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Show that $\dot x = x-y-x(x^2+y^2)$,
$\dot y=y+x-y(x^2+y^2)$ has at least one periodic orbit.

I calculated the equilibrium point which came out to be only the origin.

I converted the given system into polar coordinates:
$\dot r=r-r^3$
$\dot \theta=1$

Then I get that
$\dot r \gt0$ provided $r\lt1$
$\dot r \lt0$ provided $r\gt1$

So we get that $r=1$ is a stable limit cycle in this case, but we are not getting any annular region. So can I take the annular region to be the region enclosed by $r=\frac{1}{2}$ and $r=1$ or say, $r=1$ and $r=2$ ?

Lutz Lehmann
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Esha
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    Why would you need Poincaré–Bendixson if you already know a periodic solution explicitly?! – Hans Lundmark Nov 09 '22 at 05:33
  • Ooh! Right. Thanks for clarifying – Esha Nov 09 '22 at 06:12
  • See https://math.stackexchange.com/questions/4330898/poincare-map-for-polar-ode for the same system. Also https://math.stackexchange.com/questions/3230991/finding-alpha-and-omega-limit-set, https://math.stackexchange.com/questions/1419147/solve-doe-system-with-polar-coordinates, https://math.stackexchange.com/questions/3586968/conversion-of-ode-system-to-polar – Lutz Lehmann Nov 09 '22 at 07:57
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    Split the difference and use both, $\frac12\le r\le 2$. – Lutz Lehmann Nov 09 '22 at 07:59

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