It is well-known that on the Fr\'{e}chet space $C^{\infty}(\mathbb{R})$, the differential operator $$ D(f)\triangleq \frac{\partial f}{\partial x}, $$ is chaotic. However, what's an example of an element $f_0\in C^{\infty}(\mathbb{R})$ whose orbit under $D$ is dense?
In general, is there a "simple" criterion to determine when an element of the Fr\'{e}chet space has a dense orbit under $D$?