Preservation and destruction of periodic orbits by symplectic integrators

Abstract  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.

• Content Type Journal Article
• Category Original Paper
• DOI 10.1007/s11075-009-9352-6
• Authors
  
  
  • Robert I. McLachlan, Massey University I.F.S. Palmerston North New Zealand
  • Dion R. J. O’Neale, Massey University I.F.S. Palmerston North New Zealand
  
  

• Journal Numerical Algorithms
• Online ISSN 1572-9265
• Print ISSN 1017-1398

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Original source : http://www.springerlink.com/content/r640254185184j...