Bursting Oscillations in a Vector Field with Fold-Hopf Bifurcation at the Origin via Low-Frequency Excitation.

Purpose: To classify bursting oscillations according to different co-dimension of bifurcation is meaningful to the theory of slow-fast dynamics. The paper tried to show all possible bursting oscillations in the neighborhood of fold-Hopf bifurcation point in a vector field. Methods: The study focused on two typical cases associated with fold-Hopf bifurcation in which one of two universal unfolding parameters of the normal form was replaced by low-frequency excitation. By employing the modified slow–fast analysis method, together with the bifurcation analysis of the generalized autonomous system, we presented the dynamical evolution of the vector field. Results: Three forms of 2-D bursting attractors in the full system had been obtained, the mechanism of which was obtained by overlapping the transformed phase portrait and the bifurcation of the generalized autonomous system. Non-synchronization between the state variables was observed, while the effect of some stable equilibrium states on full dynamics disappeared because of the short window for the existence. Fold limit cycle bifurcation resulted in dramatic change of spiking amplitude, which varied according to two different stable cycles. Conclusions: By considering the influence of stable attractors in every region divided by the bifurcation sets on the slow–fast dynamics, we obtained all the types of bursting oscillations associated with the special bifurcation. The scheme can be employed to investigate the slow–fast dynamics with other high co-dimensional bifurcations. [ABSTRACT FROM AUTHOR]