What is an example of polynomial vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ such that two closed orbits $C\_1,C\_2$ of the system surrounds an annular region $R$ such that $R$ does not contain any singular point and the flow orientation of $C\_1$ is opposite to the flow-orientation of $C\_2$.


The motivation for this question is the following [counterexample of a nongeodesible flow on the torus](https://mathoverflow.net/questions/273635/finding-a-1-form-adapted-to-a-smooth-flow/273689#273689) and [the following post](https://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-or-positively-curved-space?noredirect=1&lq=1).


Note that this situation can not be occurred when the degree of $P,Q$ is at most $2$. See theorem $4$ [of the following paper.](https://www.sciencedirect.com/science/article/pii/0022039666900702).So we search for a cubic (or higher degree ) system.



A cubic system with two nested limit cycles with opposite orientations
It is not hard to concoct such an example in sufficiently high degree. For an example of degree $5$, take
$$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\end{align}
$$
The circles $x^2+y^2=1$ and $x^2+y^2=3$ are integral curves of this vector field with opposite orientations. The only zero of the vector field is at the origin $(x,y) = (0,0)$. In the annulus $1 < x^2+y^2 < 3$, the flow always increases $x^2+y^2$ and it moves in the counterclockwise direction when $1 < x^2+y^2 < 2$ but in the clockwise direction when $2 < x^2+y^2 < 3$.


I don't know whether there is a system of degree 3 or 4 that has such a property, but it wouldn't surprise me.


