  It is known that the Kuramoto model has a critical coupling strength above
which phase-locked states exist, and, by the work of Choi, Ha, Jung, and Kim
(2012), that these phase-locked states are orbitally stable. This property of
admitting orbitally stable phase-locked states is preserved under the
nonabelian generalizations of the Kuramoto model pioneered by Lohe (2009). We
provide a framework for understanding these phenomena, by formulating the
aforementioned models as special cases of a general ODE model describing
populations of particles on Lie groups with pairwise attractive Kuramoto-type
interactions. This general model can be used in producing many specific models
with orbitally stable nonabelian phase-locked states.
