Mechanisms of mixed-mode oscillations in a Rayleigh-Liénard oscillator with nonlinearities

Abstract Typically, different bifurcation structures of the multiscale systems can induce different mixed-mode oscillations, but bifurcation structures are not the only factor leading to the mixed-mode oscillations. In this paper, we aim to investigate the coupling effect of bifurcation structures and bifurcation delay phenomenon on the mixed-mode oscillations based on a Rayleigh-Liénard system excited by low-frequency harmonically parametric and external excitations. The system can be divided into fast subsystem and slow variable by regarding the cosine function as a slow variable. The stabilities and bifurcations of the fast subsystem are investigated briefly by the Jacobian matrix and characteristic equation. For the selected parameter conditions, the system can exhibit some bifurcations related to the mixed-mode oscillations, such as fold bifurcation, Hopf bifurcation, Homoclinic bifurcation and fold bifurcation of limit cycles. Based on the fast slow analysis approach, superposition diagram method and Lyapunov exponents, the generating mechanisms of different mixed-mode oscillations are studied in detail. Our results indicate that bifurcation structures are not the only factor resulting in the mixed-mode oscillations, since the same bifurcation structures may cause different mixed-mode oscillations and different bifurcation structures may induce the same mixed-mode oscillations. In fact, the generation of the mixed-mode oscillations in this paper is the result of the combined effects of the bifurcation structures and bifurcation delay phenomenon. The results of this paper may have certain reference value for the dynamic analysis, control strategy formulation and parameter identification of the nonlinear systems. Finally, we use the numerical simulations to prove the validities of the conclusions.