Bifurcation mechanism of the bursting oscillations in periodically excited dynamical system with two time scales

The main purpose of this work is to demonstrate that bursting oscillations not only occur in slow-fast systems, which have been presented in many publications, but also may take place in periodic excited systems with order gap between the excited frequency and natural frequency. Furthermore, a typical procedure to analyze the bifurcation mechanism of the bursting oscillations in periodic excited dynamical systems is presented, in which the conceptions of generalized autonomous system (GAS) and transformed phase portrait are introduced. For the periodic external excited dynamo oscillator, different types of bursting phenomena can be obtained, the mechanism of which is presented based on the bifurcation analysis of the corresponding GAS. From the evolution of the dynamics with the variation of the parameters, it is found that different forms of bifurcations, occurring at the transitions between the quiescent state and spiking state, may cause different patterns of bursting oscillations.