I would like to find the **limit set** $\omega(x)$ of the linear dynamical system

$$\mathrm f (x_1, x_2) = \begin{bmatrix} 2 & 1\\ 1 & 1\end{bmatrix} \begin{bmatrix} x_1\\ x_2\end{bmatrix}$$

I tried a few numerical values of $x_1$ and $x_2$ to see the sequence of vectors, but I'm not sure if this is right. I would appreciate your help.