It is well known that the Van der Pol equation $$\begin{cases} \dot x=y-(x^3-x)\\\dot y=-x \end{cases}$$ has no an algebraic limit cycle.
According to this fact, we search for a related question in the following form:
We consider the Lienard equation $$\begin{cases} \dot x=y-F(x)\\ \dot y=-x\end{cases}$$
where $F(x)$ is a polynomial with real coefficient.
Is there an example of a Lienard equation as the above system with two distinct limit cycles $\gamma_1, \gamma_2$ with a polynomial or rational map $g:\mathbb{R}^2 \to \mathbb{R}^2$ such that $g(\gamma_1)=\gamma_2$?