Patterns in hierarchical networks of neuronal oscillators with \(\mathbb D_3\times \mathbb Z_3\) symmetry
type: Item
instance of: scholarly article
title: Patterns in hierarchical networks of neuronal oscillators with \(\mathbb D_3\times \mathbb Z_3\) symmetry
zbMATH Open document ID: 1279.34085
author: Shangjiang Guo
author: Juanjuan Man
published in: Journal of Differential Equations
publication date: 2013-04-17T00:00:00Z
review text: The authors study a hierarchical delay-coupled network with a \(\mathbb{D}_3\times\mathbb{Z}_3\)-symmetry. Its dynamics is given by \[ \dot{x}_{(i,j)}(t) = - x_{(i,j)}(t) + af(x_{(i,j-1)}(t-\tau)) + bg(x_{(i+1,j)}(t-\tau)) + bg(x_{(i-1,j)}(t-\tau)), \] where all indices are considered modulo 3 and the sigmoid functions \(f\) and \(g\) are smooth. This system can be considered as a bidirectional ring consisting of three unidirectional rings. Explicit conditions for linear stability of the trivial solution and for its bifurcations are given. The codimension-one transcritical and Hopf bifurcations and the symmetries of the emerging states are classified. Further, codimension-two bifurcations are studied via center manifold reductions and possible generalizations are discussed.
reviewed by: Leonhard Lücken
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 34K19
zbMATH DE Number: 6154869
zbMATH Keywords: bifurcation
zbMATH Keywords: oscillator network
zbMATH Keywords: pattern generation
zbMATH Keywords: hierarchical system
zbMATH Keywords: delayed coupling
MaRDI profile type: Publication
full work available at URL: j.jde.2013.01.027
OpenAlex ID: W1968044585
DOI: 10.1016/J.JDE.2013.01.027
description: scientific article; zbMATH DE number 6154869