Let $$A=\begin{pmatrix}-2 & 0 \\ 0 &-1\end{pmatrix} $$ be a matrix and $\varphi(t,x)$ flow of the ODE $$\dot{x}= Ax, $$ $$x(0) = x. $$
It is well known that if $x = (x_1,x_2)$ with $x_2 \neq 0$, $$\frac{\varphi '(t,x)}{\|\varphi '(t,x)\|}\to (0,1)\ \text{or}\ (0,-1),\ \text{when }t\to \infty. $$
Now, consider a smooth function $F:\mathbb{R}^2\to \mathbb{R}^2$, such that $F(0) = 0$ and $\mathrm{d} F_0 = A.$
I would like to know if there exists a open neighboorhood $U$ of $0$, such that if $\varphi_F(t,x)$ is the solution of the ODE $$\dot{x} = F(x) $$ $$ x(0) = x\in U.$$ Then it is satysfied $$\frac{\varphi_F '(t,x)}{\|\varphi_F '(t,x)\|}\to (0,1)\ \text{or}\ (0,-1),\ \text{when }t\to \infty, $$ for all $x\in U$ except for points that lies in the invariant stable manifold related to the eigenvalue $-2$ (in other words $\forall\ x \in U\setminus W$, where $W$ is a 1-manifold.).
I tried to use Hartman-Grobman theorem, but this theorem only gives us a homeomorphism which is not enough to achieve the result.
Can anyone help me?
EDIT: In the paper Rodrigues, H.: Known results and open problems on $\mathcal{C}^1$ linearization in Banach spaces. São Paulo J. Math. Sci. 6, 375–384 (2012), on page $2$, the author claims the following theorem
which solves the problem. However, I could not find this Hartman's paper.