PUB - Publikationen an der Universität Bielefeld

In this thesis we investigate the exchange of energy and the evolution of phase differences in circulant networks of weakly noise-coupled commensurate oscillators. We introduce a generalized synchronization concept called eigenmode synchronization which beyond the classical notions of in-phase and anti-phase synchronization, also distinguishes between other phase-locking configurations corresponding to eigenmodes of the uncoupled system. We study the interplay of deterministic and multiplicative noise-coupling and in particular verify that the latter can amplify some of the system's eigenmodes. Such an amplification is shown to induce an asymptotic eigenmode synchronization which even persists in the presence of an additive noise perturbation. Application of the Euler-Fermat theorem from number theory, finally allows us to relate a class of circulant noise-coupling topologies to their induced synchronization patterns. Specifically, we will identify critical numbers of oscillators at which these induced synchronization patterns change. The synchronization results are obtained by studying a complex outer product process which captures all of the uncoupled system's first integrals. In the weak coupling limit, this process is shown to satisfy an averaging principle, i.e. after time-rescaling it weakly converges towards an effective limiting process governed by an averaged drift and diffusion term. This averaging result is proven by adaptation of an averaging principle based on the generalized convergence of Dirichlet forms. The effective limiting process is determined by application of the residue theorem. This allows us to identify a class of nonlinear perturbations of the drift term which yield a vanishing contribution to the evolution of the effective process.