Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter $\epsilon$ and scale one variable $x \in \mathbb{R}$. Scale $x$ such that $x \mapsto \frac{x}{\epsilon}$.
Naturally, $F$ would also be symplectic in the new scaled coordinate system. However, what if I use the smallness of $\epsilon$ to do a perturbative expansion of $F$ in terms of $\epsilon$, assuming that everything depends smoothly on $\epsilon$? So I write $F_{\epsilon} = F + \epsilon F_{1} + O(\epsilon^{2})$.
Does the map $F_{\epsilon}$ preserve symplecticity property, or not? If not, how could I ensure that the map indeed stays symplectic after a perturbative expansion in a small parameter?