All possible bursting oscillations in neighborhood of a Hopf bifurcation point induced by low-frequency excitation

Hopf bifurcation may be one of most common bifurcations, which results in the transitions between quiescence and spiking states in a bursting attractor, since it causes the alternation between an equilibrium point and a limit cycle. Though a lot of bursting attractors related to Hopf bifurcation have been obtained, while how many types of bursting attractors may appear in the neighborhood of a Hopf bifurcation point still remains an open problem. Here we try to answer the question based on the normal form up to the seventh order of a vector field with Hopf bifurcation at the origin. By introducing a low-frequency external excitation to the normal form, we derive the equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying excitation, based on which four typical cases corresponding to qualitatively different evolution of the behaviors are presented. Accordingly, four types of quasi-periodic bursting oscillations, i.e. Hopf/Hopf bursting attractors with (1) and (2) without saddle on the limit cycle bifurcation in the spiking oscillations, respectively, (3) fold/Hopf/fold/Hopf bursting attractors, and (4) bursting attractor with saddle on limit cycle bifurcation at the transition, are obtained with the increase of the exciting amplitude, the mechanism of which is obtained by overlapping equilibrium branches together with bifurcations in generalized autonomous system and transformed phase portrait. It is found that when two coexisted stable attractors are separated by an unstable attractor on the phase plane in generalized autonomous system, the influence of one of the stable attractors on the dynamics of full system may disappear. Furthermore, when the window for a stable attractor between two bifurcation points along slow-varying parameter axis is relatively short, the effect of the attractor as well as the bifurcation can only appear when the external excitation changes slowly enough. The amplitude of the repetitive spiking oscillations may vary according to two different limit cycles separated by saddle on limit cycle bifurcation, at which a sudden change of the oscillating amplitude can be observed.