Obtaining certificates for complete synchronisation of coupled oscillators
type: Item
instance of: scholarly article
title: Obtaining certificates for complete synchronisation of coupled oscillators
zbMATH Open document ID: 1218.34057
author: Elias August
author: Mauricio Barahona
published in: Physica D
publication date: 2011-03-29T00:00:00Z
full work available at URL: http://hdl.handle.net/10044/1/10411
review text: A linearly coupled system of identical oscillators has the form \[ \frac{dx}{dt} = F(x) + \kappa (C\otimes D)x, \] where \(x=[x_1,\dots,x_N]^T\), \(x_j\in\mathbb{R}^n\), \(F(x)=[f(x_1),\dots,f(x_N)]\) \(f: \mathbb{R}^n \to \mathbb{R}^n\), \(C\) and \(D\) are \(N\times N\) and \(n\times n\) coupling matrices, respectively. Previously, sufficient conditions have been obtained for the global synchronization of this system, i.e. for the stability of the invariant subspace \(S = \{x: x_1=\dots=x_N\}\). These sufficient conditions are based on finding some quadratic form, which plays the role of the Lyapunov function and guarantees the global stability of the subspace \(S\). In this paper, the authors suggest a numerical method for finding and optimizing such a quadratic form using the semidefinite programming technique. Moreover, the authors also establish new conditions for synchronization using Bendixson's criterion for higher-dimensional systems. The results are illustrated on Lorenz and van der Pol systems.
reviewed by: S. Yanchuk
Mathematics Subject Classification ID: 34D06
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 90C22
zbMATH DE Number: 5871267
zbMATH Keywords: Lyapunov theory
zbMATH Keywords: contraction theory
zbMATH Keywords: Bendixson's criterion for higher dimensions
zbMATH Keywords: semidefinite programming
zbMATH Keywords: sum of squares decomposition
zbMATH Keywords: synchronisation of coupled oscillators
describes a project that uses: SeDuMi
describes a project that uses: Sostools
describes a project that uses: YALMIP
MaRDI profile type: Publication
OpenAlex ID: W1984924768
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DOI: 10.1016/J.PHYSD.2010.12.008
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description: scientific article; zbMATH DE number 5871267