TY - JOUR

T1 - Normal forms for 4D symplectic maps with twist singularities

AU - Dullin, H.R.

AU - Ivanov, A.V.

AU - Meiss, J.D.

PY - 2006

Y1 - 2006

N2 - 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-ang

AB - 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-ang

M3 - Article

VL - 215

SP - 175

EP - 190

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 2

ER -