Normal forms for 4D symplectic maps with twist singularities

Abstract

We derive a normal form for a near-integrable, four-dimensional (4D) symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time- T mapping of a two-degree-of-freedom Hamiltonian flow. Consequently, there is an energy-like invariant. The fold Hamiltonian is similar to the well-studied one-degree-of-freedom case, but is essentially non-integrable when the direction of the singular curve in action does not coincide with curves of the resonance module. We show that many familiar features, such as multiple island chains and reconnecting invariant manifolds, are retained even in this case. The cusp Hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module. Using averaging, we approximately reduce this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.