Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

TL;DR

This paper compares the Homotopy Analysis and an improved Renormalization Group methods for analyzing Rayleigh and Van der Pol oscillators, deriving approximate formulas for limit cycle amplitudes and improving RG analysis with nonlinear time concepts.

Contribution

It introduces an enhanced RG method using nonlinear time and provides comparative analysis with the Homotopy Analysis method for these oscillators.

Findings

Derived approximate formulas for limit cycle amplitudes

Improved RG analysis with nonlinear time concept

Presented good approximate plots of limit cycles

Abstract

A comparative study of the Homotopy Analysis method and an improved Renormalization Group method is presented in the context of the Rayleigh and the Van der Pol equations. Efficient approximate formulae as functions of the nonlinearity parameter $\varepsilon$ for the amplitudes $a(\varepsilon)$ of the limit cycles for both these oscillators are derived. The improvement in the Renormalization group analysis is achieved by invoking the idea of nonlinear time that should have significance in a nonlinear system. Good approximate plots of limit cycles of the concerned oscillators are also presented within this framework.