The Phase-Shifting Limit Cycles of the van der Pol Equation

The van der Pol limit cycles are generated at small amplitudes by the computer implementation of the Poincaré-Lindstedt method. The formal algebraic solution is accomplished by manipulations of Poisson series, and the FORTRAN programming of the inductive algorithm yields the phase-shifting limit cycles to graphical accuracy over the range 0 ≤ λ ≤ 1.5. This improves upon the method of Deprit and Rom in two ways. First, because the formal solution is carried out by hand, an algebraic processor is not necessary. Second, the standard solutions which they generated are only valid for 0 ≤ λ ≤ 1.2 whereas the phase-shifting limit cycles are still valid at λ = 1.5; that is, they do not exhibit the Gibbs phenomenon even at λ = 1.5.