Formulas for the amplitude of the van der Pol limit cycle

TL;DR

This paper derives recursive formulas to accurately approximate the amplitude of the van der Pol oscillator's limit cycle across all nonlinear regimes, with less than 0.1% error, using the homotopy analysis method.

Contribution

It provides the first analytical formulas for the van der Pol limit cycle amplitude valid from weak to strong nonlinearity regimes.

Findings

Amplitude approximation error less than 0.1%

Formulas valid for entire range of epsilon

First analytical approximation covering all nonlinear regimes

Abstract

The limit cycle of the van der Pol oscillator, $\ddot{x}+ \epsilon (x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form of this limit cycle for the whole range of the parameter $\epsilon$ is obtained. These formulas generate the amplitude with an error less than 0.1%. To our knowledge, this is the first time where an analytical approximation of the amplitude of the van der Pol limit cycle, with validity from the weakly up to the strongly nonlinear regime, is given.