An elementary approach to a generalized Hopf bifurcation
type: Item
instance of: scholarly article
title: An elementary approach to a generalized Hopf bifurcation
zbMATH Open document ID: 0791.34033
DOI: 10.1007/BF00942765
author: Franz Spirig
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1994-01-19T00:00:00Z
review text: For a system of autonomous differential equations undergoing a Hopf bifurcation the bifurcation equation is derived. The zeros of the bifurcation equation correspond to periodic solutions of the differential equations. The reduction is performed by the Lyapunov-Schmidt method and by expanding the bifurcation equation into Fourier series. The coefficients are obtained by integrating a recursive set of differential equations, which is stated only symbolically. Finally the well known stability condition for the periodic solution in terms of the derivative of the bifurcation equation is stated and the procedure is compared shortly with the calculations using Center manifolds.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 488211
zbMATH Keywords: center manifolds
zbMATH Keywords: system of autonomous differential equations
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: periodic solutions
zbMATH Keywords: Lyapunov-Schmidt method
zbMATH Keywords: Fourier series
zbMATH Keywords: stability condition
reviewed by: Alois Steindl
MaRDI profile type: Publication
cites work: Q3948194
cites work: Existence of small periodic solutions of ordinary differential equations in \(\mathbb{R}^2\)
cites work: Bifurcation formulas for O.D.E.'S in n
cites work: Bifurcation equation for planar systems of differential equations
cites work: Q5182723
description: scientific article; zbMATH DE number 488211