# Poincaré-Bendixson for 3D systems?

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!