On the local spectral rigidity of sync manifold for the Justh-Krishnaprasad models

We study the local spectral rigidity of the synchronization (sync) manifold induced by the Justh-Krishnaprasad models under emergent dynamics. For each point on the sync manifold, all perturbative eigensets whose existence is guaranteed by the implicit function theorem remain to be confined within the manifold. More precisely, for a finite-particle system, we establish that under mono-cluster flocking dynamics, the sync manifold retains its local spectral rigidity. Furthermore, for an infinite system, we also show that the sync manifold retains its local spectral rigidity, when a mono-cluster flocking emerges with a central particle. In particular, we provide a sufficient framework to guarantee the mono-cluster flocking with a central particle. Our results imply that in mono-cluster regimes, the alternative solution branches of the nonlinear Laplacian spectral problem will not occur in the neighborhood of the sync manifold except at the origin in both finite and infinite systems.