Poincare-Bendixson in n-dimensions

[motherh]

Suppose that $\dot{x}$ = f(x) in $\Re$$^{n}$. There exists a bounded 2D invariant manifold M for this system. There are no critical points in M. Does it follow that there is a periodic orbit in M?

I've realized that this has it's similarities to Poincare-Bendixon's theorem but I don't believe that it holds for a system in n dimensions. Would I be correct in thinking that even with the strict conditions above we could end up with chaos? If I am correct here, does anybody know of any counterexamples for the above statement? Thanks.

 

[pasmith]

Suppose that $\dot{x}$ = f(x) in $\Re$$^{n}$. There exists a bounded 2D invariant manifold M for this system. There are no critical points in M. Does it follow that there is a periodic orbit in M?

No, as for example if $M = S^1 \times S^1$ (the 2-torus) and the reduction of the system on $M$ is (after rescaling time)
$$(\dot \theta_1, \dot \theta_2) = (1, \alpha)$$
with irrational $\alpha$. All trajectories are then dense in $M$, and there are no periodic orbits.

I've realized that this has it's similarities to Poincare-Bendixon's theorem but I don't believe that it holds for a system in n dimensions. Would I be correct in thinking that even with the strict conditions above we could end up with chaos?

No: three dimensions are necessary for chaos, so the behaviour of the system on M cannot be chaotic.