New Findings on Dynamic Systems from Kyushu University Summarized (A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model)
By a News Reporter-Staff News Editor at Journal of Mathematics -- Investigators publish new report on Dynamic Systems. According to news reporting originating in Fukuoka, Japan, by VerticalNews editors, the research stated, "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."
The news reporters obtained a quote from the research from Kyushu University, "A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength K between oscillators, which is a parameter of the system, is smaller than some threshold K-C, the de-synchronous state (trivial steady state) is asymptotically stable, while if K exceeds K-C, 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."
According to the news reporters, the research concluded: "These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto."
For more information on this research see: A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model. Ergodic Theory and Dynamical Systems, 2015;35():762-834. Ergodic Theory and Dynamical Systems can be contacted at: Cambridge Univ Press, 32 Avenue Of The Americas, New York, NY 10013-2473, USA. (Cambridge University Press - www.cambridge.org; Ergodic Theory and Dynamical Systems - journals.cambridge.org/action/displayJournal?jid=ETS)
Our news correspondents report that additional information may be obtained by contacting H. Chiba, Kyushu University, Inst Math, Fukuoka 8190395, Japan.
Keywords for this news article include: Asia, Japan, Fukuoka, Dynamic Systems
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