Discrete averaging for discrete time dynamical systems

TL;DR

This paper introduces a discrete averaging method for discrete time dynamical systems that simplifies analysis, improves approximation accuracy, and aids in identifying adiabatic invariants without the need for classical intermediate steps.

Contribution

It develops a novel discrete averaging technique that enhances classical averaging theory by eliminating complex intermediate procedures, providing explicit error bounds and broad applicability.

Findings

Provides explicit uniform bounds for approximation errors.

Demonstrates effectiveness in near-identity map dynamics.

Establishes domain of validity for adiabatic approximations.

Abstract

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 provides a simple and effective tool for finding adiabatic invariants, both numerically and analytically. It is capable of strengthening various theorems of the classical averaging theory because 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 neighbourhood of a resonant fixed point. We show that the discrete averaging provides explicit uniform bounds for approximation errors. We also show that the discrete averaging can be used to establish domain of validity of adiabatic approximations in numerical experiments.