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  "body": "\n## Summary\n\nA homoclinic orbit is a [[trajectory|07TZ2Q4C1M]] that leaves an [[equilibrium point|0N8NGKDZDZ]] and returns to the same equilibrium.\nIt lies in both the stable and the unstable manifold of the equilibrium.\nHomoclinic orbits separate regions of different behaviour in the phase plane, and they appear in global bifurcations.\nIn higher dimensions their transverse versions are a source of chaos.\n\n## Explanations\n\n### L1 human\n\nPicture a loop in the phase plane that starts at a saddle point, swings around, and comes back to the same saddle.\nA state on this loop takes infinitely long to leave the saddle and infinitely long to come back.\nFor the equation $x'' = x - x^2$, the solution $x(t) = \\frac{3}{2} \\operatorname{sech}^2(t/2)$ starts near $0$, rises to $\\frac{3}{2}$ and falls back to $0$.\nIts path in the $(x, x')$ plane is a homoclinic loop at the origin.\n",
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