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  "body": "\n## Definition\n\nLet $\\dot x = F(x, \\mu)$, $\\mu \\in \\R$, be a smooth one-parameter family of vector fields on $\\R^n$.\nAssume that for $\\mu$ near $\\mu_0$ the system has a [[hyperbolic equilibrium|01TPMXM6NM]] $x_\\mu$ of saddle type, depending smoothly on $\\mu$.\nThe family has a *homoclinic bifurcation* at $\\mu_0$ if the following hold.\n\n1. At $\\mu = \\mu_0$ there is a [[homoclinic orbit|6DBSQKZK4W]] $\\Gamma$ to $x_{\\mu_0}$: an orbit with $x(t) \\to x_{\\mu_0}$ as $t \\to +\\infty$ and as $t \\to -\\infty$.\n2. For $\\mu \\ne \\mu_0$ close to $\\mu_0$ there is no homoclinic orbit to $x_\\mu$ near $\\Gamma$.\n\n## Remarks\n\nThe orbit $\\Gamma$ lies in both the stable and the unstable manifold of $x_{\\mu_0}$, given by the [[stable manifold theorem|WF22KK7C3R]].\nIt is a global bifurcation in the sense of [[bifurcation theory|D9TW46P19Y]], since it depends on a whole orbit and not only on the equilibrium.\nIn the plane, let $\\lambda_1 \u003c 0 \u003c \\lambda_2$ be the eigenvalues at the saddle and $\\sigma = \\lambda_1 + \\lambda_2$.\nIf $\\sigma \\ne 0$ and the loop splits transversally in $\\mu$, a unique [[limit cycle|0AAQ73WRFB]] appears near $\\Gamma$ on one side of $\\mu_0$, stable if $\\sigma \u003c 0$ and unstable if $\\sigma \u003e 0$.\nExample: for $\\dot x = y$, $\\dot y = x - x^2 + \\mu y$, let $H = y^2/2 - x^2/2 + x^3/3$.\nAlong solutions $\\dot H = \\mu y^2$.\nAt $\\mu = 0$ the level set $H = 0$ contains a homoclinic loop to the saddle $(0, 0)$, in the half-plane $x \u003e 0$.\nFor $\\mu \\ne 0$, $H$ is strictly monotone along nonconstant orbits, so no orbit can leave and return to the saddle, and the loop is broken.\nHere $\\sigma = \\mu$ vanishes at $\\mu_0 = 0$, so this example is degenerate and no limit cycle is born.\nNon-example: the constant family $\\dot x = y$, $\\dot y = x - x^2$, which does not depend on $\\mu$, fails clause 2, since the loop persists for every $\\mu$.\n",
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