# Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question:

Question: For a vector field $$X$$ on the plane we define the differential operator $$D$$ on $$C^{\infty}(\mathbb{R}^2)$$ with $$D(f)=(\Delta\circ L_X-L_X\circ \Delta)(f)$$ where $$\Delta$$ is the standard Laplacian.

Is there a vector field $$X$$ on the plane with $$2$$ nested closed orbits $$\gamma_1 \subset \gamma_2$$ such that there exist a smooth function $$f$$ for which $$D(f)$$ does not vanish on the closur of the amnular region surrounded by $$\gamma_1$$ and $$\gamma_2$$?

On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Elliptic operators corresponds to non vanishing vector fields

Remark: The first words of the titles of this post is inspired by a paper by C.C. Pugh and J.P Francois " Keeping track of Limit cycles"

• I am not sure what you mean by $\gamma_1 \subset \gamma_2$: do you mean that the disc enclosed by $\gamma_1$ is a subset of the disc enclosed by $\gamma_2$? Dec 3 '19 at 18:47
• @WillieWong Yes I mean so. Dec 3 '19 at 19:06

Let $$X$$ be the vector field $$2x \partial_y - y \partial_x$$. The level sets of $$y^2 + 2x^2$$ are orbits of $$X$$, they have the shape of ellipses.
It is easy to compute $$D(f) = 2 \partial^2_{xy} f$$. So if you just let $$f(x,y) = xy$$ you in fact have $$D(f) \equiv 2 \neq 0$$, and in particular not on the annulus bounded by two level sets of $$y^2 + 2x^2$$.