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  "body": "\n## Definition\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 $t \\ge 0$. Its $\\omega$-limit set $\\omega(y)$ is the set of points $z \\in U$ such that $\\varphi_{t_k}(y) \\to z$ for some sequence $t_k \\to \\infty$.\nFor a point $y$ whose solution exists for all $t \\le 0$, the $\\alpha$-limit set $\\alpha(y)$ is defined in the same way with $t_k \\to -\\infty$.\nA limit cycle is a [[periodic orbit|01DSNP0B4R]] $\\Gamma$ such that $\\Gamma = \\omega(y)$ or $\\Gamma = \\alpha(y)$ for some point $y \\in U$ that is not on $\\Gamma$.\n\n## Remarks\n\nA limit cycle is stable if all nearby trajectories approach it as $t \\to \\infty$, and unstable if they approach it as $t \\to -\\infty$. It can also be semi-stable.\nA linear system has no limit cycles: its periodic orbits, if any, fill a region around the origin.\nThe van der Pol oscillator $x'' - \\mu (1 - x^2) x' + x = 0$ with $\\mu \u003e 0$ has exactly one limit cycle, and it is stable.\nThe [[Poincaré–Bendixson theorem|0EW1X61ZT1]] is a main tool to prove that a limit cycle exists.\n\n## Explanations\n\n### L1 human\n\nA limit cycle is an isolated loop in the phase plane that nearby motions spiral toward or away from.\nIt models a self-sustained oscillation, such as a heartbeat, whose size does not depend on the initial state.\n",
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