Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays
type: Item
instance of: scholarly article
title: Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays
zbMATH Open document ID: 1250.34058
author: Xiaochen Mao
published in: Nonlinear Dynamics
publication date: 2012-07-17T00:00:00Z
review text: The author presents a stability analysis of a trivial steady state solution in a network with delayed coupling which consists of two identical bidirectional rings of three cells with elementwise, symmetric crosscoupling. The crosscoupling delay \(\tau\) is assumed different from the ring-coupling delay \(\sigma\). The model equations are \[ \dot{x}_i (t) = -x_i(t) + af(x_{i-1}(t-\sigma)) + af(x_{i+1}(t-\sigma)) + bg(y_i(t-\tau)), \] \[ \dot{y}_i (t) = -y_i(t) + af(y_{i-1}(t-\sigma)) + af(y_{i+1}(t-\sigma)) + bg(x_i(t-\tau)) \] for \(i=1,2,3\). The linearized system is studied. Some sufficient conditions for the asymptotic stability of the steady state are given and the occurence of purely imaginary eigenvalues, i.e., Hopf bifurcations, is investigated. Numerical illustration is shown in a final chapter.
reviewed by: Leonhard Lücken
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K21
Mathematics Subject Classification ID: 34D06
Mathematics Subject Classification ID: 34K18
zbMATH DE Number: 6057985
zbMATH Keywords: time delay
zbMATH Keywords: coupled networks
zbMATH Keywords: synchronization
MaRDI profile type: Publication
full work available at URL: s11071-011-0211-3
OpenAlex ID: W2008803110
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DOI: 10.1007/S11071-011-0211-3
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description: scientific article; zbMATH DE number 6057985