In-phase synchronization is a stable state of identical Kuramoto oscillators coupled on a network with identical positive connections, regardless of network topology. However, this fact does not mean that the networks always synchronize in-phase because other attractors besides the stable state may exist. The critical connectivity μc is defined as the network connectivity above which only the in-phase state is stable for all the networks. In other words, below μc, one can find at least one network that has a stable state besides the in-phase sync. The best known evaluation of the value so far is 0.6828…≤μc≤0.7889. In this paper, focusing on the twisted states of the circulant networks, we provide a method to systematically analyze the linear stability of all possible twisted states on all possible circulant networks. This method using integer programming enables us to find the densest circulant network having a stable twisted state besides the in-phase sync, which breaks a record of the lower bound of the μc from 0.6828… to 0.6838…. We confirm the validity of the theory by numerical simulations of the networks not converging to the in-phase state.

A coupled phase-oscillator model consists of phase oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the nonsynchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality classes is reduced to one and the critical exponent is shared in the considered models having coupling functions up to second harmonics with unimodal and symmetric natural frequency distributions.

Synchronization among rhythmic elements is modeled by coupled phase-oscillators each of which has the so-called natural frequency. A symmetric natural frequency distribution induces a continuous or discontinuous synchronization transition from the nonsynchronized state, for instance. It has been numerically reported that asymmetry in the natural frequency distribution brings new types of bifurcation diagram having, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method of five types of bifurcation diagrams including the new ones, paying attention to the generality of the theory. The oscillation and the jump from partially synchronized states are discussed respectively by the linear analysis around the nonsynchronized state and by extending the amplitude equation up to the third leading term. The theoretical classification is examined by comparing with numerically obtained one.

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This paper introduces the "Masters Classic" tennis event and discusses it from the viewpoint of sports promotion. As a result of studying this event, the possibility of "spectator tennis" was suggested. At the same time, the importance of non-compelled management and continuing examination of this event in planning a strategy for the future and for making improvements were pointed out. Moreover, the necessity of deciding on a suitable admission fee was confirmed.