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 a 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 and the following post.

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..So we search for a cubic (or higher degree ) system.

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 and the following post.

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..So we search for a cubic (or higher degree ) system.