Mixed-mode bursting oscillations in the neighborhood of a triple Hopf bifurcation point induced by parametric low-frequency excitation

The mechanism of bursting oscillations in high dimensional system is one of the key topics for the theoretical framework of slow-fast dynamics, especially when higher co-dimensional bifurcation involves the vector field. This paper focuses on a six-dimensional vector field with triple Hopf bifurcation at the origin to explore possible bursting oscillations as well as the mechanism. By introducing a parametric excitation with low-frequency to the normal form, several types of bursting attractors can be obtained, the mechanism of which is presented upon the bifurcation analysis of the generalized autonomous system. With the increase of exciting amplitude, different types of bifurcations may influence the structure of the movement, which may cause a 2-D torus to evolve to 2-D bursting oscillations with single-mode, two-mode and three-mode, respectively, with different projections on three sub-planes. Because of the existence of three independent sub-planes, the bifurcations on different sub-planes may lead to the synchronization and non-synchronization between the associated state variables. Furthermore, the inertia of the movement may lead to the disappearance of the effect of the stable attractor with short window in the generalized autonomous system.