Effects of periodic parametric damping and amplitude-modulated signal on vibrational resonance and torus-doubling bifurcations occurrence in an asymmetric mixed Rayleigh-Liénard oscillator

Abstract

This research paper examines the effects of periodic parametric damping and amplitude-modulated signal on vibrational resonance and the occurrence of torus-doubling bifurcations in an asymmetric mixed Rayleigh-Liénard oscillator. The method of direct separation of the slow and fast motions is used to derive the approximate theoretical expression of response amplitude at the low frequency. The obtained results show that the presence of periodic parametric damping induces in the system multiple resonance peaks when the low frequency is varied. Moreover, the increase of carrier amplitude modulated increases or decreases the maximum amplitude value in certain range of the low frequency. However, when the periodic parametric damping coefficient is varied, one resonance peak occurs and the maximum amplitude value increases when the carrier amplitude modulated increases. The theoretical and direct numerical predictions have shown a fairly satisfactory agreement. On the other hand, the global dynamical changes of the system are numerically examined in context of vibrational resonance. It is found that, the system displays many torus attractors of different topologies, torus-doubling bifurcations, reverse torus-doubling bifurcations and torus-chaos. These observations are illustrated by plotting the phase portraits and their corresponding Poincaré maps.