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

A power series expansion in the damping coefficient a is developed for the frequency ν ( ε ) of the limit cycle of the van der Pol equation U + U = ε U( 1 - U² )( 0 ε < ∞ ). The expansion is computed to O( ε ²⁴ ) in rational arithmetic using the MACSYMA symbolic manipulation system and to O( ε ¹⁶⁴ ) in floating-point arithmetic using FORTRAN. A Padé analysis of the power series expansion indicates that the singularities in the complex-ε ² plane which are nearest the origin are branch points at radius R ≈ 3.42 and modulus ± β, where β ≈ 1 radians. Introduction of the variable w = ε ² /( ε ⁴ - 2ε ² Rcos β + R² )1/2 leads to expansions for the frequency and period which converge for 0 w < 1, i.e., for all ε The results compare very favorably with published frequencies determined numerically but disagree to some extent with reported asymptotic expressions for large ε.