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  "body": "\n## Summary\n\nA Hopf bifurcation is the birth of a [[periodic orbit|0AA97XA70K]] from an [[equilibrium point|0N8NGKDZDZ]] as a parameter crosses a critical value.\nIt happens when a pair of complex conjugate eigenvalues of the linearisation crosses the imaginary axis.\nIt is the main way in which a steady state of a system turns into a small oscillation.\nSee the [[definition|1JPGWFMAGK]].\n\n## Explanations\n\n### L1 human\n\nThink of a system resting at an equilibrium, where small disturbances spiral back to rest.\nAs a parameter changes, the spiral becomes slower, and at a critical value it turns outward.\nThen the motion no longer returns to rest but settles on a small loop, a [[limit cycle|0AAQ73WRFB]].\nIn polar coordinates, the model $r' = \\mu r - r^3$, $\\theta' = 1$ shows this.\nFor $\\mu \\le 0$ every solution spirals into the origin, and for $\\mu \u003e 0$ solutions approach the circle $r = \\sqrt{\\mu}$.\n",
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