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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)

jeq
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Ali Taghavi
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  • 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. – Malkoun Jul 23 '17 at 19:21
  • @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? – Ali Taghavi Jul 24 '17 at 10:57
  • Sorry I revise the comment ....H(2) is finite or no? – Ali Taghavi Jul 24 '17 at 11:13
  • I don't know. I haven't been following the literature surrounding this problem. – Malkoun Jul 24 '17 at 11:18
  • @Malkoun But I am (almost) sure that the problem is open even for n=2 (quadratic vector field). – Ali Taghavi Jul 24 '17 at 11:20
  • Ok. My interest in these quadratic vector fields was from another point of view. I wanted to find such vector fields for which the flow never blows up in finite time. They are rare but they exist. I suppose one may try to classify the quadratic vector fields vanishing at the origin, maybe under some assumptions, but this is a very tedious way to approach the problem you are interested in. Perhaps a less brute-force approach would be desirable, such as the one you are attempting. – Malkoun Jul 24 '17 at 11:28
  • @Malkoun If i am not mistaken you are interested in the problem of "Completeness" or Non completeness of polynomial vector fields(Of degree 2). Yes? If it is the case, I think the following paper completely classify such vector fields http://www.sciencedirect.com/science/article/pii/0022039686901130/pdf?md5=44796f6a42b1171089e1d32865e148c6&pid=1-s2.0-0022039686901130-main.pdf&_valck=1 – Ali Taghavi Jul 24 '17 at 11:34
  • Let us continue this discussion in chat. – Malkoun Jul 24 '17 at 11:51

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