Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient
type: Item
instance of: scholarly article
title: Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient
zbMATH Open document ID: 1228.34058
author: Peter E. Kloeden
published in: Communications on Pure and Applied Analysis
publication date: 2005-03-07T00:00:00Z
review text: Consider the vector differential equation \[ {dx\over dt}= A(\varepsilon)x+ b(t) H_m(x,t), \] where \(A(0)\) has the simple eigenvalue \(\mu(0)= 0\) and all other eigenvalues have negative real parts, \(b\) is an almost periodic function which is strictly positive or strictly negative, \(H_m\) is homogeneous in \(x\) of order \(m\geq 2\), \(\varepsilon\) is the bifurcation parameter. The author shows that \(x\equiv 0\) undergoes a transcritical or pitchfork bifurcation to a small nontrivial almost periodic solution when the eigenvalue \(\mu(\varepsilon)\) crosses the imaginary axis, he also determines its stability.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 37G10
zbMATH DE Number: 2141499
MaRDI profile type: Publication
full work available at URL: cpaa.2004.3.161
OpenAlex ID: W2030500054
DOI: 10.3934/CPAA.2004.3.161
description: scientific article; zbMATH DE number 2141499