Extinction of Oscillation in a Reaction-Diffusion System

Abstract

We have numerically studied a phenomenon, “the extinction of oscillation”, in a reaction-diffusion system; all the oscillators stop together in a limited region of the system size. In the numerical experiments, we adopted a model of the van der Pol oscillator with or without pacemakers. We found that the extinction of oscillation occurs when the size of the system N is comparatively small. In order to explain the numerical results for the case of no pacemakers, we have developed a theory for the system with the van der Pol oscillators, in which the Dirichlet boundary condition is satisfied. The theory predicts that the system breaks out, extincting the oscillation as expected, when the size of the system N corresponds to a critical size N_c and the oscillators never oscillate for a size N smaller than N_c. The theory also successfully explains the oscillating pattern and its maximum amplitude of the oscillation in the region of a system size larger than N_c.