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  "body": "\n## Statement\n\nLet $D \\subseteq \\R^2$ be a [[simply connected domain|0QW64ARSHB]] and let $f = (P, Q) \\colon D \\to \\R^2$ be continuously differentiable.\nLet $\\varphi \\colon D \\to \\R$ be continuously differentiable.\nAssume that the [[divergence|0MWV6ZTK6S]]\n$$\\nabla \\cdot (\\varphi f) = \\frac{\\partial (\\varphi P)}{\\partial x} + \\frac{\\partial (\\varphi Q)}{\\partial y}$$\nsatisfies $\\nabla \\cdot (\\varphi f) \\ge 0$ on $D$ and does not vanish identically on any nonempty open subset of $D$.\nThen the [[autonomous system|07BNRFQ0PJ]] $x' = P(x, y)$, $y' = Q(x, y)$ has no [[periodic orbit|0AA97XA70K]] contained in $D$.\nThe same holds when $\\nabla \\cdot (\\varphi f) \\le 0$ instead.\n\n## Remarks\n\nSuppose a periodic orbit $\\Gamma \\subseteq D$ exists, and let $R$ be the region inside $\\Gamma$; since $D$ is simply connected, $R \\subseteq D$.\nBy [[Green's theorem|0HR3N02317]], $\\iint_R \\nabla \\cdot (\\varphi f) \\, dx \\, dy$ equals the flux $\\oint_\\Gamma \\varphi \\, (P \\, dy - Q \\, dx)$.\nThe flux is $0$, because along $\\Gamma$ one has $dx = P \\, dt$ and $dy = Q \\, dt$; but the double integral is positive, since $R$ is a nonempty open set.\nThe case $\\varphi = 1$ is Bendixson's criterion, and a function $\\varphi$ that works is called a Dulac function.\nExample: for $x' = y$, $y' = -x - y + x^2$ one has $\\nabla \\cdot f = -1$, so the system has no periodic orbit in $\\R^2$.\nSimple connectivity is needed: the system $x' = x(1 - x^2 - y^2) - y$, $y' = y(1 - x^2 - y^2) + x$ has $\\nabla \\cdot f = 2 - 4(x^2 + y^2) \u003c 0$ on the annulus $0.8 \u003c \\sqrt{x^2 + y^2} \u003c 2$, yet the unit circle in this annulus is a periodic orbit.\n\n## Explanations\n\n### L1 human\n\nA periodic orbit in the plane encloses a region, and the flow neither gains nor loses area across the orbit.\nIf the flow, after reweighting by a function $\\varphi$, always expands area or always shrinks it, such a closed loop cannot exist.\nSo the theorem is a simple test that rules out cycles.\n\n### L2 human\n\nBendixson–Dulac is the standard negative criterion for periodic orbits in the plane, and it complements the [[Poincaré–Bendixson theorem|0EW1X61ZT1]], which proves their existence.\nFinding a good Dulac function is an art; for Lotka–Volterra type systems, $\\varphi = 1 / (x y)$ is a common choice.\nOn a domain with one hole, the same argument shows that there is at most one periodic orbit that goes around the hole.\n",
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