Analytical approximation for period-doubling following a Hopf bifurcation
type: Item
instance of: scholarly article
title: Analytical approximation for period-doubling following a Hopf bifurcation
zbMATH Open document ID: 0684.58030
DOI: 10.1016/0093-6413(89)90022-0
author: Richard Rand
published in: Mechanics Research Communications
publication date: 1989-01-01T00:00:00Z
review text: This work is concerned with the behavior of a system of n autonomous ordinary differential equations of the form \(\dot x=F(x,\mu)\), where x and F(x,\(\mu)\) are n-vectors, and where \(\mu\) is a scalar parameter. It frequently happens that a change in \(\mu\) causes an equilibrium point to change its stability resulting in the birth of a periodic motion called a limit cycle, a process known as a Hopf bifurcation. If the change in \(\mu\) is then continued, the limit cycle may itself undergo a change of form via a period-doubling bifurcation. This scenario is often the first step in a sequence of period-doublings leading to chaos. The goal of this work is to suggest an analytic approach which will yield an approximation for \(\mu^*\), the critical value of \(\mu\) corresponding to the first period-doubling bifurcation. The idea of the method is to use center manifold theory to approximate the newly-born limit cycle, and then to use that approximation to investigate the stability of the limit cycle. The critical value \(\mu^*\) corresponds to the change of stability of the limit cycle (and to the disintegration of the center manifold).
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 4121078
zbMATH Keywords: autonomous ordinary differential equations
zbMATH Keywords: stability
zbMATH Keywords: period-doubling bifurcation
zbMATH Keywords: center manifold
zbMATH Keywords: limit cycle
describes a project that uses: MACSYMA
MaRDI profile type: Publication
full work available at URL: 0093-6413(89)90022-0
OpenAlex ID: W2024209920
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description: scientific article; zbMATH DE number 4121078