, Volume 12, Issue 1, pp 67-87,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 22 May 2012

Bifurcations in Hamiltonian Systems with a Reflecting Symmetry

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.

Dedicated to Ken Meyer at the occasion of his 75th birthday.