Spontaneous explosive is an abrupt transition to collective behavior taking place in heterogeneous networks when the frequencies of the nodes are positively correlated with the node degree. This explosive transition was conjectured to be discontinuous. Indeed, numerical investigations reveal a hysteresis behavior associated with the transition. Here, we analyze explosive synchronization in star graphs. We show that in the thermodynamic limit the transition to (and out of) collective behavior is indeed discontinuous. The discontinuous nature of the transition is related to the nonlinear behavior of the order parameter, which in the thermodynamic limit exhibits multiple fixed points. Moreover, we unravel the hysteresis behavior in terms of the graph parameters. Our numerical results show that finite-size graphs are well described by our predictions.
We analyze star-type networks of phase oscillators by virtue of two methods. For identical oscillators we adopt the Watanabe-Strogatz approach, that gives full analytical description of states, rotating with constant frequency. For nonidentical oscillators, such states can be obtained by virtue of the self-consistent approach in a parametric form. In this case stability analysis cannot be performed, however with the help of direct numerical simulations we show which solutions are stable and which not. We consider this system as a model for a drum orchestra, where we assume that the drummers follow the signal of the leader without listening to each other and the coupling parameters are determined by a geometrical organization of the orchestra. In the various studies of synchronization of globally coupled ensembles of oscillators, usually, it is assumed that global coupling is produced by mean fields that act directly on each oscillator. However, in many natural systems (for example arrays of Josephson junctions), the global field is produced by a dynamical unit.In this paper we consider the case when this unit is a limit cycle oscillator, that can be described by a phase equation, similar to that for the other oscillators in the ensemble. Such configuration of oscillators is often called star-type network. In the first part of the paper we study the case of identical oscillators by virtue of the Watanabe-Strogatz approach that gives full analytical analysis of steady solutions. In the second part we examine inhomogeneous case, when the parameters of the coupling between each oscillator and the central element are different. Such network can be considered as a model for Japanese drums orchestra, consisting of a battery of drummers and a leader, where the drummers only follow the signal from the leader. With the help of the self-consistent approach we find solutions when the global field rotates uniformly. These solutions are obtained semi-analytically in a parametric form. In this work we analyze the case when the distribution of the parameters is determined by the geometric organization of the oscillators. However, this approach can be applied for an arbitrary distribution of the parameters.
The phenomenon of “remote synchronization” (RS), first observed in a star network of oscillators, involves synchronization of unconnected peripheral nodes through a hub that maintains independent dynamics. In the RS regime the central hub was thought to serve as a passive gate for information transfer between nodes. Here, we investigate the physical origin of this phenomenon. Surprisingly, we find that a hub node can drive remote synchronization of peripheral oscillators even in the presence of a repulsive mean field, thus actively governing network dynamics while remaining asynchronous. We study this novel phenomenon in complex networks endowed with multiple hub-nodes, a ubiquitous feature of many real-world systems, including brain connectivity networks. We show that a change in the natural frequency of a single hub can alone reshape synchronization patterns across the entire network, and switch from direct to remote synchronization, or to hub-driven desynchronization. Hub-driven RS may provide a mechanism to account for the role of structural hubs in the organization of brain functional connectivity networks.
We study synchronization properties of coupled oscillators on networks that allow description in terms of global mean field coupling. These models generalize the standard Kuramoto-Sakaguchi model, allowing for different contributions of oscillators to the mean field and to different forces from the mean field on oscillators. We present the explicit solutions of self-consistency equations for the amplitude and frequency of the mean field in a parametric form, valid for noise-free and noise-driven oscillators. As an example, we consider spatially spreaded oscillators for which the coupling properties are determined by finite velocity of signal propagation.
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