Hopf bifurcation: the appearance of virtual periods in cases of resonance
type: Item
instance of: scholarly article
title: Hopf bifurcation: the appearance of virtual periods in cases of resonance
zbMATH Open document ID: 0616.58036
DOI: 10.1016/0022-0396(86)90081-1
author: Kathleen T. Alligood
author: James A. Yorke
published in: Journal of Differential Equations
publication date: 1986-01-01T00:00:00Z
review text: The paper is concerned with an extension of the Hopf theorem in certain critical cases. The main result proved is the following. Let (0,0) be an isolated centre of (1) \(dx/dt=f(x,\lambda)\), where \(f\in C^ 1(R^ n\times R,R^ n)\), and let \(i\beta\) be an eigenvalue of \(A=D_ xf(0,0)\). If the crossing number \(\chi(i\beta)\neq 0\) and if \(\pi/\beta\) is not a centre period of (0,0), then there exists a family Q of orbits of (1) such that Q bifurcates from (0,0) starting with virtual period \(2\pi/\beta\). Several examples are included.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 3998836
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: resonance
zbMATH Keywords: virtual periods
MaRDI profile type: Publication
full work available at URL: 0022-0396(86)90081-1
OpenAlex ID: W1992111712
cites work: Global Bifurcations of Periodic Orbits
cites work: Q3330179
cites work: Q4167627
cites work: Global Hopf bifurcation from a multiple eigenvalue
cites work: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems
cites work: Q5843611
cites work: Snakes: oriented families of periodic orbits, their sources, sinks, and continuation
cites work: Periodic orbits near an equilibrium and a theorem by Alan Weinstein
cites work: Q4070368
cites work: Periodic solutions near a resonant equilibrium of a Hamiltonian system
cites work: Normal modes for nonlinear Hamiltonian systems
Recommended article: Hopf bifurcations in resonance 2:1
Recommended article: Q4941050
Recommended article: Resonant parametric perturbations of the Hopf bifurcation
Recommended article: Hopf bifurcations of nonautonomous systems at resonance
Recommended article: Hopf bifurcation at 1:1 resonance
Recommended article: Hopf bifurcations in nonautonomous systems at points of resonance
Recommended article: Q3684447
Recommended article: Q3698515
Recommended article: Hopf bifurcation with non-semisimple 1:1 resonance
Recommended article: Parametric resonance of Hopf bifurcation
description: scientific article; zbMATH DE number 3998836