A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is sin θ, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not sin θ, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.

We show that the mathematical structure of Tsallis entropy underlies an important and ubiquitous problem in nonlinear science related to an efficient synchronization of weakly forced nonlinear oscillators. The maximization of the locking range of oscillators with the use of phase models is analyzed with general constraints that encompass forcing waveform power, magnitude, or area. The optimization problem is then recasted as a general form of Tsallis entropy maximization. The solution of these optimization problems is shown to be a direct consequence from Hölder's inequality. The resulting new maximization principle is confirmed in numerical simulations and experiments with chemical oscillations with nickel electrodissolution. While weakly nonlinear oscillators have generic optimal waveforms (sinusoidal, 50% duty cycle square wave, and equally paced bipolar pulses for power-, area-, and magnitude-constraints, respectively), strongly nonlinear oscillators require more complex waveforms such as smooth, square, and pulse ones.

We investigate the diffusion coefficient of the time integral of the Kuramoto order parameter in globally coupled nonidentical phase oscillators. This coefficient represents the deviation of the time integral of the order parameter from its mean value on the sample average. In other words, this coefficient characterizes long-term fluctuations of the order parameter. For a system of N coupled oscillators, we introduce a statistical quantity D, which denotes the product of N and the diffusion coefficient. We study the scaling law of D with respect to the system size N. In other well-known models such as the Ising model, the scaling property of D is D∼O(1) for both coherent and incoherent regimes except for the transition point. In contrast, in the globally coupled phase oscillators, the scaling law of D is different for the coherent and incoherent regimes: D∼O(1/N(a)) with a certain constant a>0 in the coherent regime and D∼O(1) in the incoherent regime. We demonstrate that these scaling laws hold for several representative coupling schemes.

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