Preservation and destruction of periodic orbits by symplectic integrators
The paper is interested in what happens to periodic orbits and lower dimensional invariant tori of a Hamiltonian system when they are discretized by a symplectic integrator. The main question addressed is whether the numerical solution preserve the periodic orbits of the original vectorfield or not. The results are directly related with two cases studied in sections 2 and 3 of the paper. The first case considers non resonant step sizes, this is, when none of the integer multiples of the step size of the numerical integrator coincides with the period of the periodic orbit. In this case the periodic orbits generally persist but slightly perturbed. Similar to what happens with invariant tori in KAM theory, section 2 starts commenting a previous result of Shang where for full dimensional invariant tori the frequencies of the numerical algorithm are the same ones of the original system but multiplied by the step size. Then it continues with its goal of developing a similar result for the preservation of periodic orbits, and what the authors prove, is that under suitable hypothesis (essentially of analyticity, reducibility and isotropy) the result is the same as in the full dimensional case but with the set of frequencies shifted from the original ones. To prove this result, the main tool that the authors use is an embedding of the symplectic map produced by the numerical integrator into an analytic symplectic flow and a KAM result of \textit{Á. Jorba} and \textit{J. Villanueva} [J. Nonlinear Sci., 7, 427--473 (1997; Zbl 0898.58044)]. The resonant case is studied in section 3. Here it is assumed that the period of a periodic orbit divided by the step size of the numerical integrator is an integer (\(T/h=n\)). As it should be expected from the results of the previous section, in this case the periodic orbit is destroyed by the numerical integrator leaving \(n\) elliptic and \(n\) hyperbolic periodic points under the numerical map. This fact is clearly illustrated numerically considering some periodic orbits in the two degrees of freedom Hénon-Heiles system, but a full description of the numerical flow in the neighborhood of a periodic orbit under resonant symplectic discretization, and a proof of the behaviors conjectured in the paper remains to be undertaken.
scientific article; zbMATH DE number 5675069
scientific article