Talk:Poincaré–Bendixson theorem

Assumption

Doesn't this make the assumption that the 2 dimensional space involved has a nice clean topology and geometry? I ask this because it intuitively appears that if you have a closed 2 dimensional hyperbolic space, two close rays would be sensitive to their initial differences and separate sharply. This would show up as the rays producing different homotopies as they traversed the space. (Of course, this could be represented as a discrete system equivalent to the continuous one, but the continuous one is still available.)

Good point about the topology, I edited the article to reflect this.--Experiment123 03:25, 3 March 2006 (UTC)

I think, one should mention topology also here: "planar two-dimensional continuous dynamical system cannot give rise to a strange attractor", since the the theorem doesn't prohibit strange attractor in 2D systems with different topologies. —Preceding unsigned comment added by 129.206.110.147 (talk) 10:25, 26 May 2010 (UTC)

In fact, on surfaces other than the plane and the 2-sphere, the dynamics can be complicated. Not too complicated (nothing like Lorentz attractor), but however the limit set can be transversely Cantor. This is called the Cherry flow: look here. And on surfaces of higher genus you can define dynamics corresponding to the interval exchange maps, that would not be too simple neither. --Burivykh (talk) 13:53, 26 May 2010 (UTC)

Orbits

This page really needs to be rewritten, the mathematics is simply wrong. I have no time to do this myself, sorry for not being more helpful. What the Poincaré-Bendixson Theorem says is that if an orbit stays in a region that does not contain a fix point, then it approaches a periodic orbit. This does not imply that "any orbit which stays in a compact region ... approaches either a fixed point or a periodic orbit". In general, the attractor could be some arbitrarily complicated combination of homoclinic and heteroclinic orbits. 129.16.29.35 (talk) 10:35, 24 February 2009 (UTC)

Not sure I follow that. Don't homoclinic orbits and heteroclinic orbits approach a fixed point ? Can you explain your point further - maybe with an example ? Gandalf61 (talk) 11:52, 24 February 2009 (UTC)

The anonymous commenter is quite right; the mathematics is wrong. For a counterexample, have a look at my lecture notes (page 30 in the A5 version). There is a simple way to state the theorem (again, see my lecture notes): If the omega set of the orbit does not contain a fixed point, then the omega set is a cycle, and the orbit approaches this cycle. And there is a more general form of the theorem, stating that the omega set is either a fixed point, a cycle, or a graphic, i.e., a finite set of fixed points together with heteroclinic and homoclinic orbits connecting these. I sometimes see the latter referred to as the generalized Poincaré–Bendixson theorem, but some pictures in Bendixson's paper indicates that he was most likely aware of the possibility. I am not sure if he did prove the more general version though, as my French and time are both limited. I'll try to correct the main article now, but it may require more work yet. (That this has stood uncorrected for such a long time, is embarrasing.) Hanche (talk) 03:35, 19 November 2009 (UTC)

I edited the page and added the Expert-subject tag. As stated above, I am not really sure that Bendixson had the theorem in its most general form, and if not, I am not sure what is a good reference. If one cannot be found, maybe we should limit the statement to just the case where the ω-limit set contains no fixed point. Hanche (talk) 04:09, 19 November 2009 (UTC)

Oh rats – I just realized that the third paragraph restates the theorem in its more restricted form, and this time almost correctly. The whole thing needs to be restructured. Well, I am going to let it ride for a while and see if others want to pitch in. Hanche (talk) 04:15, 19 November 2009 (UTC)