Cooperative Phenomena in Two-Dimensional Active Rotator Systems

Abstract

Phase transitions of active rotator systems with short-range coupling are discussed. The constituents of the system which we call active rotators are represented by a phase model of a limit-cycle oscillator or an excitable element, i.e., dt/J/dt = w-bsint/J, (Iw/bl 1, we have a stable-unstable pair of fixed points ¢s and ¢u, given by ¢s=arcsin(w/b) and bcos¢s >0, (1· 2) (1· 3) If Ib/wl is slightly greater than 1, the rotator is sensitive to external perturbations. This is because the system is then easily kicked out of its stable equilibrium thus making a long tour before coming back to its original equilibrium. Such a feature is known to be basic to excitable elements. Equation (1·1) may also be looked upon as an equation describing the heavily damped motion of a particle in a periodic potential under the action of a constant driving force. The population model of the elements like (1·1) has been employed in the study of a driven and heavily damped sine-Gordon chain!) and also of the surface roughening transition. 2) In those cases, the elements are supposed to be under the effect of external noises, and nearest-neighbour mutual coupling is assumed. However, the coupling assumed is linear in phase difference, so that the transformation ¢-4 ¢ + 2][ for individual elements does not