The complicated behaviors as well as the bifurcation mechanism of the dynamical systems with different time scales have become one of the hot subjects at home and abroad, since they often behave in bursting attractors characterized by the combinations between large-amplitude oscillations and small-amplitude oscillations. Since the slow-fast analysis was employed to investigate the mechanism of the special forms of movements, a lot of results related to the bursting oscillations in autonomous systems with two scales in time domain have been obtained. Recently, based on the transformed phase portraits, different types of bursting oscillations as well as the mechanism in the vector fields with single periodic excitation have been presented. However, few works has been published related to the systems with multiple periodic excitations, the dynamics of which still remains an open problem. The main purpose of the manuscript is to explore effect of the multiple scales in such systems. As a example, based on a relatively simple four-dimensional Chua’s circuit, by introducing two periodically changed electric sources, when the two exciting frequencies are strictly resonant, both of which are far less than the natural frequency of the system, a dynamical model with scales under two periodic excitations is established. Note that the combination of the two exciting terms can be transformed as a function of a periodic term with single frequency, which can be regarded as a slow-varying parameter. The equilibrium branches as well as the associated bifurcations with the variation of the slow-varying parameter can be derived by employing the characteristics analysis of the equilibrium points. It is found that the distribution of the equilibrium branches as well as the bifurcation details may changed with the variation the amplitudes of the excitations, which may influence the attractors of the whole dynamical system. Three typical cases corresponding to the different situations of the equilibrium branches are considered, in which different forms of bursting oscillations are observed. Based on the transformed phase portraits, the bifurcation mechanism of the bursting oscillations has been presented. It is found that the trajectory may move almost strictly along one of the stable equilibrium branches, while jumping to another stable equilibrium branch may occur at the fold bifurcation points, the transient process of which leads to the large-amplitude oscillations corresponding to spiking states. Furthermore, it is pointed out that when more fold bifurcation points involve the behaviors of the system, more complicated bursting oscillations may appear.

2018, 50(2): 362-372.
doi: 10.6052/0459-1879-18-017