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  "body": "\n## Definition\n\nLet $I, J \\subseteq \\R$ be open intervals and let $(x, \\mu) \\mapsto f_\\mu(x)$ be a map $I \\times J \\to \\R$ of class $C^3$.\nWrite $f_x, f_{xx}, f_{xxx}$ for the partial derivatives in $x$.\nThe family $(f_\\mu)$ has a *period-doubling bifurcation* (or *flip bifurcation*) at $(x_0, \\mu_0) \\in I \\times J$ if the following hold:\n\n1. $f_{\\mu_0}(x_0) = x_0$, so $x_0$ is a [[fixed point|R6QVS6P21N]] of $f_{\\mu_0}$.\n2. $f_x(x_0, \\mu_0) = -1$.\n3. Let $x(\\mu)$ be the branch of fixed points through $x_0$, given by the implicit function theorem. Then $\\mu \\mapsto f_x(x(\\mu), \\mu)$ has a nonzero derivative at $\\mu_0$.\n4. $\\frac{1}{2} f_{xx}(x_0, \\mu_0)^2 + \\frac{1}{3} f_{xxx}(x_0, \\mu_0) \\neq 0$.\n\n## Remarks\n\nUnder these conditions, for $\\mu$ near $\\mu_0$ on one side of $\\mu_0$ there is exactly one orbit of [[period|T75CZ5H47N]] $2$ near $x_0$.\nOn the other side there is none, and the period-$2$ orbit tends to $x_0$ as $\\mu \\to \\mu_0$.\nThe sign of the quantity in clause 4 decides whether the bifurcation is supercritical, with an attracting period-$2$ orbit, or subcritical, with a repelling one.\n\nExample: the logistic family $f_r(x) = r x (1 - x)$ at $r_0 = 3$ and $x_0 = 2/3$.\nClause 1: $3 \\cdot \\frac{2}{3} \\cdot \\frac{1}{3} = \\frac{2}{3}$.\nClause 2: $f_x = r(1 - 2x) = 3 \\cdot (-\\frac{1}{3}) = -1$.\nClause 3: on the branch $x(r) = 1 - 1/r$ one has $f_x = 2 - r$, with derivative $-1$ in $r$.\nClause 4: $f_{xx} = -2r = -6$ and $f_{xxx} = 0$, so the quantity is $18 \\neq 0$.\nFor $r \u003e 3$ the period-$2$ orbit is $\\frac{r + 1 \\pm \\sqrt{(r - 3)(r + 1)}}{2r}$.\n\nNon-example: the family $f_\\mu(x) = \\mu - x$ satisfies clauses 1 and 2 at $(x_0, \\mu_0) = (0, 0)$, but $f_x = -1$ for every $\\mu$, so clause 3 fails.\nHere every point other than the fixed point $\\mu/2$ has period $2$ for every $\\mu$, and no isolated orbit is born.\n\nA fixed point with derivative $+1$ instead of $-1$ gives a [[saddle-node bifurcation|MAP5J6HK0T]] or one of its degenerate relatives.\n",
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