Double flip bifurcations in $\mathbb{Z}/2\mathbb{Z}$-symmetric Hamiltonian systems

Abstract

In this paper we introduce a new bifurcation in Hamiltonian systems, which we call the double flip bifurcation. The Hamiltonian depends on two parameters, one of which controls the double flip bifurcation. The result of the bifurcation is the occurrence of two Hamiltonian flip bifurcations with respect to the other parameter. The two Hamiltonian flip bifurcations are simultaneous with respect to the first parameter, and are connected by a curve-segment of singular points. We find a normal form for Hamiltonians describing systems going through double flip bifurcations, and compute said normal form for some examples.