High order analysis of the limit cycle of the van der Pol oscillator

TL;DR

This paper applies advanced analytical techniques to precisely analyze the limit cycle of the van der Pol oscillator, significantly improving the accuracy of previous results through high-order series expansions and resummation methods.

Contribution

It introduces a high-order series analysis combined with Hermite-Padé and Padé approximants to accurately determine the van der Pol oscillator's limit cycle characteristics.

Findings

Series coefficients computed up to order 859.

Location of the branch cut determined with 100-digit accuracy.

Resummation approach captures exact asymptotic behaviors.

Abstract

We have applied the Lindstedt-Poincar\'e method to study the limit cycle of the van der Pol oscillator, obtaining the numerical coefficients of the series for the period and for the amplitude to order $859$. Hermite-Pad\'e approximants have been used to extract the location of the branch cut of the series with unprecendented accuracy ($100$ digits). Both series have then been resummed using an approach based on Pad\'e approximants, where the exact asymptotic behaviors of the period and the amplitude are taken into account. Our results improve drastically all previous results obtained on this subject.