Can someone sketch some ideas of how to use the Poincaré-Bendixson Theorem to prove that there must be a fixed point contained inside a periodic orbit?
Poincare-Bendixson Theorem
3
$\begingroup$
ordinary-differential-equations
dynamical-systems
-
0I have a feeling that that statement is false. I do not have a counterexample though. – 2011-04-06
-
0I cant see the question, the image link isn't showing either. – 2011-04-06
-
1This certainly doesn't hold for discrete or higher dimensional dynamical systems (ex: billiards in polygons), so I take it we can assume the system is continuous and 2 dimensional? – 2011-04-06
-
0@Alex: is there even a Poincare-Bendixson theorem in more than 2 dimensions? In anycase, certain topological assumptions must be made about the domain, else just consider the rotations on $\mathbb{S}^1\times \mathbb{R}$. – 2011-04-06
-
0@Brad: please flesh out your question. Your statement is presumably true under certain assumptions, none of which you stated in the question. Please put more effort into describing what exactly it is that you want to know. – 2011-04-06
-
0@Willie: Certainly not without limiting assumptions. – 2011-04-07