We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are "conformists"; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are "contrarians"; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population's mean natural frequency.

We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters which measure synchronization of phases and frequencies are introduced and analyzed by means of dynamic simulations and finite-size scaling. Phase synchronization is observed to emerge in the presence of even a tiny fraction P of shortcuts and to display saturated behavior for P > ∼ 0.5. This indicates that the same synchronizability as the random network (P = 1) can be achieved with relatively small number of shortcuts. The transient behavior of the synchronization, obtained from the measurement of the relaxation time, is also discussed. Systems of coupled nonlinear oscillators, which serve as prototype models for various oscillatory systems in nature, have attracted much attention. Those systems exhibit remarkable phenomena of collective synchronization, which have been observed in a variety of physical, biological, and chemical systems [1]. Up to date, existing studies on collective synchronization have mostly been performed either on the local regular networks such as d-dimensional cubic lattices or on the globally connected geometry. In recent years, there has been suggested the possibility that a number of diverse systems in nature may have the same topological structure as the smallworld networks [2], which are intermediate of the local regular networks and the fully random networks. Such small-world networks are usually characterized by two interesting features: high clustering, which is a characteristic of regular networks, and the short path length, which is typically observed in random networks [2]. Most studies on small-world networks have been focused on the geometrical and topological characterization of the networks, with little attention paid to dynamics defined on them. Recently, some studies have considered dynamical systems put on small-world networks [3,4], where such desirable features as faster propagation of information, better computational power, and stronger synchronizability have been observed. In Ref.[3], frequency synchronization on the small-world network has been noticed in the presence of a small amount of randomly rewired connections and the possibility of the transition to global entrainment with the mean-field nature has been pointed out. However, quantitative analysis has not been performed and proper understanding is still lacking. For example, the critical rewiring probability beyond which true long-range order is present at finite coupling strength has not been addressed.In this paper we study the detailed aspects of the collective synchronizations on small-world networks, as the rewiring probability and the coupling strength are varied.In general, frequency synchronization can be attained without synchronization of phases, and we explore both to investigate the synchronizationdesynchronization transition. Via careful finite-size scaling, we find the following: (i) Phase synchronization as well as frequency one, which is absent in one-dimensional regula...

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Synchronization occurs in many natural and technological systems, from cardiac pacemaker cells to coupled lasers. In the synchronized state, the individual cells or lasers coordinate the timing of their oscillations, but they do not move through space. A complementary form of self-organization occurs among swarming insects, flocking birds, or schooling fish; now the individuals move through space, but without conspicuously altering their internal states. Here we explore systems in which both synchronization and swarming occur together. Specifically, we consider oscillators whose phase dynamics and spatial dynamics are coupled. We call them swarmalators, to highlight their dual character. A case study of a generalized Kuramoto model predicts five collective states as possible long-term modes of organization. These states may be observable in groups of sperm, Japanese tree frogs, colloidal suspensions of magnetic particles, and other biological and physical systems in which self-assembly and synchronization interact.