Mixed-Mode Oscillation in a Class of Delayed Feedback System and Multistability Dynamic Response

In this paper, a class of two-parameter mixed-mode oscillation with time delay under the action of amplitude modulation is studied. The investigation is from four aspects. Firstly, a parametric equation is considered as a slow variable. By the time-history diagram and phase diagram, we can find that the system generates a cluster discovery image. Secondly, the Euler method is used to discrete the system and obtain the discrete equation. Thirdly, the dynamic characteristics of the system at different time scales are discussed when the ratio of the natural frequency and the excitation frequency of the system is integer and noninteger. Fourthly, we discuss the influence of time delay on the discovery of clusters of this kind of system. The research shows that the time lag does not interfere with the influence of the cluster image, but the dynamics of the upper and lower parts of the oscillation in each period will be delayed. So, we can improve peak performance by adjusting the time lag and obtain the desired peak. Finally, we explore the multistate dynamic response of a two-dimensional nonautonomous Duffing system with higher order. According to bifurcation diagram and time-history curve, bistable state will appear in the system within the critical range. With the gradual increase of parameters, the chaotic attractor will suddenly disappear which will lead to the destruction of the bistable state.