Power Series Expansions for the Frequency and Period of the Limit Cycle of the Van Der Pol Equation

Abstract

A power series expansion in the damping coefficient a is developed for the frequency $\nu ( \varepsilon )$ of the limit cycle of the van der Pol equation $\ddot U + U = \varepsilon \dot U( 1 - U^2 )( 0\leqq \varepsilon < \infty )$. The expansion is computed to $O( \varepsilon ^{24} )$ in rational arithmetic using the MACSYMA symbolic manipulation system and to $O( \varepsilon ^{164} )$ in floating-point arithmetic using FORTRAN. A Padé analysis of the power series expansion indicates that the singularities in the complex-$\varepsilon ^2 $ plane which are nearest the origin are branch points at radius $R \approx 3.42$ and modulus $ \pm \beta $, where $\beta \approx 1$ radians. Introduction of the variable $w = \varepsilon ^2 /( \varepsilon ^4 - 2\varepsilon ^2 R\cos \beta + R^2 )^{1/2} $ leads to expansions for the frequency and period which converge for $0\leqq w < 1$, i.e., for all $\varepsilon $ The results compare very favorably with published frequencies determined numerically but disagree to some extent with reported asymptotic expressions for large $\varepsilon $.