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Normal forms for 4D symplectic maps with twist singularities
- Source :
- Physica D: Nonlinear Phenomena. 215:175-190
- Publication Year :
- 2006
- Publisher :
- Elsevier BV, 2006.
-
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.
- Subjects :
- Cusp (singularity)
Hamiltonian mechanics
010102 general mathematics
Invariant manifold
Mathematical analysis
Statistical and Nonlinear Physics
Condensed Matter Physics
01 natural sciences
010305 fluids & plasmas
symbols.namesake
Singularity
0103 physical sciences
symbols
Gravitational singularity
0101 mathematics
Hamiltonian (quantum mechanics)
Symplectomorphism
Mathematics
Symplectic geometry
Subjects
Details
- ISSN :
- 01672789
- Volume :
- 215
- Database :
- OpenAIRE
- Journal :
- Physica D: Nonlinear Phenomena
- Accession number :
- edsair.doi...........0dddcb5c9adcc72ad18a32ed3de5aa96