Bursting dynamics and the zero-Hopf bifurcation of simple jerk system

Abstract

We characterize the zero-Hopf bifurcation at a singular point of a parameter jerk system. By employing the second order averaging theory, we demonstrate that up to three periodic orbits generated as disturbance parameters tend to zero. Again, both the bifurcation mechanism and bursting dynamics of the 3D jerk system with external periodic excitation are systematically explored. While an order difference exists between the frequency of external excitation and the average frequency of the system, the system exhibits bursting oscillations. The mechanisms of different bursting oscillations are investigated by means of the equilibrium point curve and the transformed phase portraits. As the amplitudes of the excitation change, the system displays “delayed symmetric pitchfork/point, delayed symmetric pitchfork/supHopf, and delayed pitchfork/supHopf/homoclinic connection bursting”. Finally, an analog circuit is designed to verify the complex bursting phenomena of the system.