On Synchronous Steady States and Internal Models of Diffusively Coupled Systems

We investigate the problem of synchronizing nonidentical nonlinear dynamical systems by means of generalized diffusive couplings. The focus is not on the actual solution to the problem but on the derivation of necessary conditions for the existence of such a solution despite the systems possessing nonidentical models. We show that for the problem to be solvable a synchronous steady state needs to exist. This condition leads to the requirement that all individual system models need to embed an internal model of some common endosystem. The latter condition is expressed in terms of nonlinear partial differential equations. The conditions derived in this paper are related to those known from the theory of output regulation.