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 $\\omega(x)\eq\\emptyset$ 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)\\\umerical id&4\\\\Author&jarino (552)\\\\Entry type&Theorem\\\\Classification&msc 34C05\\\\Classification&msc 34D23\\\\\\hline\\end{tabular}\\end{flushright}\\end{document}