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I am having difficulty using Poincare-Bendixson to show that a particular system of differential equations has a periodic orbit. The system is $$ \left\{\begin{array}{rcl} \dot{x} & = & -3y+x(1-x^2-y^2+y) \\ \dot{y} & = & \phantom{-}2x+y(1-x^2-y^2-4x) \end{array}\right. $$ Now, the expression $1-x^2-y^2$ made me investigate the unit circle:

  • However, I found that some solutions enter through the unit circle while others leave it, so it wasn't a useful "boundary".
  • So, more generally, I converted to polar and consider the circle $x^2+y^2=r^2$.
  • I was able to verify that for sufficiently large $r$, no solutions leave through the circle of radius $r$ (i.e. $\dot{r}<0$).
  • Thus, any solution starting in the circle of radius $r$ remains in it.
  • Thus, if I take a solution starting within this circle, I can consider its forward $\omega$-limit set, which will be nonempty, compact, and connected.
  • The issue is that there is at least one fixed point within this circle (at the origin) and there could be more. - I am not sure how to find the fixed points and argue that there is a solution not converging to any of them in forward time.

Any ideas $?$.

Felix Marin
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Anon
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  • It is also not too hard to show that for sufficiently small $r$, $\dot{r}>0$, so there is a bounded trapping region away from the origin. However, plotting seems to indicate that there are two more critical points in this region: one saddle point and one stable. – whpowell96 May 14 '24 at 00:40
  • @whpowell96 I see, that makes sense. Then I suppose all that is left is to figure out how to deal with the other equilibria. – Anon May 14 '24 at 01:17

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