Poincare-Bendixson theorem

Let $M$ be an open subset of $\mathbb{R}^{2}$ , and $f\in C^{1}(M,\mathbb{R}^{2})$ . Consider the planar differential equation

$x^{\prime}=f(x)$

Consider a fixed $x\in M$ . Suppose that the omega limit set $\omega(x)\neq\emptyset$ is compact, connected, and contains only finitely many equilibria. Then one of the following holds:

$roman]{enumerate}\item\omega(x)isafixedorbit(aperiodicpointwithperiodzero,i.e.% ,anequilibrium).\item\omega(x)isaregularperiodicorbit.\item\omega(x)consistsof% (finitelymany)equilibria\{x_{j}\}andnon-closedorbits\gamma(y)suchthat\omega(y)% \in\{x_{j}\}and\alpha(y)\in\{x_{j}\}(where\alpha(y)isthealphalimitsetofy).% \end{enumerate}Thesameresultholdswhenreplacingomegalimitsetsbyalphalimitsets.% \par \par Sincefwaschosensuchthatexistenceandunicityhold,% andthatthesystemisplanar,% theJordancurvetheoremimpliesthatitisnotpossiblefororbitsofthesystemsatisfyingthehypothesestohavecomplicatedbehaviors% .% Typicaluseofthistheoremistoprovethatanequilibriumisgloballyasymptoticallystable% (afterusingaDulactyperesulttoruleoutperiodicorbits).\begin{flushright}\begin{% tabular}[]{|ll|}\hline Title&Poincare-Bendixson theorem\\ Canonical name&PoincareBendixsonTheorem\\ Date of creation&2013-03-22 13:18:40\\ Last modified on&2013-03-22 13:18:40\\ Owner&jarino (552)\\ Last modified by&jarino (552)\\ Numerical id&4\\ Author&jarino (552)\\ Entry type&Theorem\\ Classification&msc 34C05\\ Classification&msc 34D23\\ \hline\end{tabular}\end{flushright}\end{document}$