Normal forms for 4D symplectic maps with twist singularities

Abstract

We derive a normal form for a near-integrable, four-dimensional 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 nonintegrable 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 reduced this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.

Untitled

“…Our results for the close-to-integrable case generalize the results in [22,23]. In [22] an approach similar to that taken here is used to study the fold singularity in the standard non-twist family.…”

“…The parameters of the Hamiltonian correspond to characteristics of the system that can be tuned by the designer to obtain the desired effect of the system. For example [2,23,24], in the design of plasma confinement devices, it has been heuristically argued that nontwist invariant tori are very efficient barriers for the undesired effect of transport. Similar considerations have appeared in mixing of fluids [20,21,59].…”

“…Our methodology is based on the theory developed in parts 1 and 2 of this monograph and gives a robust way to construct bifurcation diagrams of KAM tori. It generalizes the BNF procedure used in [23] and the available persistence results obtained using the classical Parametric KAM Theory.…”

“…In [22] an approach similar to that taken here is used to study the fold singularity in the standard non-twist family. In [23], using different techniques, the fold and the cusp were studied for the case of two degrees of freedom.…”

Resonances and Twist in Volume-Preserving Mappings

“…Indeed, introducing the lifted coordinate s ∈ R by η = αs, then this system has the Hamiltonian H(η, ζ) = 1 2 ζ 2 + V (s) , where V is a quasi-periodic function of s with derivative DV (s) = −ḡ(α 1 s, α 2 s). This model also occurs as a normal form for symplectic maps near a twistless singularity in the frequency map, see [DIM06].…”