A SELF-ORGANIZING NETWORK IN THE WEAK-COUPLING LIMIT

We prove the existence of spatially localized ground states of the dif fusive Haken model. This model describes a self-organizing network who se elements are arranged on a d-dimensional lattice with short-range d iffusive coupling. The network evolves according to a competitive grad ient dynamics in which the effects of diffusion are counteracted by a localizing potential that incorporates an additional global coupling t erm. In the absence of diffusive coupling, the ground states of the sy stem are strictly localized, i.e. only one lattice site is excited. Fo r sufficiently small non-zero diffusive coupling alpha, it is shown an alytically that localized ground states persist in the network with th e excitations exponentially decaying in space. Numerical results estab lish that localization occurs for arbitrary values of ct in one dimens ion but vanishes beyond a critical coupling alpha(c)(d), when d > 1. T he one-dimensional localized states are interpreted in terms of instan ton solutions of a continuum version of the model.