Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales

A controlled Lorenz model with fast-slow effect has been established, in which there exist order gap between the variables associated with the controller and the original Lorenz oscillator, respectively. The conditions of fold bifurcation as well as Hopf bifurcation for the fast subsystem are derived to investigate the mechanism of the behaviors of the whole system. Two cases in which the equilibrium points of the fast subsystem behave in different characteristics have been considered, leading to different dynamical evolutions with the change of coupling strength. Several types of bursting phenomena, such as fold/fold burster, fold/Hopf burster, near-fold/Hopf burster, fold/near-Hopf buster have been observed. Theoretical analysis shows that the bifurcations points which connect the quiescent state and the repetitive spiking state agree well with the turning points of the trajectories of the bursters. Furthermore, the mechanism of the period-adding bifurcations, resulting in the rapid change of the period of the movements, is presented.