COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM

Bursting oscillation behavior induced by multiple time-scale coupling effect is one of the important topics in nonlinear dynamics research. In this paper, complicated bursting oscillation behaviors as well as their generation mechanism of a three-dimensional nonlinear dynamo system with slowly changing parametric excitation are revealed when the excitation frequency is much smaller than the natural frequency. The system can be used to describe the dynamic behaviors of two kinds of self-exciting homopolar dynamo systems, which are mathematically equivalent. By treating the parametric excitation as a slow-varying parameter, the generalized autonomous system corresponding to the nonautonomous system as well as the fast subsystem and the slow variable are got based on the fast-slow analysis method. Then, the stabilities and bifurcations of the fast subsystem are investigated theoretically, and the correctness of the theoretical analysis is verified by a one-parameter bifurcation diagram related to a typical parameter. With the help of the overlapping of the transformed phase diagram and bifurcation diagram, the mechanism of the symmetric delayed subHopf/fold cycle bursting oscillation as well as its dynamic transitions, i.e. delayed subHopf/fold cycle bursting oscillation, symmetric delayed pitchfork bursting oscillation of focus/focus type and delayed pitchfork bursting oscillation of focus/focus type are analyzed. The result shows that, two different forms of bifurcation delay phenomenon will appear under different parameter conditions, one is the subcritical Hopf bifurcation delay, and the other one is the pitchfork bifurcation delay. In addition, our research indicates that the stabilities of the equilibrium points and the width of the pitchfork bifurcation delay interval are both influenced by the control parameter. Meanwhile, we also find that the symmetry of the dynamic behaviors is affected by the choice of the initial values. The study of this paper further deepens the understanding and the comprehending of the different bursting patterns induced by bifurcation delay phenomenon.