Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

Abstract

Relaxation oscillations are pervasive in diverse areas of natural sciences and engineering, and exploring the dynamical mechanisms of relaxation oscillations is one of the most significant issues. Typical relaxation oscillations can be observed in the Duffing system. Recently, amplitude modulation has emerged as a novel control mechanism for investigating the behavior of fast-slow dynamics in systemic tension oscillations. It has demonstrated the ability to prolong the quasi-static slow process of the system and increase the number of bifurcation points. However, the exploration of the mechanistic aspects of amplitude modulation is still in its early stages, with many unreported dynamical mechanisms. Among these, investigating the modes of relaxation oscillations induced by amplitude modulation is one of the most important issues. Therefore, this manuscript focuses on studying the effect of amplitude modulation on relaxation oscillations, using the classical forced Duffing system as a representative model. Significantly, we report an intriguing finding for the first time, revealing a new amplitude-modulated mechanism by which the disappearance of relaxation oscillations can be induced. By employing the fast-slow analysis, we have examined the underlying dynamical mechanisms, revealing a strong correlation with the modulation index of amplitude modulation. Notably, when the system operates under low amplitude modulation, an extension of the quasi-static process is observed, manifesting as a prolonged slow process. Conversely, under high amplitude modulation, relaxation oscillations suddenly disappear. Our results serve to enrich the potential mechanisms of amplitude modulation, and our analysis provides a reference for investigating the dynamical behavior induced by amplitude modulation in other dynamical systems.