Propagation of small perturbations in synchronized oscil- lator networks

– We study the propagation of a harmonic perturbation of small amplitude on a network of coupled identical phase oscillators prepared in a state of full synchronization. The perturbation is externally applied to a single oscillator, and is transmitted to the other oscillators through coupling. Numerical results and an approximate analytical treatment, valid for random and ordered networks, show that the response of each oscillator is a rather welldefined function of its distance from the oscillator where the external perturbation is applied. For small distances, the system behaves as a dissipative linear medium: the perturbation amplitude decreases exponentially with the distance, while propagating at constant speed. We suggest that the pattern of interactions may be deduced from measurements of the response of individual oscillators to perturbations applied at different nodes of the network. Synchronization phenomena have recently attracted a great deal of attention [1]. The emergence of synchronous behaviour in large ensembles of interacting dynamical elements is a paradigmatic form of collective self-organization, typical of a vast class of natural systems ranging from complex chemical reactions to large-scale biological processes [2, 3]. Many of these phenomena are successfully reproduced by relatively simple mathematical models. The most basic form of synchronized dynamics, which can be achieved in an ensemble of identical periodic oscillators subject to attractive coupling, is full synchronization [4]. In such state, the individual motions of all oscillators coincide. A network of fully-synchronized coupled oscillators may be viewed as an active medium in a highly coherent collective state. A relevant question regarding the dynamics of this system is how the medium responds to an external perturbation which affects its coherence. Due to the coupling between oscillators, the perturbation should spread through the medium. Moreover, if dissipative mechanisms are acting and the state of full synchronization is stable, the perturbation should die out as the distance from the point where it is applied becomes larger. The properties of this propagation phenomenon provide a dynamical characterization of the synchronized medium, much like the propagation of a defect in an extended system. This is the problem that we address in this Letter. Kuramoto’s model of coupled phase oscillators [5] provides the basis for our model. The state of each oscillator is described by a phase variable φi ∈ [0, 2π) which, in the absence of