On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds

We study the collective relaxation dynamics appearing in weakly coupled Lohe oscillators in a large coupling regime. The Lohe models on the unit sphere and unitary group were proposed as a nonabelian generalization of the Kuramoto model on the unit circle and their emergent dynamics has been extensively studied in previous literature for some restricted class of initial data based on the Lyapunov functional approach and order parameter approach. In this paper, we extend the previous partial results to cover a generic initial configuration via the detailed analysis on the order parameter measuring the modulus of the centroid. In particular, we present a detailed relaxation dynamics and structure of the resulting asymptotic states for the Lohe sphere model. We also present new gradient flow formulations for the Lohe matrix models with the same one-body Hamiltonians on some group manifolds. As a direct application of this new formulation, we show that every bounded Lohe flow which originated from any initial configuration converges asymptotically.