Modulated motion and infinite-period bifurcation for two non-linearly coupled and parametrically excited van der Pol oscillators

Abstract

We investigate the slow flow resulting from the application of the asymptotic perturbation method to a system of two non-linearly coupled van der Pol oscillators under the effect of a harmonic parametric excitation. The slow flow consists of four non-linear coupled equations on the amplitudes and phases of the oscillators. Their fixed points correspond to two-period quasiperiodic motion for the starting system and we show parametric excitation–response and frequency–response curves. We use energy considerations to study existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a three-period modulated motion for the van der Pol oscillators. Modulated motion can be also observed for very low values of the parametric excitation and an approximate analytic solution is constructed. Moreover, if the parametric excitation increases, the oscillation period of the limit cycle for the slow flow equations becomes infinite, the limit cycle disappears and an infinite-period bifurcation occurs. In the starting system the three-period quasiperiodic motion has been substituted with a two-period quasiperiodic motion. Analytical results are verified with numerical simulations.