Geometric Shadowing in Slow-Fast Hamiltonian Systems

TL;DR

This paper investigates how slow-fast Hamiltonian systems with hyperbolic periodic orbits can shadow arbitrary curves in the slow phase space, extending previous results to more general scenarios with multiple degrees of freedom.

Contribution

It introduces a new assumption on the actions of hyperbolic periodic orbits that allows shadowing of any continuous curve in the slow phase space for systems with multiple degrees of freedom.

Findings

Shadowing of arbitrary curves in slow phase space achieved

Extension of previous shadowing results to systems with multiple degrees of freedom

Reparameterization of slow dynamics to match geometrical curves

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 $% \epsilon =0$ 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. Br\"annstr\"om, 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 $t=O(\epsilon^{-1})$. 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.