Antiphase Versus In-Phase Synchronization of Coupled Pendulum Clocks and Metronomes

The paper reduces the dynamics to slow amplitude–phase equations (their Eqs. (1), (3), (5)) and states the synchronized fixed points at A1 = A2 = Ac with ψ ∈ {0, π}, after defining α and Ac via A_c^{(σ)}; it then gives the exact stability thresholds r1, r2 and asserts the corresponding stability of in-phase and antiphase synchronization (and that the smaller-amplitude fixed points at A_c^{(-1)} are unstable) . The candidate solution performs the same Jacobian-based linearization at (Ac, Ac, ψ*), exhibits the block structure, isolates the symmetric eigenvalue a < 0 at Ac = A_c^{(+1)}, and derives the two-by-two block determinant condition det > 0, yielding r1, r2 and their transversality via dp/dr = −Ac/8. It also reconciles notation by showing β = 8/Ac^2, which matches the paper’s r1, r2 formulas when expressed in terms of α and Ac. Hence both the result and the proof approach (Jacobian eigenvalue analysis about synchronized equilibria) align closely with the paper’s method and conclusions .