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

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