Vibrational tuning of subharmonic entrainment and limit-cycle stability: A unified control mechanism

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. While the conventional studies use forcing strength as the control parameter, we use the signal strength as the tuning element, appearing as a more efficient and physically observable measure. Our analytical estimations are well supported by numerical simulations.