Complicated behavior and mechanism of Chen system with slowly variable control

Since most of the work relevant to the control of Chen system is based on the same time scale, and the results associated with the coupled systems with two time scales are mainly for the cases with only one slow variable, in this paper we investigate the dynamical evolution of the Chen system with the controller described by Duffing oscillator on a slow time scale, which implies two slow variables may be involved in the coupled vector field. Different types of bursting oscillations such as the symmetric fold/fold bursting, symmetric fold/Hopf bursting, symmetric homoclinic/homoclinic bursting, and mechanism are presented, revealing that the mutual influence between the two slow variables may cause the bursting behaviors similar to those in the periodic excited systems.