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

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.