Asymptotic Formation and Orbital Stability of Phase-Locked States in Kuramoto–Lohe Type Synchronization Models on Lie Groups

The paper formulates the generalized Kuramoto–Lohe model on a Lie group (equation (1.1)) under Hypothesis (H) and proves Theorem 1.1, which includes (1) existence/near-uniqueness of a phase-locked state, (2) global existence and local stability for clustered data, (3) asymptotic phase-locking, (4) orbital stability, and (5) synchronization of normalized speeds . The proof proceeds by passing to relative coordinates Y_{ij} = log(X_i X_j^{-1}) using functional calculus (formulas (2.5)–(2.8)) and deriving the reduced system (3.6) and two Grönwall inequalities ((3.11), (3.14)) that yield an exponential diameter decay and convergence of two flows, respectively . Orbital stability is then obtained by constructing time-dependent normalizations Zi(t) that converge, producing a phase-locked state {Z_i^∞ exp(M t)} and exponential decay of (dL_{X_i}^{-1})_e Ẋ_i(t) − M . The candidate model solution proves the same claims: it derives the exact log-coordinate evolution ż_{ij} = dexp_{z_{ij}}^{-1}(V_j − Ad_{Y_{ij}} V_i), obtains a one-sided Dini derivative inequality D^+ r ≤ −κ c_2 r + C‖H‖∞, and then uses an implicit function theorem (on the gauge slice Σ_i log X_i^∞ = 0) to construct the phase-locked state and show orbital stability. This aligns with the paper’s structure and conclusions. The main methodological difference is that the paper constructs the phase-locked state via limits of solutions (using (3.6), (3.11), (3.14)) , whereas the candidate solution uses a direct IFT on the algebraic system (1.2) after gauge-fixing. Both arguments are sound given (H), the local BCH/functional-calculus expansions, and a coercive inner product ensuring ⟨(dφ)_e v, v⟩ ≥ λ|v|^2, which the paper enforces by a suitable choice of right-invariant metric (2.1) and the candidate enforces via a Lyapunov inner product P. The conclusions (1)–(5) match exactly .