Exponential stable manifold for some linear system with application to the mean field models

This paper investigates the exponential stability of abstract mean field systems in their synchronized state. We analyze stability by studying the linearized system and demonstrate the existence of an exponentially stable invariant manifold. Our focus is on the equilibrium stability under synchronization. We provide a comprehensive analysis of both linear and nonlinear cases of the system. Additionally, we prove the existence of stable limit cycles and establish a relation between the dynamics in linear and nonlinear frameworks.