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The Poincaré-Bendixson theorem completely characterizes the $\omega$-limit sets of planar systems.

I would like to know whether extensions exist to 3D systems which tend to 2D systems in the following sense: Suppose that as the independent variable increases, the motion approaches a plane, as in the following system: $$\begin{align} \dot{x}&=f(x,y,z), \\ \dot{y} &=g(x,y,z), \\ \dot{z} &=-z. \end{align} $$ The (bounded) $\omega$-limit set of any point are necessarily on the plane $\{z=0 \}$, and they are invariant sets of the flow $$\begin{align} \dot{x} &= f(x,y,0), \\ \dot{y} &= g(x,y,0). \end{align} $$ Must these $\omega$-limit sets be, as in the conclusion of Poincaré-Bendixson, be:

  • a fixed point
  • a periodic orbit or
  • a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these?

Thank you!

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  • $\begingroup$ What else? Since you say that the plane is invariant, this is an immediate consequence of the usual theorem. Still, there are even appropriate infinite-dimensional versions of the Poincaré-Bendixson (for reaction diffusion equations). $\endgroup$ – John B Nov 29 '18 at 2:12
  • $\begingroup$ @JohnB I'm talking about orbits that start outside the plane. Could their $\omega$-limit set exhibit other behaviors (such as orbits crossing in a way forbidden by the conclusion of PB)? $\endgroup$ – user1337 Nov 29 '18 at 2:48
  • $\begingroup$ Now I understand, I thought you were talking only about the plane. Still, the meaning of your second centered equations is that the plane is invariant. No matter what happens outside, in that plane it will be the usual. Speaking now of something coming from outside, since you are assuming that all goes to the plane, what else could happen? $\endgroup$ – John B Nov 29 '18 at 11:11
  • $\begingroup$ @JohnB Maybe as the orbit is approaching the plane it could go back and forth, resulting in two curves on the plane that intersect at a non fixed point? This would still be an invariant set of the flow. $\endgroup$ – user1337 Nov 29 '18 at 11:29
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    $\begingroup$ Still the $\omega$-limit set would have to be a connected set, but this doesn't seem to go well with the orbits approaching the plane. $\endgroup$ – John B Nov 29 '18 at 12:03

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