From period-doubling bursting to chaotic–periodic bursting in a modified Chua’s circuit

The aim of this paper is to reveal the dynamical mechanism of bursting oscillations in non-smooth dynamical systems, and the interest is focused on the effects of period-doubling bifurcation and chaotic attractor on the bursting oscillations. By modifying a fourth-order Chua’s circuit, a dynamical system possessing non-smooth switching manifold and multiple scale variables, is established. By taking the slow-varying variable as a bifurcation parameter, Subcritical non-smooth Hopf bifurcation, C-bifurcation, and period-doubling bifurcation are obtained in the fast subsystem. Further more, chaotic attractors, which are generated from a series of period-doubling bifurcations, are also observed. Based on the numerical simulations and bifurcation analysis of the fast subsystem, eight typical bursting patterns are obtained, and their dynamical mechanism is revealed. Study shows that period-2 limit cycle in the fast subsystem may cause the slow–fast system to oscillates in alternating small and large amplitudes; A complex spiking attractors made up of limit cycles and chaotic attractors may lead to spiking states changing from chaotic oscillations to periodic ones, and the trajectory may jump directly from a chaotic attractor to other stable one without undergoing any bifurcation of the fast subsystem.