Can a harmonic vector field possess a limit cycle?

Question

Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)?

Note that the Laplacian of a vector field is defined via natural correspondence between the space of vector fields and the space of $1$-forms.(The natural correspondence arising from the Riemannian metric). If this correspondence is denoted by $i$ then the Laplacian of a vector field $X$ is defined as $\Delta X=i^{-1} \Delta (i(X))$. Where the latter Laplacian is the natural Laplacian on the space of differential forms.

In particular is there a quadratic vector field $$\begin{cases} x'= ax+by+\lambda(x^2-y^2)+txy \\ y'=cx+dy+\mu (x^2-y^2)+sxy\end{cases}$$

which has at least one limit cycle?

The second question: Assume that $X$ is a vector field on a Riemannian surface Assume that $\gamma$ is a periodic orbit of $X$. Is it true to say that there is a point $p\in \gamma$ such that $\Delta X$ is tangent to $\gamma$ at $p$?

Answer 1

Consider the infinite strip, $[0,1]\times\mathbb{R}$ with $[0,y]$ identified with $[1,y]$ for all $y\in\mathbb{R}$, and with the Riemannian metric $g=dxdx+dydy$. Consider $X=\partial_x +f(y)\partial_y$. The Laplacian is $\Delta X =f''(y) \partial_y$.

For the first question, let $f=-y$, so that the line $y=0$ is a limit cycle and $\Delta X=0$. For the second question, let $f=y^2$, so that $y=0$ is again a periodic orbit, but now, everywhere on the curve, $\Delta X=2\partial_y$ is orthogonal to the tangent $\partial_x$.