence -- 3.7.1 Regular graphs -- 3.7.1.1 Construction 1: graph with low 2 and r -- 3.7.1.2 Construction 2: graph with high 2 and r -- 3.7.2 Graphs with prescribed degree sequence -- 3.7.2.1 Construction 1: graph with low 2 and r -- 3.7.2.2 Construction 2: graph with high 2 and r -- 3.8 Further reading -- 4. Synchronization in Networks of Nonlinear Continuous-time Dynamical Systems -- 4.1 Static coupling topology -- 4.1.1 Properties of (G) -- 4.1.2 Computing (G) -- 4.1.3 Zero row sums matrices -- 4.1.4 Matrices in W -- 4.1.5 Synchronization and algebraic connectivity -- 4.2 Coupling topology with a spanning directed tree -- 4.3 Time-varying coupling topology -- 4.4 Coupling between delayed state variables -- 4.4.1 Choosing the factorization B1B2 = UG V D
  • 4.4.2 Choosing the matrix U 2 Ws -- 4.4.3 Choosing the matrix K -- 4.4.4 The case D = 0 -- 4.5 Synchronization criteria based on algebraic connectivity -- 4.6 Further reading -- 5. Synchronization in Networks of Coupled Discrete-time Systems -- 5.1 Synchronization of coupled scalar maps via contractivity of operators -- 6. Synchronization in Network of Systems with Linear Dynamics -- 6.1 Autonomous coupling -- 6.2 Nonautonomous coupling: continuous-time case -- 6.2.1 Slowly varying coupling -- 6.3 Nonautonomous coupling: discrete-time case -- 6.3.1 A discrete-time consensus problem -- 6.4 Ergodicity of inhomogeneous Markov chains -- 6.5 Contractive matrices -- 6.5.1 Pseudocontractivity and scrambling stochastic matrices -- 6.5.2 Set-nonexpansive and set-contractive operators -- 6.5.3 Set-contractivity under the max-norm -- 6.5.4 Set-contractivity under the Euclidean norm -- 6.5.5 Set-contractivity under a weighted Euclidean norm -- 6.5.6 Set-contractivity and coefficient of ergodicity -- 6.6 Further reading -- 7. Agreement and Consensus Problems in Groups of Interacting Agents -- 7.1 Continuous-time models -- 7.1.1 Rate of exponential convergence -- 7.1.2 Dynamic coupling topology -- 7.2 Discrete-time models -- 7.2.1 Follow the leader dynamics and leadership in coordinated agents -- 7.3 A nonlinear model for consensus -- 7.4 Agreement in random networks -- 7.4.1 Synchronization in random networks without the scrambling condition -- 7.5 Further reading -- Appendix A Algebraic Connectivity and Combinatorial Properties of a Graph -- A.1 Properties of algebraic connectivity -- A.2 Vertex and edge connectivity -- A.3 Graph partitions -- A.3.1 Maximum directed cut -- A.3.2 Edge-forwarding index -- A.3.3 Bisection width -- A.3.4 Isoperimetric number -- A.3.5 Minimum ratio cut -- A.3.6 Independence number -- A.4 Semibalanced graphs -- Bibliography -- Index