Unstable attractors with active simultaneous firing in pulse-coupled oscillators

Unstable attractors whose nearby points will almost leave the neighborhood have been observed in pulse-coupled oscillators. In this model, an oscillator fires and sends out a pulse when reaching the threshold. In terms of these firing events, we find that the unstable attractors have a simple property hidden in the event sequences. They coexist with active simultaneous firing events. That is, at least two oscillators reach the threshold simultaneously, which is not directly caused by the receiving pulses. We show that the split of the active simultaneous firing events by general perturbations can make the nearby points leave the unstable attractors. Furthermore, this structure can be applied to study the bifurcation of unstable attractors. Unstable attractors can bifurcate due to the failure of establishing active simultaneous firing events.