Overmodulation causes a variation in the number of jumps in the Duffing system

Amplitude modulation, involving normal and overmodulation scenarios, is crucial for information transmission. However, the complex dynamics of how modulation phase shift affects relaxation oscillations, especially under overmodulation conditions, has not been fully elucidated. Thus, this paper aims to explore the dynamical mechanism of relaxation oscillations affected by modulation phase shift under overmodulation conditions. The result shows that minor phase adjustments in low overmodulation phases can change the time series of the signal. Notably, at critical modulation phase thresholds, the number of transitions in each period of relaxation oscillations increases, and this phenomenon can be observed across a range of parameter values. However, further increase in phase will lead to the decrease in the number of transitions in relaxation oscillations, which demonstrates a clear correlation between phase adjustments and fold bifurcations affecting oscillation patterns. Based on the tri-parametric and bi-parametric bifurcation analysis, we explore the effect of overmodulation index on the number of transitions, and find that higher indices induce complex variations in it. These findings highlight the intricate interplay between modulation phase and modulation index in determining relaxation oscillation patterns, which are crucial for understanding amplitude modulation diversity and can serve as a reference for future research on other modulation scenarios.