Synchronization in directionally coupled systems: Some rigorous results
type: Item
instance of: scholarly article
title: Synchronization in directionally coupled systems: Some rigorous results
zbMATH Open document ID: 0999.37006
author: Valentin Afraimovich
author: A. Cordonet
author: Jean-René Chazottes
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2002-12-05T00:00:00Z
review text: This paper concerns dynamical systems \((x,y)\in \mathbb{R}^m\times \mathbb{R}^n\mapsto (x',y')=F(x,y) \in\mathbb{R}^m \times\mathbb{R}^n\), which are generated by unidirectional coupling of the type \(x'=f(x)\), \(y'=g(x,y)\), where \(f\) is a homeomorphism and \(g\) is continuous. It is assumed that there exists a ball of dissipation \(B\subset\mathbb{R}^m\times\mathbb{R}^n\) and that master-slave synchronization occures in \(B\). Let \({\mathcal A}=\cap^\infty_{\ell =0}F^\ell(B)\) be the maximal attractor in \(B\), and denote by \(P_1:\mathbb{R}^m \times\mathbb{R}^n \to \mathbb{R}^m\) and \(P_2:\mathbb{R}^m \times\mathbb{R}^n \to\mathbb{R}^n\) the canonical projections. It is proved that there exists a map \(h:P_1({\mathcal A}) \to\mathbb{R}^n\) with \({\mathcal A}=\text{graph}(h)\), \(h\circ P_1\circ F^\ell= P_2\circ h\circ F^\ell\) and that \(h\) is Hölder continuous or even Lipschitz continuous (depending on the coupling strength). In a second part the authors prove almost everywhere synchronization for coupling of two identical maps of the interval that are neither continuous nor invertible.
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37B05
zbMATH DE Number: 1736541
zbMATH Keywords: Hölder continuous synchronization function
zbMATH Keywords: fractal dimensions
zbMATH Keywords: master-slave synchronization
zbMATH Keywords: almost everywhere synchronization
reviewed by: Lutz Recke
MaRDI profile type: Publication
full work available at URL: dcdsb.2001.1.421
OpenAlex ID: W2077620381
DOI: 10.3934/DCDSB.2001.1.421
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description: scientific article; zbMATH DE number 1736541