3-torus, quasi-periodic bursting, symmetric subHopf/fold-cycle bursting, subHopf/fold-cycle bursting and their relation

In this paper, the dynamical behaviors of a perturbed hyperchaotic system is studied. The fast subsystem is examined using local stability and bifurcations, including simple bifurcation, Hopf bifurcation, and fold bifurcation of limit cycle. The results of these analysis are applied to the perturbed hyperchaotic system, where two types of periodic bursting, i.e., symmetric subHopf/fold-cycle bursting and subHopf/fold-cycle bursting, can be observed. In particular, the symmetric subHopf/fold-cycle bursting is new and has not been reported in previous work. With variation of the parameter, subHopf/fold-cycle bursting with symmetric structure may bifurcate into two coexisted subHopf/fold-cycle bursting symmetric to each other. Moreover, 3-torus and quasi-periodic bursting (2-torus) are presented. The relation among 3-torus, quasi-periodic bursting, and symmetric subHopf/fold-cycle bursting is discussed, which suggests that 3-torus may develop to quasi-periodic bursting, while quasi-periodic bursting may further evolve to symmetric subHopf/fold-cycle bursting.