This work shows a new approach to the study of dynamic systems that act on a graph $ \mathcal{G} = (V, E) $ and that synchronize. As a first example, we take a simple linear system, known as the Laplacian associated with the adjacency matrix of $\mathcal{G}$ the ODE on $\mathbb{R}^{|V|}$

$\frac{d}{dt}x=L_\mathcal{G}(x),$

where $|V|=n$ and $L_\mathcal{G}$ is the Laplacian matrix of the adjacency matrix of $\mathcal{G}$. Which can also be written as:

$\frac{d}{dt}x_k=\sum_{j\in \mathcal{V}(k)}(x_j-x_k),$

where $x_k$ are the coordinates of the vector $x$ and $\mathcal{V}(k)$ denotes the set of closest neighbors of vertex $k$.

This system is such that the diagonal

$\Delta=\{x\in\mathbb{R}^{n}:x_{i}=x_{j}\ \forall\ 0\leq i\leq j\leq n\}$

is a \textbf{global attractor}, that is, such that $x(t)\rightarrow\Delta$ as $t\to\infty$ for all initial condition $x\in\mathbb{R}^{n}$.

The second system that we analyze is the nonlinear system, known as the Kuramoto Model which is the ODE on $\mathbb{R}^{|V|}$

$\frac{d}{dt}\varphi_k=\omega_k+\sigma\sum_{j\in \mathcal{V}(k)}\sin (\varphi_j-\varphi_k),$

where $\mathcal{V}(k)$ denotes the set of closest neighbors of node $ k $, the natural frequencies are distributed according to some probability density $\omega\mapsto g(\omega)$ and $\sigma$ is the coupling strength with a suitable scale, so that the model has a good behavior when $|V|=n\rightarrow\infty$. The conditions under which we observe synchronization behaviors are well known.

We make a comparative study of both systems with the approach proposed here.