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We propose an infinite Kuramoto model for a countably infinite set ofKuramoto oscillators and study its emergent dynamics for two classes of networktopologies. For a class of symmetric and row(or column)-summable networktopology, we show that a homogeneous ensemble exhibits completesynchronization, and the infinite Kuramoto model can cast as a gradient flow,whereas we obtain a weak synchronization estimate, namely practicalsynchronization for a heterogeneous ensemble. Unlike with the finite Kuramotomodel, phase diameter can be constant for some class of network topologieswhich is a novel feature of the infinite model. We also consider a second classof network topology (so-called a sender network) in which coupling strengthsare proportional to a constant that depends only on sender's index number. Forthis network topology, we have a better control on emergent dynamics. For ahomogeneous ensemble, there are only two possible asymptotic states, completephase synchrony or bi-cluster configuration in any positive coupling strengths.In contrast, for a heterogeneous ensemble, complete synchronization occursexponentially fast for a class of initial configuration confined in a quarterarc.

On the emergent dynamics of the infinite set of Kuramoto oscillators