Stable phase-locked periodic solutions in a delay differential system.
type: Item
instance of: scholarly article
title: Stable phase-locked periodic solutions in a delay differential system.
zbMATH Open document ID: 1044.34024
DOI: 10.1016/S0022-0396(03)00187-6
author: Jianhong Wu
published in: Journal of Differential Equations
publication date: 2003-11-19T00:00:00Z
review text: The subject of the paper is the following system with delay \[ \dot x_i(t) = - \mu x_i(t) +a f(x_i(t-\tau)) + b[f(x_{i-1}(t-\tau)) + f(x_{i+1}(t-\tau))], \quad i(\text{mod}) \; 3, \] where \(x_i \in \mathbb{R}\), \(f: \mathbb{R}\to \mathbb{R}\), \(\mu>0\), \(b>0\), and \(a<0\). This system appears in the theory of neural networks. The nonlinearity \(f\) is assumed to be close to the step function. The author obtains existence, stability, and the limiting profile of phase-locked periodic solutions \(P(t)=(p_1(t),p_2(t),p_3(t))\) with period \(T\), which have the following symmetry properties: \(p_1(t)=p_2(t+T/3)\), \(p_2(t)=p_3(t+T/3)\), \(p_3(t)=p_1(t+T/3)\).
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 92B20
zbMATH DE Number: 2005879
zbMATH Keywords: periodic solutions
zbMATH Keywords: stability
zbMATH Keywords: phase-locking
zbMATH Keywords: delay
zbMATH Keywords: neural networks
zbMATH Keywords: rotating wave
reviewed by: S. Yanchuk
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2005879