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Emergent behaviors of high-dimensional Kuramoto models on Stiefel manifolds

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.