Study the stability of the limit cycle r=1 for the system given in polar coordinates by the equations $\dot{r}=(r^2−1)(2x−1), \dot{\phi}=1$, where $x=r\cos \phi$.
I've been trying to solve this problem by estimating the return function, but haven't made any progress. Can anyone give me some hints?
Just found out that this is a problem in Arnold's ODE book.
The limit cycle is asymptotically stable. This can be seen as follows:
Whenever the trajectory starts at some point with $x>\frac12$ it will be pushed away from the periodic orbit $r=1$ until it reach the line $x=\frac12$. After that it is pushed to the periodic orbit until it reach again $x=\frac12$. But since since the line $x=\frac12$ divides the plane asymmetrically, the time that the orbit spends in the region $x<\frac12$ is more than the time that it passed in the region $x<\frac12$ in the last "iteration".