Extension of Poincaré-Bendixson Theorem to $\mathbb{R}^3$

Question

Hartman mentioned in his ODE book (chapter 7) that Poincaré-Bendixson Theorem is limited to $\mathbb{R}^2$ or $2$-manifold because of Jordan Curve Theorem. Since there is generalization for Jordan Curve Theorem, why did no one extend Poincaré-Bendixson to higher dimensions?

Answer 1

Trajectories of ODE systems are curves, no matter what the dimension of the ambient space. They also do not intersect, by the uniqueness theorem.

In two dimensions the above facts severely restrict the possibilities for the set of trajectories. For example, once we have a closed trajectory, the Jordan separation theorem implies that the rest of trajectories are divided into "inner" and "outer" ones. And even when a trajectory is not closed, it is still a serious obstacle for other trajectories: they'd have to a long way to get to the other side.

In three or more dimensions, smooth closed curves do not separate the space, and curves can easily pass by each other, like skew lines do. This allows for much richer behavior, like in the Lorentz system:

(Image by Dan Quinn, from Wikipedia).

The fact that topological spheres separate the space does not matter much: trajectories are not spheres.