Subharmonic entrainment and limit cycle modulation by high frequency excitation: A Renormalization group approach

In this article, we explore the possibility of a sub-harmonic $(1{:}2)$ entrainment and supercritical Hopf bifurcation in a van der Pol-Duffing oscillator that has been excited by two frequencies, comprising a slow parametric drive and a fast external forcing, through the variation of the amplitude of the external fast signal. We also deduce the condition for the threshold parametric strength required to generate sub-harmonic oscillation. The Blekhman perturbation (direct partition of motion) and the Renormalization group technique have been employed to study how the signal amplitude plays a pivotal role in modulating the limit cycle dynamics as well as the subharmonic generation. Studies of nonlinear responses and bifurcations of such driven nonlinear systems are usually done by treating the strength of the fast drive as the control parameter. Here we show that, beyond its role in allowing one to study the dynamics with the slow and fast components nicely separated, the amplitude of the high-frequency signal can also be treated as an independent control parameter for controlling both the limit cycle behavior and the onset of subharmonic oscillation in the oscillator. Our analytical estimations are well supported by numerical simulations.