Preservation and destruction of periodic orbits by symplectic integrators

• Published in: Numerical algorithms Vol. 53; no. 2-3; pp. 343 - 362
• Main Authors: McLachlan, Robert I., O’Neale, Dion R. J.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2010; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computer Science; Degrees of freedom; Hamiltonian functions; Invariants; Mathematical analysis; Mathematical models; Nonlinearity; Numeric Computing; Numerical Analysis; Orbits; Original Paper; Theory of Computation; Toruses; Hénon-Heiles; Periodic orbit; Lower dimensional invariant tori; Symplectic integrators; KAM theory; Geometric numerical integration
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-009-9352-6

We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997 ) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.