Bursting oscillations with adding-sliding structures in a Filippov-type Chua’s circuit

Abstract

In this paper, a modified Chua’s circuit, possessing an external excitation current and a piecewise nonlinear resistor, is considered to investigate bursting oscillations and the dynamical mechanism in the Filippov system, focusing on the effects of sliding bifurcations on the bursting dynamics. Five typical representative bursting oscillations are observed when there is a gap between the external excitation frequency and the natural one. Codimension-1 bifurcations of the fast subsystem are discussed by regarding the whole excitation term as a bifurcation parameter. The necessary conditions of conventional bifurcations of the equilibria and the bifurcations of boundary equilibria are obtained via theoretical analysis. Both the local adding-sliding bifurcation, crossing-sliding bifurcation, non-smooth fold bifurcation, the fold bifurcation of non-smooth limit cycle in the fast subsystem and the adding-sliding bifurcation in the slow–fast coupling system are observed via numerical method. Based on slow–fast analysis method and the bifurcation analysis, the dynamical mechanism is discovered. Research found that the non-smooth fold bifurcation of the boundary equilibrium can lead to jumping behaviors of the trajectory from the switching manifold to the attractors of the subsystem. The crossing-sliding bifurcation may not cause the transition between spiking state and quiescent state. The local adding-sliding bifurcation in the fast subsystem can lead to adding-sliding bifurcation of non-smooth limit cycle in the coupling system. There may exist adding-sliding structure in bursting oscillations before or after the adding-sliding bifurcation.