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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?

MrYouMath
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lulu
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    Be sure to check http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference for formatting function in mathjax. – Arbuja Jan 31 '16 at 23:33
  • Nice problem. Hint: the line $x=1/2$ causes that there is only one periodic orbit. You should really avoid writing "limit cycle" before knowing that it is an isolated periodic orbit. – John B Feb 01 '16 at 00:07

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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".

John B
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