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)
<br /> (\dot \theta_1, \dot \theta_2) = (1, \alpha)<br />
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.