Dynamical analysis of bursting in a system with a discontinuous boundary involving two state variables

Abstract

This paper investigates the bursting oscillations and the underlying dynamical mechanisms of a non-smooth system with a discontinuous boundary that involves two state variables. A slow-fast non-smooth system is established after modifying the Chua's circuit. By applying the variable substitution, the system with a boundary involving two state variables is transformed into a system where the boundary involves only a single state variable. The topological equivalence between the systems before and after the transformation is demonstrated. The stability analysis is performed on the transformed system, and the bifurcation conditions are given. Six types of bursting oscillation patterns under different parameter settings are presented. The bifurcation transition mechanisms of these bursting patterns are revealed using two-dimensional transformed phase diagrams. It is found that an increase in the excitation amplitude leads to changes in the transition patterns of the system trajectory between point and cycle, resulting in the formation of a 'sticking' special spiking state. This spiking state involves multiple limit cycle attractors and an equilibrium branch. The magnitude of the excitation amplitude also affects the slow passage effect of the system, determining whether the trajectory undergoes periodic oscillations at the Hopf bifurcation point.