Statement: Use the Poincaré-Bendixson Theorem to show that the Van der Pol oscillator $$ \ddot{x}+\varepsilon(x^2-1)\dot{x}+x = 0,\; \; \varepsilon >0 $$ has at least one stable limit cycle for sufficiently small values of $\varepsilon$.
Approach: First, I rewrite the equation as a 2D system: $$\begin{aligned} \dot{x}=&y\\ \dot{y}=&-x-\varepsilon (x^2-1)y \end{aligned} $$ We observe that $(0,0)$ is the unique fixed point.
Doing a change of variables to polar coordinates, $(x,y)=(r\cos\theta,\: r \sin\theta)$, the system has the following expression: $$\begin{aligned} \dot{r}=&-\varepsilon r \sin^2\theta(r^2\cos^2\theta-1)\\ \dot{\theta}=&-1-\varepsilon (r^2\cos^2\theta-1)\sin\theta \cos\theta \end{aligned} $$ Now, I need to find a positively invariant compact set with no fixed point (so I can apply Poincaré-Bendixson theorem and state it exists a periodic orbit). To avoid the fixed point (0,0), I consider a small circle of radius $r^-<1$. In the boundary of the circle, $\dot{r}>0$, so the flux goes in outside direction. But now I don't know how to close the set to do it positively invariant. I would appreciate any help.
Thanks!