Effects of overmodulation on the slow progresses of relaxation oscillations in the Duffing system

Amplitude modulation (AM) is a key technique for transmitting information by altering the amplitude of a carrier signal, including both conventional modulation and overmodulation strategies. Conventional AM focuses on maintaining signal integrity and minimizing distortion despite lower power efficiency, whereas overmodulation enhances signal strength and efficiency, albeit with potential increases in distortion. Crucially, modulation phase plays a pivotal role in AM, impacting signal representation, demodulation, and the processing of multiple signals. This paper aims to report on the dynamics of relaxation oscillations caused by changes in modulation phase under overmodulation. Typically, at low modulation phases, the introduction of modulation phase leads to changes in the system’s vector field structure and a shift in the time series. Particularly, when modulation phase reaches a certain critical value, it causes a sudden extension of the quasi-static slow process in the system’s upper branch, a pattern that persists within a specific parameter range. Furthermore, as modulation phase increases to another critical value, the quasi-static process in the system’s upper branch suddenly extends again. Interestingly, as the modulation phase continues to increase to two additional critical values, this pattern of extending the quasi-static slow process on the upper branch transforms into a pattern of shortening, with each occurring suddenly within their respective parameter ranges. Lastly, using three-parameter and two-parameter bifurcation diagrams, we illustrate the impact of the overmodulation index on the system’s dynamic behavior. Changes in the overmodulation index alter relaxation oscillation patterns: low indices cause sudden extensions, while high indices reduce oscillations. Our research highlights the significant influence of the overmodulation index on dynamic behavior and provides a reference for other modulation conditions.