Synchronization within synchronization: transients and intermittency in ecological networks

The paper and the model solution both use network symmetry to define an invariant cluster-synchrony manifold, reduce the variational dynamics into tangential/transverse blocks, and use conditional (transverse) Lyapunov exponents to locate thresholds for cluster and global synchronization. They agree on the specific 10-node range-2 ring reflection partition, on the presence of global phase synchronization with intermittent cluster synchronization near ε≈0.038, on the T^{-3/2} laminar-time law at the cluster blowout, on the monotone steepening of the tail under added noise, and on the numerical thresholds ε_CS^c≈0.039 and ε_GS^c≈0.073. The model adds detail (e.g., explicit Laplacian eigenvalues and a reflecting-floor picture for noise) consistent with, though somewhat more specific than, the paper’s random-walk/bias narrative.