Bursting oscillations in a piecewise system with time delay under periodic excitation: Theoretical and experimental observation of real electrical bursting signals using microcontroller

In this paper, we report the existence of bursting oscillations in systems governed by a second-order differential equation with only one signum nonlinearity term and time-delay feedback, driven by the slowly varying external force. Depending on the sign of the strength of signum nonlinearity, two cases are studied: two-well and single-well potentials. The external force acts as the control parameter, the stability of equilibrium points is first discussed and the condition for Hopf bifurcation is established. Secondly, we present the effect of time delay on bursting oscillations when periodic forcing changes slowly. Our results show that the time delay is responsible for the appearance or disappearance of the bursting phenomenon. It is also found that the amplitude, the number of peaks and period of bursting oscillation depend on the value of the time delay. The bursting shapes obtained theoretically are exhibited experimentally using the real microcontroller simulation.