Show that a system has no limit cycle

Question

$\dot{x_1} = -x_1(1 - x_1^2 - x_2^2)$, $\dot{x_2} = -x_2(1 - x_1^2 - x_2^2)$

The question is to prove this system has no limit cycle.

I changed to polar co-ordinates to get: $\dot{r} = -r(1-r^2) , \dot{\theta} = 0 $

Can any conclusions be made from here?

Answer 1

Following Gerhard S’s comment, can you have a limit cycle when $\dot{\theta}=0$?

My original comment on $r$ dynamics, based on having read your question as $\dot{\theta}=1$ was referring to the fact that $r=1$ would have been an unstable limit cycle. (See comments below).

Note: what you have is a circle of unstable fixed points, not a limit cycle.