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  "body": "\n## Statement\n\nLet $\\dot x = F(x)$ be a smooth vector field on $\\R^3$.\nLet $x_0$ be a [[hyperbolic equilibrium|01TPMXM6NM]] whose Jacobian matrix $DF(x_0)$ has [[eigenvalues|0BQAA2BK3Q]] $\\lambda \u003e 0$ and $-\\rho \\pm i \\omega$ with $\\rho \u003e 0$ and $\\omega \\ne 0$.\nLet $\\Gamma$ be a [[homoclinic orbit|6DBSQKZK4W]] to $x_0$.\nAssume the Shilnikov condition $\\rho \u003c \\lambda$.\nThen every neighbourhood of $\\Gamma \\cup \\set{x_0}$ contains infinitely many [[periodic orbits|0AA97XA70K]] of saddle type.\n\n## Remarks\n\nSuch an equilibrium is called a saddle-focus: orbits near it spiral in along a plane and leave along a line.\nThe proof studies the return map to a small section transverse to $\\Gamma$.\nNear $x_0$ the flow is close to linear, and the passage time near the equilibrium makes the returning image a spiral that winds infinitely often.\nWhen $\\rho \u003c \\lambda$ this spiral is stretched rather than contracted, and it crosses the section so as to form infinitely many [[Smale horseshoes|N63Z6EA6YS]].\nEach horseshoe carries infinitely many periodic orbits, which gives the conclusion.\nThe condition $\\rho \u003c \\lambda$ is needed: if $\\rho \u003e \\lambda$, a sufficiently small neighbourhood of $\\Gamma$ contains at most one periodic orbit.\nThe theorem is due to Leonid Shilnikov.\n\n## Explanations\n\n### L1 human\n\nImagine a flow in space with a point that pulls nearby orbits inward along a spiralling disc and pushes them out along a line.\nIf an orbit leaves along the line and comes back spiralling into the point, the theorem says that complicated behaviour is forced near this loop.\nWhen the push outwards is stronger than the pull inwards, infinitely many closed orbits appear near the loop.\nThis is a simple test for chaos that can be checked from the eigenvalues at one point.\n\n### L2 human\n\nThe theorem is a main source of chaos in three-dimensional flows, because it reduces chaos to the existence of a single homoclinic orbit plus an inequality on eigenvalues.\nA [[homoclinic bifurcation|JQYCTY3Q8V]] to a saddle-focus that satisfies the Shilnikov condition is therefore a route to chaos in families of flows.\nIn the plane, by contrast, the breaking of a homoclinic loop creates at most one limit cycle.\n",
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