Creating bound states in excitable media by means of nonlocal coupling

Abstract

We consider pulses of excitation in reaction-diffusion systems subjected to nonlocal coupling. This coupling represents long-range connections between the elements of the medium; the connection strength decays exponentially with the distance. Without coupling, pulses interact only repulsively and bound states with two or more pulses propagating at the same velocity are impossible. Upon switching on nonlocal coupling, pulses begin to interact attractively and form bound states. First we present numerical results on the emergence of bound states in the excitable Oregonator model for the photosensitive Belousov-Zhabotinsky reaction with nonlocal coupling. Then we show that the appearance of bound states is provided solely by the exponential decay of nonlocal coupling and thus can be found in a wide class of excitable systems, regardless of the particular kinetics. The theoretical explanation of the emergence of bound states is based on the bifurcation analysis of the profile equations that describe the spatial shape of pulses. The central object is a codimension-4 homoclinic orbit which exists for zero coupling strength. The emergence of bound states is described by the bifurcation to 2-homoclinic solutions from the codimension-4 homoclinic orbit upon switching on nonlocal coupling. We stress that the high codimension of the bifurcation to bound states is generic, provided that the coupling range is sufficiently large.