Bursting oscillations in a slow-varying periodically excited vector field with Bogdanov–Takens bifurcation

Abstract

The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations.