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

Dynamical Systems