| Abstract: |
| Known as a relatively simple mean-field model of coupled phase oscillators, the Kuramoto model plays an important role in studying the collective synchronization phenomenon. In this talk, the linear stability of incoherent state to the continuum limit of the Kuramoto model with time delay is investigated. Here, it is known that due to the presence of continuous spectrum on the imaginary axis, stability analysis meets obstacles even if no eigenvalue exists on the right complex half-plane. To address it, a generalized spectral theory on a Gelfand triple is utilized. Under some analyticity condition, resolvent operator is extended to a generalized resolvent. It is shown that continuous singularities (continuous spectrum) disappear from the Riemann surface of the generalized resolvent due to the change of topology via the Gelfand triple. Then, the contour deformation is applied in the inverse Laplace representation to show stability in a weak topology. |
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