Foundations of synergetics. II: Complex patterns
type: Item
instance of: scholarly article
title: Foundations of synergetics. II: Complex patterns
zbMATH Open document ID: 0729.58001
DOI: 10.1007/978-3-642-97294-2
author: Alexander S. Mikhailov
author: Alexander Loskutov
published in: Springer Series in Synergetics
publication date: 1992-09-18T00:00:00Z
review text: This textbook is the second of two volumes based on a lecture course in synergetics given at the University of Moscow. The first volume [the first author, I: Distributed active systems (1990; Zbl 0712.92001)] concerned deterministic coherent structures in active media, e.g. propagating wave fronts and pulses, spiral waves, and localized stationary dissipative patterns. The distinctive feature of a coherent pattern is the presence of long-range order in space and time. Chaotic patterns, treated in this second volume, are characterized by short-range order with respect to time and/or spatial coordinates. Chaotic behavior may be an autonomous property of an active system which does not require external random perturbations or it may be the result of selective amplification of fluctuations induced by weak external noise. Both processes are discussed extensively. In essence, volume 2 of Foundations of Synergetics is a didactic review which links the nonlinear nondeterministic ``new math'' of the last 25 years to classical studies in dynamics and noise theory. The authors present such topics as strange attractors, fractals, bifurcations, the Fokker-Planck equation, Brownian motion, population growth and extinction, and catastrophe theory in the context of chaos. Even today the relationships among these and kindred subjects are far from clear, but this commendable book represents one of the few efforts to unite them to the extent that it is possible to do so. Chapters 1-7 give a concise introduction to chaotic dynamics in small systems. Chapters 8-12 treat noise-induced complex patterns. The first chapter is introductory. A chapter-by-chapter summary of the rest of the book follows. Chapter 2. Unpredictable dynamics. The phase-space representation of Hamiltonian (non-dissipative) systems is reviewed. Ergodicity is a necessary but not sufficient condition for chaotic motion. Mixing, in which the accessible volume of phase space ejects long irregular unbroken tentacles, is t…
Mathematics Subject Classification ID: 92-02
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 92C20
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 82C31
zbMATH DE Number: 50431
zbMATH Keywords: strange attractors
zbMATH Keywords: fractals
zbMATH Keywords: bifurcations
zbMATH Keywords: Fokker-Planck equation
zbMATH Keywords: Brownian motion
zbMATH Keywords: population growth
zbMATH Keywords: extinction
zbMATH Keywords: catastrophe theory
zbMATH Keywords: chaotic dynamics
zbMATH Keywords: noise-induced complex patterns
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-97294-2
OpenAlex ID: W4249508414
description: scientific article; zbMATH DE number 50431