Feed-forward networks, center manifolds, and forcing
type: Item
instance of: scholarly article
title: Feed-forward networks, center manifolds, and forcing
zbMATH Open document ID: 1257.34025
author: Martin Golubitsky
author: Claire Postlethwaite
published in: Discrete and Continuous Dynamical Systems
publication date: 2012-07-04T00:00:00Z
reviewed by: Alois Steindl
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6053022
zbMATH Keywords: center manifolds
zbMATH Keywords: feed-forward networks
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: periodic excitation
zbMATH Keywords: generalized Duffing equation
MaRDI profile type: Publication
full work available at URL: dcds.2012.32.2913
OpenAlex ID: W2002972958
DOI: 10.3934/DCDS.2012.32.2913
review text: The authors investigate the dynamics of a feed-forward network consisting of three cells in the vicinity of a Hopf bifurcation. By calculating the center manifold, they can show that the reduced bifurcation equations inherit the feed-forward structure from the original system and decouple into the normal form equations for a Hopf bifurcation and a periodically excited Duffing type equation close to primary resonance, where the excitation is introduced by the solution of the Hopf subsystem. The combined system has the remarkable feature that there is a branch of periodic solution, which grows like \(\lambda^{1/6}\).NEWLINENEWLINEIn the second part, the authors consider a resonant external excitation of their system and find an exciting variety of bifurcation diagrams: for certain combinations of parameters it is possible that several closed branches separate from the well-known shape of the bifurcation diagram of the Duffing equation.
description: scientific article; zbMATH DE number 6053022