We have studied in this paper the dynamics of globally coupled phase oscillators having the Lorentzian frequency distribution with zero mean in the presence of both time delay and noise. Noise may be Gaussian or non-Gaussian in characteristics. In the limit of zero noise strength, we find that the critical coupling strength (CCS) increases linearly as a function of time delay. Thus the role of time delay in the dynamics for the deterministic system is qualitatively equivalent to the effect of frequency fluctuations of the phase oscillators by additive white noise in absence of time delay. But for the stochastic model, the critical coupling strength grows nonlinearly with the increase of the time delay. The linear dependence of the critical coupling strength on the noise intensity also changes to become nonlinear due to creation of additional phase difference among the oscillators by the time delay. We find that the creation of phase difference plays an important role in the dynamics of the system when the intrinsic correlation induced by the finite correlation time of the noise is small. We also find that the critical coupling is higher for the non-Gaussian noise compared to the Gaussian one due to higher effective noise strength. 1. Introduction In this paper we have investigated the synchronization behavior of globally coupled phase oscillators. Recent trends in physics [1–6] imply that it is one of the important issues in basic science such as the Brownian motion, nanomaterials, and biophysics. Biological clocks [7, 8], chemical oscillators [9–11], coupled map lattices [12, 13], and coupled random frequency oscillators [14] are examples where the phenomena of synchronization have been observed. To account the phenomenon, coupled phase oscillators model was introduced by Kuramoto [9–11], and it is popularly known as the Kuramoto model. After that, synchronization in nonlinear systems has been systematically studied and attracted much attention. Several reviews on the developments can be found in [1, 2, 5, 6]. Recently there has been considerable interest in some stochastic systems, whose dynamics are determined by both the present state and the state in the past with the time delay ( ). It has been considered in visual feedback [15, 16] and brain activity [17, 18] to mention a few. Delay is also studied in the coupled oscillator (CO) model [19–23]. In [19] authors showed that intrinsic frequency of the network of limit cycle oscillators decreases as the time delay grows, and for greater delay there is a metastable synchronized state. However, Nakamura
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