Compound relaxation oscillations in Duffing oscillator driven by a low frequency amplitude-modulated excitation

Low frequency amplitude-modulated excitation is a recently reported excitation that can induce relaxation oscillations and prolongs the system's slow processes. This investigation aims to research the effects of different modulation modes of low frequency amplitude-modulated excitation on relaxation oscillations. Generally, intriguing dynamical characteristics can be observed in compound relaxation oscillations when amplitude-modulated modulation is introduced. Specifically, the quasi-static processes of the system exhibit different relaxation oscillations while the active phase of the relaxation oscillations remains relatively constant. We investigate the dynamical behaviors of the entire excitation through the fast-slow analysis method. For this purpose, a three-dimensional visualization is proposed to explore the relaxation oscillations induced by different modulation modes, providing insights into the system's dynamical behaviors from multiple perspectives. The results show that the oscillations depend on the modulation mode of the amplitude-modulated excitation. Modulation frequency changes extend the quasi-static processes and increase the number of oscillations. Besides, variations in the modulation index can trigger amplitude variations that lead to an increased number of bifurcation points during overmodulation. Based on these findings, we reveal the dynamic mechanism of composite relaxation oscillations. Our study demonstrates that the introduction of amplitude-modulated excitation can induce compound relaxation oscillations in the system, and different modulation modes significantly influence the fast-slow dynamics. Moreover, our results provide a valuable reference for investigating other dynamic systems that incorporate amplitude-modulated excitation.