Averaging of time-periodic systems without a small parameter

TL;DR

This paper introduces a novel averaging method for non-Hamiltonian systems with periodic forcing that does not rely on a small parameter, using Lie transforms to derive formal coordinate transformations for higher-order averaging.

Contribution

It develops a new averaging approach applicable to a broad class of systems without small parameters, utilizing Lie transforms for formal coordinate changes.

Findings

Effective higher-order averaging formulas derived

Method successfully applied to atmospheric chemistry problem

Explicit inverse transformations for initial data estimation

Abstract

In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones. Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation. Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.