Routes to bursting in active control system with multiple time delays

This paper investigates the generation of some novel bursting patterns in active control oscillator with multiple time delays. We present the bursting patterns, including symmetric codimension one and codimension two bursters with the slow variation of periodic excitation item. We calculate the bifurcation conditions of fast subsystem as well as its stability related to the time delay. We also identify some regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Time delay can enhance the spiking performance and obtain the remarkable spiking dynamics even in a very simple model, which enriches the routes to bursting dynamics. Also some numerical simulations are included to illustrate the validity of our study.