Complete Entrainment of Lohe Oscillators under Attractive and Repulsive Couplings

Abstract

The dynamic behavior of Lohe oscillators on the unit sphere is examined under attractive and repulsive couplings. We introduce an order parameter measuring the degree of synchronization, which is defined by the modulus of a centroid of positions, and study the dynamics of this parameter. It is found that this order parameter completely characterizes equilibria up to constant motion for identical oscillators. Considering these identical oscillators, we show that the order parameter evolves from nonzero values toward the unit value or zero exponentially fast, depending on the nature of the couplings. This improves and generalizes our earlier work [D. Chi, S.-H. Choi, and S.-Y. Ha, J. Math. Phys., 55 (2014), 052703] on the complete synchronization for Lohe oscillators under attractive couplings. For identical oscillators with repulsive couplings, we prove that the state to the Lohe system converges to the ground state with zero order parameter. For nonidentical oscillators under the attractive force, we show that the ensemble exhibits a practical synchronization as the coupling strength goes to infinity, enabling us to push the configuration asymptotically close to the completely synchronized state.