Bursting oscillations with multiple modes in a vector field with triple Hopf bifurcation at origin

The mechanism of bursting oscillations in high dimensional systems with the coupling of two scales is an open problem in nonlinear dynamics, since there may coexist stable attractors in the fast subsystem with different bifurcations. Here we consider the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at origin. When slow-varying external excitation is introduced, with the increase of the exciting amplitude, Hopf bifurcations in the three sub-planes occur in turn, which leads to three stable limit cycles. The dynamics may evolve from a 2-D torus bursting oscillations with two-mode and three-mode, respectively, the mechanism of which can be presented by overlapping the transformed phase portrait and the equilibrium states as well as the bifurcations of the fast subsystem. Synchronization and non-synchronization between the state variables in different sub-planes can be observed, which is explained by the bifurcations. An interesting phenomenon is found that the three-mode bursting oscillations still behave in quasi-periodic type instead of high dimensional torus. The projections of the trajectory in the sub-planes alternate between quiescence and repetitive spiking state with the oscillating frequency approximated at the frequency of the corresponding limit cycle.