Dynamics of bow-tie shaped bursting: Forced pendulum with dynamic feedback

A detailed study is performed on the parameter space of the mechanical system of a driven pendulum with damping and constant torque under feedback control. We report an interesting bow-tie shaped bursting oscillatory behaviour, which is exhibited for small driving frequencies, in a certain parameter regime, which has not been reported earlier in this forced system with dynamic feedback. We show that the bursting oscillations are caused because of a transition of the quiescent state to the spiking state by a saddle-focus bifurcation, and because of another saddle-focus bifurcation, which leads to cessation of spiking, bringing the system back to the quiescent state. The resting period between two successive bursts (T_rest) is estimated analytically.