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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an interval of parameters and $X$ a state space.\nLet $(\\phi_\\mu)_{\\mu \\in I}$ be a one-parameter family of dynamical systems on $X$, flows or maps.\nFor each $\\mu$, let $A_\\mu \\subseteq X$ be a chosen invariant set of $\\phi_\\mu$.\nTypical choices are the set of [[equilibria|0N8NGKDZDZ]] and periodic points, or the [[attractor|7FWX24X5MW]] reached from a fixed initial point.\nThe *bifurcation diagram* of the family for this choice is the set\n$$\\set{(\\mu, x) \\in I \\times X : x \\in A_\\mu}.$$\n\n## Remarks\n\nExample: for $\\dot x = \\mu x - x^3$ on $\\R$, take $A_\\mu$ to be the set of equilibria.\nThe equilibria solve $x(\\mu - x^2) = 0$, so the diagram is the line $x = 0$ together with the parabola $\\mu = x^2$.\nThe branches meet at $(0, 0)$, the [[pitchfork bifurcation|P88X34MSVV]] of [[bifurcation theory|D9TW46P19Y]].\nUsually stable branches are drawn solid and unstable ones dashed.\nFor the [[logistic map|QN9X9BNKTX]], one takes for $A_\\mu$ the limit points of the orbit of a typical initial point.\nNon-example: a plot of a single orbit $x_n$ against the time $n$, at one parameter value, is a time series, not a bifurcation diagram.\n",
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