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!