Mixed-mode Oscillations in Filippov System

Abstract

The mechanism of the mixed mode oscillations of a class of non-smooth Filippov systems under multistable coexistence was studied in this paper. Based on a Lorenz-type chaotic model with multi-attractor coexistence, the Filippov system was established by introducing non-smooth terms as well as an external excitation. With multiple stable attractors in the discontinuous vector field, the parameter changes have led to complex transition patterns between the attractors and the non-smooth interface, or between the attractors. When an order gap exists between the exciting frequency and the natural frequency, implying the mixed-mode oscillations. Here we have taken several excitation amplitudes to cover different coexistence regions, a set of mixed mode oscillation patterns were obtained. Besides, the bifurcation set of two generalized autonomous subsystems and the coexistence region of attractors were discussed. Combined with the transformed phase diagram method, the bifurcation mechanism of bursting oscillation and the sliding dynamical behaviors of the system at the discontinuous interface has revealed with slow varying parameters access in different regions of multistable attractors coexistence. The alternations between quiescent and spiking states become more frequent and complex, leading to the change of the structure of the bursting oscillation modes. Moreover, the non-smooth partition interface of the system yields multiple non-smooth bifurcations, which will also affect the oscillation modes of the generalized autonomous system.