A MODEL FOR THE DYNAMICAL PROCESS OF CLUSTER FORMATION.

The formation of several clusters, arising from attracting forces between non-identical entities or agents, is a phenomenon observed in diverse fields. Think of people gathered through a mutual interest, pricing policy of different distributors in an economic market or clustering of oscillators in brain cells. We introduce a dynamical model of mutually attracting agents for which we prove that the long term behavior consists of agents organized into several groups or clusters. We have completely characterized the cluster structure by means of a set of inequalities in the parameters of the model and have identified the intensity of the attraction as a key parameter governing the transition between different cluster structures. We illustrate the relation with the Kuramoto model on interconnected oscillators and we discuss an application on compartmental systems.