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.