Effects of time delay and connection probability on self-sustained oscillations and synchronization transitions in excitable Erdös–Rényi random networks

Abstract In this paper we have systematically investigated the effects of time delay and connection probability on self-sustained oscillations and synchronization transitions in excitable Erdos–Renyi random networks (EERRNs) via order parameters and space-time plots. At certain connection probabilities, time delay can not only sustain the oscillations effectively in some cases, but also enhance synchronization performance remarkably and can induce synchronization transitions. The oscillatory behaviour of order parameter is discovered. It is induced by the phenomenon of bistability, which contains the states of synchronous oscillation and double-period oscillation. Importantly, the mechanism and characteristic of double-period oscillation are discussed in detail. At certain time delays, three distinct parameter regions of connection probability are revealed for self-sustained oscillations, i.e., the oscillation region, the transition region and the damping region. The synchronization performance can be remarkably enhanced by increasing connection probability in the oscillation region. Furthermore, the oscillatory behaviour of order parameter is detected in the transition region, which is also induced by the phenomenon of bistability. Finally, the optimal combinations of time delay and connection probability on self-sustained oscillations and synchronization transitions in EERRNs are exposed explicitly.