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

TL;DR

This paper rigorously analyzes the bifurcation structure of the infinite dimensional Kuramoto model, proving the existence of a finite-dimensional center manifold and confirming Kuramoto's bifurcation conjecture for synchronization.

Contribution

It develops spectral theory on a space of generalized functions and proves the existence of a finite-dimensional center manifold for the infinite dimensional Kuramoto model.

Findings

De-synchronous state is stable when coupling strength is below threshold.

Synchronization bifurcates from the de-synchronous state when coupling exceeds threshold.

Spectral decomposition enables stability analysis of the system.

Abstract

The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as a coupled phase oscillators. In this paper, a bifurcation structure of the infinite dimensional Kuramoto model is investigated. For a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid a 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 is calculated by using the spectral decomposition to prove the linear stability of a 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 dimensional because of the continuous spectrum on the imaginary axis. The results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto. If the coupling strength $K$ between oscillators is smaller than some threshold $K_c$, the de-synchronous state proves to be asymptotically stable, and if $K$ exceeds $K_c$, a nontrivial solution, which corresponds to the synchronization, bifurcates from the de-synchronous state.