Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

Abstract

This paper deals with the transitions through Melnikov thresholds and the corresponding fast–slow dynamics in a family of bi-parametric mechanical oscillators subjected to an amplitude modulation force both analytically and numerically. Applying Melnikov analytical method, the threshold condition for the occurrence of mixed-mode oscillations is obtained. First, the Melnikov threshold curves are drawn in different parameter space in the unperturbed system. Next, as the control variable crosses above the threshold, two states about mixed-mode oscillations can be identified: the rest state corresponding to local limit cycle from nontrivial equilibrium point and the spiking state corresponding to large amplitude cycle produced by saddle–node bifurcation of limit cycles. Such mixed-mode oscillations are created since the amplitude modulation force slowly and periodically switches between the rest states and spiking states. Furthermore, we elucidate the effect of modulation frequency on phase portraits and dynamical behaviors, and some numerical simulations are also included to illustrate the validity of our study.