Functional relations in renormalization group methods for a class of ordinary differential equations

TL;DR

This paper introduces a renormalization group (RG) perturbation approach for certain ordinary differential equations, revealing exact functional relations that ensure secular term elimination and provide explicit amplitude relations.

Contribution

It develops a unified RG framework for a broad class of ODEs, extending previous results and enabling explicit inversion of amplitude relations.

Findings

RG-based perturbation scheme for ODEs developed

Exact functional relations for secular coefficients established

RG equations for slow dynamics derived directly

Abstract

We develop a renormalization group (RG)-based perturbation scheme for a class of ordinary differential equations, including first-order systems with semisimple or nilpotent linear parts, as well as scalar higher-order equations. The key observation is that the secular coefficients arising in naive perturbation theory satisfy an exact functional relation. This yields, in a unified manner, several fundamental features of the RG method: the renormalized amplitudes satisfy a closed functional relation with a group-like structure, the RG equation governing their slow dynamics is obtained directly, the absence of secular terms is ensured to all orders, and the relation between bare and renormalized amplitudes admits an explicit inversion. The results extend earlier ones for second-order scalar equations.