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  "body": "\n## Statement\n\nLet $U$ be an open subset of $\\mathbb{R}^2$, let $f : U \\to \\mathbb{R}^2$ be continuously differentiable, and let $\\varphi$ be the [[flow|02903SSA48]] of $x' = f(x)$.\nLet $y \\in U$ be a point whose solution exists for all $t \\ge 0$. Let $\\omega(y)$ be its $\\omega$-limit set, as in the definition of a [[limit cycle|0EERDDPSVT]].\nAssume that $\\omega(y)$ is non-empty and compact, and that it contains only finitely many [[equilibrium points|0G4X36AE0X]].\nThen $\\omega(y)$ is one of the following three kinds of set.\n(1) It is a single equilibrium point.\n(2) It is a [[periodic orbit|01DSNP0B4R]].\n(3) It is a connected set of finitely many equilibrium points and of trajectories that tend to these points as $t \\to \\infty$ and as $t \\to -\\infty$.\n\n## Remarks\n\nA common corollary: let $K \\subseteq U$ be compact, contain no equilibrium point, and contain the positive half-trajectory of $y$. Then $\\omega(y)$ is a periodic orbit.\nTo apply it, one often finds an annulus that trajectories enter but cannot leave. Then the annulus contains a periodic orbit.\nThe theorem is special to the plane. It fails in $\\mathbb{R}^3$, where chaotic behaviour is possible, as for the Lorenz system.\nThe trajectories in case (3) are homoclinic or heteroclinic orbits.\n\n## History\n\nThe theorem is named after Henri Poincaré and Ivar Bendixson.\n\n## Explanations\n\n### L1 human\n\nIn the plane, a bounded motion has few options in the long run.\nIt settles at an equilibrium, approaches a closed loop, or follows a chain of paths between equilibria.\nTrajectories in the plane cannot cross, and this is what limits the possibilities.\n",
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