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Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$ Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center singularity at the origin. This means that the origin is surrounded by a band of closed orbits.

Is there a Riemannian metric on $\mathbb{R}^2 \setminus C$ with zero curvature such that all closed orbits of the vector field are closed geodesics? Here $C$ is the algebraic curve $$C=\{(x,y)\mid yP(x,y)-xQ(x,y)=0\}$$

The motivation comes from the idea of consideration of "Limit cycles" of polynomial vector fields as "Closed Geodesics" of a Riemannian metric on the phase space. This situation is discussed in the following MO posts and item 5 of page 3 of the this preprint.

Finding a 1-form adapted to a smooth flow

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

Limit cycles as closed geodesics(in negatively or positively curved space)

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  • $\begingroup$ I find it surprising, and highly interesting, that problems concerning polynomial vector fields in dimension 2, which are vector fields that can each easily be drawn, remain open to this date. I am not very familiar with the literature, but I know that Date had classified the homogeneous quadratic fields in dimension 2. Perhaps the quadratic vector fields vanishing at the origin have also been classified by now? I mean, in principle, you are only adding a linear vector field to the homogeneous quadratic one, but I suspect the classification to be very tedious. Just a small question/comment. $\endgroup$
    – Malkoun
    Jul 23, 2017 at 19:21
  • $\begingroup$ @Malkoun Thank you for your comment. While a homogeneous quadratic vector field does not have a limit cycle, but I think that the problem for a general quadratic system is still open.That is it is open to decide whether $H92)$ is finite or no? $\endgroup$
    – Ali Taghavi
    Jul 24, 2017 at 10:57
  • $\begingroup$ Sorry I revise the comment ....H(2) is finite or no? $\endgroup$
    – Ali Taghavi
    Jul 24, 2017 at 11:13
  • $\begingroup$ I don't know. I haven't been following the literature surrounding this problem. $\endgroup$
    – Malkoun
    Jul 24, 2017 at 11:18
  • $\begingroup$ @Malkoun But I am (almost) sure that the problem is open even for n=2 (quadratic vector field). $\endgroup$
    – Ali Taghavi
    Jul 24, 2017 at 11:20

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