All Possible Bursting Attractors in the Neighborhood of Hopf Bifurcation Point Under Periodic Excitation

Abstract

For a vector field with Hopf bifurcation at the origin, when periodic excitation is introduced in which the frequency of the excitation is far less than the natural frequency, bursting oscillations, characterized by the alternations between the large-amplitude and the small-amplitude oscillations, can often be observed. The paper tries to answer the common question of how many types of bursting attractors may appear in such a vector field. Based on the normal form of Hopf bifurcation up to the third order, by regarding the whole exciting term as a slow-varying bifurcation parameter, all possible equilibrium branches of the generalized autonomous system are derived, the bifurcation sets of which divide the parameter space into regions associated with different types of dynamical behaviors. It is found that, there exist totally four possible combinations of the regions that the trajectory of the full system may visit as the slow-varying parameter varies. Accordingly, four qualitatively different behaviors, i.e. periodic symmetric oscillations, periodic symmetric mixed-mode oscillations, fold/Hopf/Hopf/fold and fold-Hopf/fold-Hopf bursting attractors, are presented, the mechanism of which is obtained by overlapping the transformed phase portrait and the equilibrium branches in the generalized autonomous system. Furthermore, the dynamics of the generalized autonomous system in different regions in the parameter space may influence the structures of the bursting oscillations when the trajectory enters the corresponding regions, while the bifurcations at the boundaries of the regions may lead to changes between the quiescent states and spiking states in the bursting attractors.