Consider the differential equation $$\dot x = Ax +F(x)=:G(x)$$ defined locally around the origin in $\mathbb{R}^n$, where $A \in \mathbb{R}^{n \times n}$ has eigenvalues with strictly negative real parts and $F(x) = o(|x|)$, for $x \to 0$, and $\frac{\partial F}{\partial x}=0$.
A theorem of Hartman [Perko, Differential Equations and Dynamical Systems, page 127] says that in this case, there exists a $C^1$ map $y=x+\phi(x)$, $\phi = o(|x|), \frac{\partial \phi}{\partial x}=0$ so that the trajectories of $$\dot y = Ay$$ are conjugate to that of the system above.
Can we say anything about the dependence of $\phi$ of $G$? Namely, does $\phi$ depend continuously on $G$ for the $C^r$ topology, $r\geq 1$?