Yoshiki Kuramoto

Spiral Waves in a Nonlinear Dissipative System

We have shown recentlyll that spiral waves similar in many respects to the observed ones in the Belousov-ZhabotinskyZaikin reaction~ 1 can arise in a self-oscillatory medium. Two basic assumptions, namely, the existence of a stable limit cycle and slow spatial variation of phase, led to a simple time-evolution equation for the phase in a closed form. In the course of the analysis, however, it became clear that the latter assumption leads to a contradiction in describing the central region of the spiral. Though this difficulty was avoided by a phenomenological cutoff procedure, a more adequate nonperturbative approach was hoped to be developed, which we shall do here for a simple two-component model. Let x and y be concentrations (measured from an unstable steady state), and define w=p exp (ifJ) =.x+iy. The model equation we shall now consider