Stability of singular Hopf bifurcations
type: Item
instance of: scholarly article
title: Stability of singular Hopf bifurcations
zbMATH Open document ID: 1144.37437
author: Yang Lijun
author: Zeng Xianwu
published in: Journal of Differential Equations
publication date: 2004-11-18T00:00:00Z
review text: Consider the singularly perturbed system \[ \frac{dx}{dt}=\varepsilon f(x,y,\lambda),\quad\frac{dy}{dt}= g(x,y,\lambda)\tag{*} \] with \(x\in\mathbb{R}^n\), \(y\in\mathbb{R}^m\), \(\varepsilon,\lambda\in\mathbb{R}\), \(\varepsilon\) small. It is assumed that there exists a curve \(\lambda=\lambda(\varepsilon)\) in the parameter plane associated with the occurence of Hopf bifurcation. The goal of the authors is to derive an explicit formula for the so-called Lyapunov constant \(\sigma\) whose sign determines the stability of the bifurcating small limit cycle. The derivation of this formula is based on a reduction technique. It requires that \(f\) and \(g\) are sufficiently smooth and that the corresponding Jacobi matrix satisfies some spectral conditions.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 2116217
zbMATH Keywords: singular Hopf bifurcation
zbMATH Keywords: singularly perturbed system
zbMATH Keywords: Lyapunov constant
zbMATH Keywords: stability constant
zbMATH Keywords: stability formula
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: j.jde.2004.08.002
OpenAlex ID: W1996838494
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DOI: 10.1016/J.JDE.2004.08.002
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description: scientific article; zbMATH DE number 2116217