Quasi-periodic structure in chaotic bursting attractor for a controlled jerk oscillator

Abstract

The main purpose of the paper is to show that parts of the trajectory for chaotic bursting oscillations may oscillate according to different limit cycles, which implies orderly movement exists in chaos. For a typical jerk oscillator with a cubic nonlinearity, when a slow-varying controlling term is introduced, bursting oscillations may occur, appearing in the combination of large-amplitude and small-amplitude oscillations. Two coexisting stable equilibrium points may evolve to chaos via different cascades of period-doubling bifurcation, respectively, which means different types of stable attractors may coexist with the variation of slow-varying parameter, while several periodic windows exist in the chaotic region. It is found that the spiking oscillations of the full system may change according to different cycles not only for 2-D torus bursting oscillations but also for the chaotic bursting attractor. Furthermore, whether the influence of the short periodic window of the fast subsystem on the bursting oscillations appears or not depends on the size of the slow scale, since the inertia of the movement may cause the delay effect of the bifurcations at both two edges of the window. The findings suggest that coupling of multiple scales may be one of key factors for the diversity of real world.