BURSTING OSCILLATIONS AS WELL AS THE DELAYED PITCHFORK BIFURCATION BEHAVIORS IN A CLASS OF CHAOTIC SYSTEM

Abstract

Due to wide existence of multiple-time-scale problems in practical engineering, the complicated dynamic behaviors and their generation mechanism have become one of the hot topics at home and abroad. The systems with multiple time scales can often exhibit bursting oscillations with the bifurcation delay phenomenon. In order to investigate the bifurcation mechanism of bursting oscillations caused by bifurcation delay in a nonlinear system, a parametric excitation is introduced in a novel three-dimensional chaotic system. When the exciting frequency is far less than the natural frequency, the coupling of two time scales involves the vector field, which leads to the bursting oscillations. By considering the whole exciting term as a slow-varying parameter, the original system can be considered as a generalized autonomous system, which can be regarded as the fast subsystem. Upon the analysis of equilibrium points and bifurcation conditions of the fast subsystem, combining with the transformed phase portraits, the bifurcation mechanisms of bursting oscillations is presented. Four typical cases with different parameter conditions are discussed to reveal the evolution of the bursting oscillations. It is found that when the slow-varying exciting term passes across the bifurcation points, the delayed behaviors of super-critical pitchfork bifurcation can be observed. With the increase of the exciting amplitude, the occurring needed for the bifurcation delay is increased gradually. When the delayed behaviors end in different parameter regions, different types of bursting oscillations which may surround different attractors such as equilibrium points and limit cycles appear.