Dynamics and bifurcation in networks : theory and applications of coupled differential equations / Martin Golubitsky, The Ohio State University, Ian Stewart, University of Warwick.

Format:

Book

Author/Creator:

Golubitsky, Martin, 1945- author.

Stewart, Ian, 1945- author.

Contributor:

Society for Industrial and Applied Mathematics, publisher.

Series:

Other titles in applied mathematics.

Other titles in applied mathematics

Language:

English

Subjects (All):

Differential equations.

Differential equations--Qualitative theory.

Bifurcation theory.

Physical Description:

1 PDF (xxxii, 834 pages).

Place of Publication:

Philadelphia, Pennsylvania : Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), [2023]

System Details:

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader.

Summary:

In recent years, there has been an explosion of interest in network-based modeling in many branches of science. This book synthesizes some of the common features of many such models, providing a general framework analogous to the modern theory of nonlinear dynamical systems. How networks lead to behavior not typical in a general dynamical system and how the architecture and symmetry of the network influence this behavior are the book's main themes. Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations is the first book to describe the formalism for network dynamics developed over the past 20 years. In it, the authors introduce a definition of a network and the associated class of "admissible" ordinary differential equations, in terms of a directed graph whose nodes represent component dynamical systems and whose arrows represent couplings between these systems; develop connections between network architecture and the typical dynamics and bifurcations of these equations; and discuss applications of this formalism to various areas of science, including gene regulatory networks, animal locomotion, decision-making, homeostasis, binocular rivalry, and visual illusions.

Contents:

Why networks?

Examples of network models

Network constraints on bifurcations

Inhomogeneous networks

Homeostasis

Local bifurcations for inhomogeneous networks

Informal overview

Synchrony, phase relations, balance, and quotient networks

Formal theory of networks

Formal theory of balance and quotients

Adjacency matrices

ODE-equivalence

Lattices of colorings

Rigid equilibrium theorem

Rigid periodic states

Symmetric networks

Spatial and spatiotemporal patterns

Synchrony-breaking steady-state bifurcations

Nonlinear structural degeneracy

Synchrony-breaking Hopf bifurcation

Hopf bifurcation in network chains

Graph fibrations and quiver representations

Binocular rivalry and visual illusions

Decision making

Signal propagation in feedforward lifts

Lattices, rings, and group networks

Balanced colorings of lattices

Symmetries of lattices and their quotients

Heteroclinic cycles, chaos, and chimeras

Epilogue

Appendix A. Liapunov-Schmidt reduction

Appendix B. Center manifold reduction

Appendix C. Perron-Frobenius theorem

Appendix D. Differential equations on infinite networks.

Notes:

Description based on title page of print version.

Includes bibliographical references (pages 773-812) and index.

ISBN:

1-61197-733-9

Publisher Number:

OT185 SIAM