  We study emergent asymptotic dynamics for the first and second-order
high-dimensional Kuramoto models on Stiefel manifolds which extend the previous
consensus models on Riemannian manifolds including several matrix Lie groups.
For the first-order consensus model on the Stiefel manifold proposed in
[Markdahl et al, 2018], we show that the homogeneous ensemble relaxes the
complete consensus state exponentially fast. On the other hand for a
heterogeneous ensemble, we provide a sufficient condition leading to the
phase-locked state in which relative distances between two states converge to
definite values in a large coupling strength regime. We also propose a
second-order extension of the first-order one by adding an inertial effect, and
study emergent behaviors using Lyapunov functionals such as an energy
functional and an averaged distance functional.
