Geometric Shadowing in Slow-Fast Hamiltonian Systems

Abstract

We study a class of slow-fast Hamiltonian systems with any finite number of degrees of freedom, but with at least one slow one and two fast ones. At $% ε=0$ the slow dynamics is frozen. We assume that the frozen system (i.e. the unperturbed fast dynamics) has families of hyperbolic periodic orbits with transversal heteroclinics. For each periodic orbit we define an action $J.$ This action may be viewed as an action Hamiltonian (in the slow variables). It has been shown in (N. Brännström, V.Gelfreich (2008)) that there are orbits of the full dynamics which shadow any \emph{finite} combination of forward orbits of $J$ for a time $t=O(ε^{-1})$. We introduce an assumption on the mutual relationship between the actions $J.$ This assumption enables us to shadow any continuous curve (of arbitrary length) in the slow phase space for any time. The slow dynamics shadows the curve as a purely geometrical object, thus the time on the slow dynamics has to be reparameterised.