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  "body": "\n## Summary\n\nA Bogdanov–Takens bifurcation is a bifurcation of an [[equilibrium point|0N8NGKDZDZ]] in a family with two parameters, at which the linearisation has a double zero eigenvalue.\nIt is a point where a [[saddle-node bifurcation|MAP5J6HK0T]] curve and a [[Hopf bifurcation|F5H8GW1XYS]] curve meet.\nNear it, the family also has a curve of homoclinic bifurcations.\nIt is one of the basic codimension-two bifurcations.\nSee the [[definition|H3GWX277EF]].\n\n## Explanations\n\n### L1 human\n\nMost bifurcations happen when one parameter reaches a critical value.\nA Bogdanov–Takens point needs two parameters to be tuned at once.\nIn the plane of the two parameters, three curves start from this point.\nOn one of them two equilibria collide, on another a small oscillation is born, and on the third this oscillation grows until it touches a saddle and disappears.\nSo a single point organizes several kinds of change in the behaviour of the system.\n",
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