Bursting phenomenon as well as the bifurcation mechanism of self-excited Lorenz system
Abstract
The Lorenz system under the influence of self-excited oscillator, which behaves in stable periodic limit cycle, may exhibit rich special nonlinear phenomena when different time scales are introduced. Bifurcation conditions of the fast subsystem have been derived via the analysis of the equilibrium points as well as the corresponding characteristics. Upon the investigation of the dynamical evolution of coupled the system, it is pointed out that, fold/fold bursting may appear when the excitation increases to meet with the conditions of fold bifurcation of the fast subsystem, in which the quiescent state corresponds to the equilibrium state of the fast subsystem, while the spiking state is related to the fluctuation around the foci of the fast subsystem. The bifurcation mechanism is presented, which is applied to the exploration of the bursting evolution with the variation of the parameters. It is found that, with the increase of the excitation, bursters may change in the forms that, near the center of the spiking oscillation, the trajectories of the whole system move along with the equilibrium states of the fast subsystem, the distance of which is approximated at the difference between amplitude of the self-excitation and the values of the slow variable at the fold bifurcation.
