On the Emergence and Orbital Stability of Phase-Locked States for the Lohe Model

Abstract

We study the emergence and orbital stability of phase-locked states of the Lohe model, which was proposed as a non-abelian generalization of the Kuramoto phase model for synchronization. Lohe introduced a first-order system of matrix-valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase-locked states of the Lohe model. In this paper, we provide an analytical framework to confirm Lohe’s observations of emergent phase-locked states. This extends earlier special results on lower dimensions to any finite dimension. For the construction and orbital stability of phase-locked states, we introduce Lyapunov functions to measure the ensemble diameter and dissimilarity between two Lohe flows, and using the time-evolution estimates of these Lyapunov functions, we present an admissible set of initial states, and show that an admissible initial state leads to a unique phase-locked asymptotic state.