Resonances and Power Series Solutions of the Forced van der Pol Oscillator

Abstract

To the first approximation, the nonresonant steady-state motion of the forced van der Pol oscillator is a combination of a free and a forced response. When the forcing amplitude increases beyond a certain critical value, which depends on the forcing frequency, the free response will decay and the motion is represented by just a particular solution that has the same frequency as the forcing term. In the first part of this paper, a general expression is determined for all of the higher-order resonant frequencies associated with a formal multiple timescale perturbation expansion of the solution. These frequencies are shown to be dense on the real axis. Then a formal power series expansion is developed for a particular solution in terms of the small damping parameter $\varepsilon $, which is shown to be valid when the forcing frequency is not close to a certain subset of the resonant frequencies. Using Padé approximants, information is obtained about the location and nature of the singularities in the complex $\varepsilon $-plane that limit the convergence of the series. Three pairs of singularities whose locations change as the forcing amplitude varies are discovered. In particular, when the forcing amplitude exceeds a certain value, a pair of singularities along the imaginary axis becomes dominant. The emergence of these singularities as the dominant ones is discussed in relation to the quenching of the free oscillation in the total response.