Solution of reduced equations derived with singular perturbation methods

TL;DR

This paper demonstrates that solutions of reduced equations from various singular perturbation methods are exactly the sum of the most divergent secular terms in naive expansions, using Lie symmetry group theory for proof.

Contribution

It shows that all common singular perturbation methods produce solutions equal to the sum of the dominant secular terms, providing a unified understanding.

Findings

Solutions of reduced equations match the most divergent secular terms.

Lie symmetry group theory is used to prove the equivalence.

Provides a new construction method for perturbation solutions.

Abstract

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, the renormalization group method are well known. In this paper, it is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of Lie symmetry group.