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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $f \\colon \\R^n \\times \\R \\to \\R^n$ be [[smooth|0V33BNKZ9S]], and consider the family of systems $x' = f(x, \\mu)$ with parameter $\\mu$.\nAssume $f(0, \\mu) = 0$ for all $\\mu$ near $0$, so that $0$ is an [[equilibrium point|0G4X36AE0X]] for each such $\\mu$.\nLet $A(\\mu)$ be the [[Jacobian matrix|047NDTEPCV]] of $x \\mapsto f(x, \\mu)$ at $0$.\nThe family has a *Hopf bifurcation* at $\\mu = 0$ if:\n\n1. $A(\\mu)$ has a pair of simple complex conjugate [[eigenvalues|0BQAA2BK3Q]] $\\alpha(\\mu) \\pm i \\beta(\\mu)$ for $\\mu$ near $0$, with $\\alpha(0) = 0$ and $\\beta(0) \u003e 0$;\n2. no other eigenvalue of $A(0)$ has zero real part;\n3. $\\alpha'(0) \\ne 0$.\n\n## Remarks\n\nUnder these conditions, a family of [[periodic orbits|0AA97XA70K]] branches off the equilibrium at $\\mu = 0$, with periods close to $2\\pi / \\beta(0)$.\nIf moreover a certain coefficient $\\ell_1$, the first Lyapunov coefficient, is nonzero, the orbits exist for $\\mu$ on one side of $0$ only and have amplitude of order $\\sqrt{\\abs{\\mu}}$.\nThe bifurcation is supercritical when $\\ell_1 \u003c 0$; in the plane the orbits are then stable limit cycles.\nIt is subcritical when $\\ell_1 \u003e 0$; the orbits are then unstable.\n\nExample: $x' = \\mu x - y - x(x^2 + y^2)$, $y' = x + \\mu y - y(x^2 + y^2)$.\nThe Jacobian at $0$ is $\\begin{pmatrix} \\mu \u0026 -1 \\\\ 1 \u0026 \\mu \\end{pmatrix}$, with eigenvalues $\\mu \\pm i$.\nSo $\\alpha(\\mu) = \\mu$, $\\beta(\\mu) = 1$, and all three clauses hold.\nIn polar coordinates the system is $r' = \\mu r - r^3$, $\\theta' = 1$, and the circle $r = \\sqrt{\\mu}$ is a stable limit cycle for $\\mu \u003e 0$.\n\nNon-example: $x' = \\mu^2 x - y$, $y' = x + \\mu^2 y$.\nThe eigenvalues are $\\mu^2 \\pm i$, so $\\alpha(\\mu) = \\mu^2$ and $\\alpha'(0) = 0$; clause 3 fails.\n",
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