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

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