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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 $p$?

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  • $\begingroup$ I think the answers to both the questions depend on the de Rham cohomology of M - that is, whether or not the limit cycle bounds a simply connected domain. Could you clarify this point? As for the second question, the 1-form i(X) is closed and co-closed. Hence, the inner product of i(X) with its Laplacian ( as defined by the metric g) can be seen to be positive. $\endgroup$ – Chaitanya Nov 23 '17 at 12:38
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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 $\partial_x$.

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  • $\begingroup$ A limit cycle is in priori a compact set but the line $x=0$ is not a compact set on the cylinder. Moreover is your vector field , a well defind vector field on the cylinder? $\endgroup$ – Ali Taghavi Jan 6 '18 at 20:14
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    $\begingroup$ I had the roles of x and y reversed in the second paragraph. I've now fixed this. Is this now correct? $\endgroup$ – user119324 Jan 6 '18 at 22:06

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