Sincroniza\c{c}\~ao em Redes Complexas: Estabilidade e Persist\^encia

TL;DR

This paper studies the stability and persistence of synchronization in complex networks of identical oscillators, using graph theory and dynamical systems to derive conditions based on spectral properties and coupling parameters.

Contribution

It introduces a criterion for synchronization stability in diffusively coupled networks using uniform contractions theory, linking it to spectral properties and the global coupling parameter.

Findings

Derived a stability criterion based on spectral properties and coupling strength.

Identified the critical global interaction parameter for synchronization.

Linked the stability conditions to the isolated oscillator dynamics and network Laplacian eigenvalues.

Abstract

We investigate emergence of the global collective behavior in networks of diffusively coupled identical oscillators, which in the established model is an invariant manifold of the motion equations. The interaction is modeled with the graph theory and dynamical systems theory. We use the uniform contractions theory in non-autonomous linear differential equations to address the criterion under the global coupling parameter, which it turns defines the synchronized motion and it stability under small linear and non-linear perturbations. The critical global interaction parameter is given only by the isolated dynamics, by the spectral properties of the coupling function and the second eigenvalue network laplacian.