Exponential stable manifold for the synchronized state of the abstract mean field system

TL;DR

This paper studies the exponential stability of synchronized states in abstract mean field systems, establishing the existence of stable manifolds and limit cycles through linear and nonlinear analysis.

Contribution

It introduces a novel framework for analyzing exponential stability and invariant manifolds in mean field systems, linking linear and nonlinear dynamics.

Findings

Existence of an exponentially stable invariant manifold

Proof of stable limit cycles in the system

Relationship established between linear and nonlinear dynamics

Abstract

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.