A single step integrator for an ordinary differential equation may be considered as a high frequency perturbation of the original system. An averaging method may be applied to check that there is a coordinate change, wich eleminate this time-dependance up to an exponentially small remainder. In general this remainder is not zero.
Applying this idea to an autonomous Hamiltonian system with 1 degree of freedom, we see that the truncated system is also autonomos, and it is $h^n$ close to the original system, where $h$ stends for the integrator step and $n$ is the order of the integrator. In this case the autonomous systems have a quite simple dynamics: every trajectory is either periodic, (be)asymptotic to periodic trajectories, or goes to infinity. On the other hand, the remainder may be a source of a chaotic behavior for numerical solutions. Chaotic means here, that the dynamics on some close invariant subset is semiconjucated to the Bernulli schift on sequences of two symbols. In other words, the symbolic dynamics may be constructed. The creation of chaotic trajectories may be modelled by the famous Smale horseshue. In Hamiltonian systems horseshues appear near transversal homoclinic or heteroclinic points. It is easy to see that the original autonomous system has no trajectories of that type, and the exponentially small remainder can only result in exponentially small transversality of the separatrices. Thus the latter may not be detected by the classical perturbation theory, which is based on the expansion in powers of the small parameter. We discuss some latest result on the exponentially small splitting of separatrices, especially, the methods for obtaining upper and lower bounds for the splitting. Finally, some numerical methods, specially desined to study exponentially small phenomena, are discussed.
Jan 27 1998