Classification and unfoldings of \(1:2\) resonant Hopf bifurcation
type: Item
instance of: scholarly article
title: Classification and unfoldings of \(1:2\) resonant Hopf bifurcation
zbMATH Open document ID: 0879.34041
DOI: 10.1007/BF02206623
author: W. Langford
author: Victor G. LeBlanc
published in: Archive for Rational Mechanics and Analysis
publication date: 1998-01-29T00:00:00Z
review text: The paper concerns the interaction of two Hopf bifurcations in the case of 1:2 resonance, i.e. the bifurcation of periodic solutions from a stationary solution of an autonomous differential equation whose linearization has eigenvalues \(\pm i\) and \(\pm 2i\) and no other eigenvalues of the form \(ki\), where \(k\) is an integer. No other restrictions on the equation (such as requiring it to be Hamiltonian or time-reversible or equivariant) are assumed. Working in spaces of periodic functions and using the Lyapunov-Schmidt reduction, only periodic solutions (near the stationary one) with period near \(2\pi\) are investigated (and not such with larger minimal period or homoclinic or heteroclinic solutions), but for those solutions the results are rigorous and complete. Under a specific non-degeneracy condition, the generic cases of equations with no distinguished control parameter and such with one distinguished bifurcation parameter are completely classified and unfolded.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 58E09
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
zbMATH DE Number: 969892
zbMATH Keywords: Hopf-Hopf mode interaction
zbMATH Keywords: strong resonance
zbMATH Keywords: bifurcation of periodic solutions from a stationary solution
zbMATH Keywords: autonomous differential equation
zbMATH Keywords: Lyapunov-Schmidt reduction
MaRDI profile type: Publication
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description: scientific article