Discrete averaging for discrete time dynamical systems

In this paper we develop the theory of discrete averaging designed to study discrete time dynamical systems defined by iterates of a map. The discrete averaging uses weighted averages over a segment of trajectory to find an autonomous vector field that approximates the original map. The method also provides a simple and effective tool for finding adiabatic invariants, both numerically and analytically. It eliminates two intermediate steps used in the classical averaging: the suspension procedure that assigns a rapidly oscillating flow to the map and time-dependent coordinate changes that eliminate the dependence on time. We discuss two applications of the discrete averaging — to the dynamics of a near-identity map and to the dynamics of a map in a neighborhood of a resonant fixed point — and derive explicit uniform bounds for approximation errors. We illustrate the method by analyzing domain of validity of adiabatic approximations in a numerical experiment involving the area preserving Hénon map.