Bursting oscillations with delayed C-bifurcations in a modified Chua’s circuit

Abstract

In this research, a typical Chua’s circuit with a piecewise nonlinear resistor and a slow-varying periodic excitation is considered to investigate the dynamical mechanisms of bursting oscillations in the piecewise-smooth dynamical system. A set of new bursting oscillations is observed when the amplitude of the excitation is changed. By regarding the excitation term as a bifurcation parameter, the codimension-1 conventional bifurcations and non-smooth bifurcations of the fast subsystem are explored. Fold bifurcation, supercritical Hopf (sup-Hopf) bifurcation, non-smooth Hopf bifurcation, grazing bifurcation, and C-bifurcation are discovered via theoretical and numerical methods. The C-bifurcation connects the stable limited cycle bifurcated from non-smooth Hopf bifurcation with the stable limited cycle bifurcated from the sup-Hopf bifurcation. When the fast subsystem driven by the slow subsystem passes through the bifurcation points, slow passage effect near the non-smooth Hopf bifurcation and delay of the C-bifurcation take place. The delayed C-bifurcation may lead to multiple transition patterns between different attractors, including two transition patterns of reverse direction near the fold and sup-Hopf bifurcations. The delayed transition to other attractor creates a non-smooth hysteresis loop and enables the generation of bursting oscillations.