Cramer Asymptotics in the Averaging Method for Systems with Fast Hyperbolic Motions

Abstract

A dynamical system $w'=S(w,z,ε)$, $z'=z+εv(w,z,ε)$ is considered. It is assumed that slow motions are determined by the vector field $v(w, z, ε)$ in the Euclidean space and fast motions occur in a neighborhood of a topologically mixing hyperbolic attractor. For the difference between the true and averaged slow motions, a limit theorem is proved and sharp asymptotics for the probabilities of large deviations (that do not exceed $ε^δ$) are calculated; the exponent $δ$ depends on the smoothness of the system and approaches zero as the smoothness increases.