Hopf bifurcation from infinity
type: Item
instance of: scholarly article
title: Hopf bifurcation from infinity
zbMATH Open document ID: 0644.34037
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108079
review text: The following result is proved: ``Let \(\{\pi_{\mu}\}_{\mu}\) be a one- parameter continuous family of flows on X, a locally compact metric space. If \(\pi_ 0\) is negatively ultimately bounded, and for \(\mu >0\pi_{\mu}\) is ultimately bounded, then there is a family of compact, invariant, asymptotically stable sets \(M_{\mu}\) bifurcating from infinity'', If \(X={\mathbb{R}}^ 2 \)and all the fixed points of \(\pi_{\mu}\) are contained in a disk independent of \(\mu\), then \(M_{\mu}\) contains a cycle, for \(\mu\) positive. The above result is applied to prove the existence of periodic solutions bifurcating from infinity for second order differential equations of generalized Liénard and Rayleigh type.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 4050050
zbMATH Keywords: second order differential equations
zbMATH Keywords: Liénard
zbMATH Keywords: Rayleigh
author: Marco Sabatini
reviewed by: Marco Sabatini
MaRDI profile type: Publication
cites work: Prolongations and stability in dynamical systems
cites work: Q5614853
cites work: Boundedness theorems for some second order differential equations I
cites work: Q4134266
cites work: Q3848373
cites work: Q4180923
cites work: On the generalized Lienard equation with negative damping
cites work: An introduction to infinite dimensional dynamical systems -- geometric theory. With an appendix by Krzysztof P. Rybakowski
cites work: Q4135884
cites work: Liapunov direct method in approaching bifurcation problems
cites work: Q3240935
cites work: Q5520199
description: scientific article; zbMATH DE number 4050050