Perturbation Analysis of the Limit Cycle of the Free van der Pol Equation

A power series expansion in the damping parameter $\varepsilon $ of the limit cycle $U(t;\varepsilon )$ of the free van der Pol equation $\ddot U + \varepsilon (U^2 - 1)\dot U + U = 0$ is constructed and analyzed. Coefficients in the expansion are computed up to $O(\varepsilon ^{24} )$ in exact rational arithmetic using the symbolic manipulation system MACSYMA and up to $O(\varepsilon ^{163} )$ using a FORTRAN program. The series is analyzed using Pade approximants. The convergence of the series for the maximum amplitude of the limit cycle is limited by two pairs of complex conjugate singularities in the complex $\varepsilon $-plane. These singularities are the same as those which limit the convergence of the series expansion of the frequency of the limit cycle. A new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of $\varepsilon $. Amplitudes computed from this transformed series agree very we...