Effects of pulse amplitude modulation on relaxation oscillations in the Duffing system

This paper reports on the fast-slow dynamics induced by pulse amplitude modulation (PAM). Typically, PAM transmits information signals by altering the amplitude of the pulse signal, with the core concept being that the amplitude of the pulse sequence varies according to the input data. It was observed that introducing additional pulse excitation in the Duffing system results in a relaxation oscillation mode characterized by catastrophic jumps. Using the fast-slow analysis method, the generation mechanism of these relaxation oscillations was elucidated, wherein PAM results from the superposition of periodic cosine and periodic pulse signals. Subsequently, the cases of periodic cosine excitation, pulse excitation, and PAM excitation for a single slow variable were separately discussed. The study demonstrates that PAM plays a critical role in the fast-slow dynamics of the Duffing system. Specifically, changes in amplitude under pulse modulation lead to varying degrees of relaxation oscillation modes. For instance, low PAM induces a mode where conventional oscillations connected by catastrophic jumps alternate with a resting state. Moreover, when the amplitude crosses a certain critical value, it leads to a mode where relaxation oscillations connected by catastrophic jumps alternate with a resting state. Furthermore, when the modulation amplitude exceeds another critical value, it causes a transition in the resting state position, which is associated with the attraction domains at different amplitudes. Based on these findings, compound relaxation oscillation modes were revealed. Finally, comparisons of circuit simulation experiments validated the results.