Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

TL;DR

This paper extends and unifies various singular perturbation methods for ordinary differential equations using the renormalization group approach, providing theoretical foundations, error estimates, and convergence conditions.

Contribution

It demonstrates that the RG method unifies traditional perturbation techniques and establishes new theoretical results on error bounds, invariant manifolds, and convergence.

Findings

Proves error estimates for approximate solutions

Shows inheritance of symmetries in RG equations

Provides conditions for convergence of the infinite order RG equation

Abstract

The renormalization group (RG) method is one of the singular perturbation methods which is used in search for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, inheritance of symmetries from those for the original equation to those for the RG equation, are proved. Further it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper-) normal forms theory, the center manifold reduction, the geometric singular perturbation method and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.