A remark on renormalization group theoretical perturbation in a class of ordinary differential equations

TL;DR

This paper refines the renormalization group perturbation method for oscillator-type second-order ODEs, providing a simple relation among secular coefficients and an inversion formula, applicable to various classical equations.

Contribution

It introduces a straightforward functional relation among secular coefficients and an inversion formula, enhancing the understanding of RG perturbation theory for a broad class of oscillator equations.

Findings

Derived a simple functional relation among secular coefficients.

Established an inversion formula between bare and renormalized amplitudes.

Proved the absence of secular terms in all orders of the RG series.

Abstract

We revisit the renormalization group (RG) theoretical perturbation theory on oscillator-type second-order ordinary differential equations. For a class of potentials, we show a simple functional relation among secular coefficients of the harmonics in the naive perturbation series. It leads to an inversion formula between bare and renormalized amplitudes and an elementary proof of the absence of secular terms in all orders of the RG series. The result covers nonautonomous as well as autonomous cases and refines earlier studies, including the classic examples of Van der Pol, Mathieu, Duffing, and Rayleigh equations.