Bursting oscillations with bifurcations of chaotic attractors in a modified Chua’s circuit

Abstract

This paper aims at the bursting oscillations as well as the dynamical mechanism in non-smooth systems. A fourth-order Chua’s circuit is modified to establish a non-smooth system by introducing a periodic exciting current and a piecewise nonlinear resistor. Under the condition that the dimensionless excitation frequency is far less than 1, the system is a slow–fast system, and five representative bursting patterns are observed via changing the excitation amplitude. By regarding the whole term of periodic exciting current as a bifurcation parameter, several kinds of smooth or non-smooth bifurcations are obtained in the fast subsystem via numerical or theoretic method. Particularly, a so-called saddle-focus-induced catastrophic bifurcation(SFIC), which is generated by collision between a non-smooth chaotic attractor and a saddle-focus, is observed. In addition, bifurcation diagrams and the associated maximal Lyapunov exponent spectrum are also obtained numerically, which can illustrate the scenario from periodic solutions to chaotic solutions via period-doubling bifurcations. Based on the bifurcation analysis, the generation mechanism of the bursting patterns is revealed. Research shows that the SFIC bifurcation in the fast subsystem may result in a sudden jumping behavior in the slow–fast system from a chaotic attractor to another. A spiking attractor may be constructed by chaotic attractors and limit cycle attractors together, which can lead to chaotic–periodic bursting oscillations. Slow passage effect induced by Hopf bifurcation can lead to delay phenomenon of other bifurcation in the slow–fast system.