Hayato Chiba

A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model

The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as coupled phase oscillators. In this paper, a bifurcation structure of the infinite-dimensional Kuramoto model is investigated. A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

K

Center manifold reduction for large populations of globally coupled phase oscillators.

between oscillators, which is a parameter of the system, is smaller than some threshold

C 1 Approximation of Vector Fields based on the Renormalization Group Method

{K}_{c}

Approximation of center manifolds on the renormalization group method

, the de-synchronous state (trivial steady state) is asymptotically stable, while if

Stability of an [N/2]-dimensional invariant torus in the Kuramoto model at small coupling

K

The Third, Fifth and Sixth Painlevé Equations on Weighted Projective Spaces

exceeds

Reduction of weakly nonlinear parabolic partial differential equations

{K}_{c}

Lie equations for asymptotic solutions of perturbation problems of ordinary differential equations

, a non-trivial stable solution, which corresponds to the synchronization, bifurcates from the de-synchronous state. One of the difficulties in proving the conjecture is that a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, has the continuous spectrum on the imaginary axis. Hence, the standard spectral theory is not applicable to prove a bifurcation as well as the asymptotic stability of the steady state. In this paper, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid the continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator will be estimated with the aid of the spectral theory on a rigged Hilbert space to prove the linear stability of the steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite-dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimension because of the continuous spectrum on the imaginary axis. These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto.

A center manifold reduction of the Kuramoto-Daido model with a phase-lag

The renormalization group (RG) method is one of the singular perturbation methods which is used in searching for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, and inheritance of symmetries from those for the original equation to those for the RG equation are proved. Further, it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper)normal forms theory, the center manifold reduction, the geometric singular perturbation method, and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.