Compound relaxation oscillations in a modified Rayleigh–Duffing system with periodic non-smoothness

Yi Zhang, Jin Song, Wenjie Zuo and Zhengdi Zhang

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 243, issue C, 82-94

Abstract: This paper aims to explore compound relaxation oscillations and underlying mechanisms in the dynamical system with periodic non-smoothness, focusing on the effect of non-smooth bifurcations on compound relaxation oscillations. Based on the Rayleigh–Duffing system with external excitation, a modified non-smooth dynamical system is developed by introducing a periodic term that represents discontinuous external influences, such as wave-induced forces in ship rolling dynamics. Various non-smooth bifurcation phenomena are systematically investigated, including non-smooth homoclinic bifurcation, C-bifurcation, persistence bifurcation, and non-smooth fold bifurcation. Five different oscillation modes are demonstrated through numerical simulations, and their mechanisms are revealed in combination with the slow–fast analysis. It is found that the non-smooth homoclinic bifurcation significantly alters the oscillation process and induces transitions between stable states. The C-bifurcation has less effect on the oscillation mode even though it changes the topology of limit cycles. Different types of boundary equilibrium bifurcations lead to substantial changes in the stability and structure of compound relaxation oscillations. In addition, two types of coexisting attractors are identified through the basin of attraction, indicating multistability that gives rise to different oscillation modes.

Keywords: Compound relaxation oscillations; Slow-varying dynamics; Periodic non-smoothness; Non-smooth homoclinic bifurcation (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:243:y:2026:i:c:p:82-94

DOI: 10.1016/j.matcom.2025.11.027

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