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Coupled limit-cycle oscillators exhibit interesting collective phenomena, like synchronization and pattern formation. Effective models of the classical phase dynamics in these systems have been very successful in describing these effects. Important examples are the canonical Kuramoto model and the Kuramoto-Sakaguchi model. In this thesis, we study a closely related, slightly more general effective phase model, which we call Hopf-Kuramoto model. We focus on the phase dynamics in one-dimensional and two-dimensional lattices.
One central topic is the pattern formation in the deterministic model. As a main result, we present the pattern phase diagram for two-dimensional arrays. This diagram illustrates which patterns are relevant in the long-time dynamics, after starting from random initial conditions, in dependence on the parameters of the model. We examine details of important stationary and non-stationary patterns. This includes the shape and movement of spiral structures, as well as their influence on correlations. Regarding one-dimensional systems, we find smooth stationary patterns with characteristic defects and solitary-wave-like structures in different limiting cases of our model.
Subsequently, we discuss the stochastic dynamics. We study the effects of noise on some of the patterns found in the deterministic case. We then continue with an analysis of a limiting case of the Hopf-Kuramoto model, the noisy Kuramoto-Sakaguchi model. For smooth phase fields, this model is related to the Kardar-Parisi-Zhang model of surface growth. This enables us to explain scaling properties of the phase field with time, as well as a sudden desynchronization process which we find in simulations.
As an example of a system where our model is applicable, we discuss future optomechanical arrays. Moreover, we show that the derivation of the Hopf-Kuramoto model is based on very general assumptions about the dynamics of nonlinear oscillators close to their limit cycle. Hence, our results are relevant for a large class of experiments on arrays of locally coupled oscillators.