Dynamics of globally coupled rotators with multiplicative noise
Seunghwan Kim 
, Seon Hee Park , Chang Soo Ryu ,
and Seung Kee Han 
Research Department, Electronics and Telecommunications Research
Institute
P.O. Box 106, Yusong-gu, Taejon 305-600, Korea
Department of Physics, Chungbuk National University
Cheongju, Chungbuk 360-763, Korea
Globally coupled active rotators have been studied as a model system to understand the synchronous oscillations found in the visual cortex and dynamics of Josephson-junction arrays and charge-density waves. The dynamics of the globally coupled active rotators in the weak coupling limit has usually been investigated in the reduced model with effective interaction given by the first Fourier mode. It has been claimed, however [1,2], that higher Fourier mode interactions are indispensible for interesting collective dynamics such as clustering of rotators.
Recently, it has been shown that in this system randomly fluctuating interaction induces the interesting nonequilibrium phenomena [3]: at a critical noise intensity the system undergoes a noise-induced phase transition and is split into clusters. These are pure noise effects and show a route to the clustering phenomena without introducing higher Fourier mode interactions which have usually been considered to be necessary for clustering [4].
In this paper we study the effects of the time-delayed interaction on the system. The inclusion of time delay is natural in the realistic considerations of finite transmission of interactions. In the case of information propagation through a neural network time delay has been demonstrated to have a substantial influence on the temporal characteristics of oscillatory behavior of neural circuits [5]. It has also been shown that even small delay times affect the global dynamics of two-dimensional systems of limit cycle oscillators [6].
We show that in the system of the globally coupled active rotators time delay affects the picture of the phase transitions induced by multiplicative noise suppressing the clustering of the rotators and generates the switching phenomena between the clusters. We discuss the nature of the transitions in detail. Switching in this system can be seen directly in the motion of each rotator: A rotator joins the other cluster right after escape from the cluster to which it belongs. This is a different notion of switching from the one considered in Ref. [4] where the switching occurs between the two two-cluster states.