Title: 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 $\varepsilon =0$ 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 (Br\"annstr\"om,Gelfreich 2008) that there are orbits of the full
dynamics which shadow any \emph{finite} combination of forward orbits of
$J$ for a time $t=O(\varepsilon^{-1})$.
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.