Qualitative Theory of Dynamical Systems

, Volume 12, Issue 1, pp 67–87

Bifurcations in Hamiltonian Systems with a Reflecting Symmetry

Open AccessArticle

DOI: 10.1007/s12346-012-0075-z

Cite this article as:
Bosschaert, M. & Hanßmann, H. Qual. Theory Dyn. Syst. (2013) 12: 67. doi:10.1007/s12346-012-0075-z

Abstract

A reflecting symmetry \({q \mapsto -q}\) of a Hamiltonian system does not leave the symplectic structure \({{\rm d}q \wedge {\rm d}p}\) invariant and is therefore usually associated with a reversible Hamiltonian system. However, if \({q \mapsto -q}\) leads to \({H \mapsto -H}\) , then the equations of motion are invariant under the reflection. Such a symmetry imposes strong restrictions on equilibria with q = 0. We study the possible bifurcations triggered by a zero eigenvalue and describe the simplest bifurcation triggered by non-zero eigenvalues on the imaginary axis.

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© The Author(s) 2012

Authors and Affiliations

  1. 1.Mathematisch Instituut, Universiteit UtrechtUtrechtThe Netherlands