A note on the calculation of paths of Hopf bifurcations
type: Item
instance of: scholarly article
title: A note on the calculation of paths of Hopf bifurcations
zbMATH Open document ID: 0684.65057
DOI: 10.1016/0377-0427(89)90151-9
author: A. D. Jepson
author: Alastair Spence
author: K. A. Cliffe
published in: Journal of Computational and Applied Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A two-parameter nonlinear system is considered. The classical extended system for Hopf bifurcation points is analyzed. Namely, the system is shown to be \(Z_ 2\)-equivariant. Under certain nondegeneracy conditions, a Bogdanov-Takens bifurcation point is shown to be a symmetry-breaking bifurcation point (pitchfork) of this extended system. Moreover, the symmetric and asymmetric branches parametrize locally the folds and Hopf bifurcation points, respectively, in a neighbourhood of the above pitchfork. The \(Z_ 2\)-equivariance reflects the conjugacy of complex eigenvalues. Numerical applications are hinted.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65H17
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 4121322
zbMATH Keywords: approximation of branches
zbMATH Keywords: Banach space
zbMATH Keywords: two-parameter nonlinear system
zbMATH Keywords: extended system
zbMATH Keywords: Hopf bifurcation points
zbMATH Keywords: Bogdanov-Takens bifurcation point
zbMATH Keywords: symmetry-breaking bifurcation point
zbMATH Keywords: pitchfork
zbMATH Keywords: complex eigenvalues
MaRDI profile type: Publication
full work available at URL: 0377-0427(89)90151-9
OpenAlex ID: W2042905181
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description: scientific article; zbMATH DE number 4121322