# Algebraic independence of limit cycles of Lienard equation

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 $g(\gamma_1)=\gamma_2$?

• @Mahdi I assume $F(x)$ is a polynomial function in $x$. In the classical Lienard equation, $F$ is a polynomial. – Ali Taghavi Aug 11 '18 at 11:33
• @Mahdi But the system has no LIMIT CYCLE when F=x. So actually we start from degree F=3. For deg(F)=2 we have no limit cycle. There is a LNM paper I think(LNM 598) by Pugh, Lins Neto and W. de melo the title is "On Lienard equation". In that paper one can find more explanation on this system. – Ali Taghavi Aug 11 '18 at 13:06
• @Mahdi This is the exact address (1977) Lins Neto, de Melo and Pugh [Lectures Notes in. Math. 597, 335–357] – Ali Taghavi Aug 11 '18 at 13:12
• Thanks, for responses. I will delete my disturbing comments, soon. – Mahdi Aug 11 '18 at 13:28
• @Mahdi I thank you too for your attention to my question. Your comment was not disturbing. – Ali Taghavi Aug 11 '18 at 13:40