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  "body": "\n## Summary\n\nA bifurcation in a family of maps $f_\\mu$ is a parameter value $\\mu_0$ at which the qualitative behaviour of the iterates of $f_\\mu$ changes.\nTypical signs are fixed points or periodic orbits that appear, disappear or change their stability.\nIt is the setting of [[bifurcation theory|D9TW46P19Y]] for [[discrete-time dynamical systems|5AA9AJEARJ]], that is, for difference equations $x_{n+1} = f_\\mu(x_n)$.\nFor one-dimensional maps the basic local types are the saddle-node, the transcritical, the pitchfork and the [[period-doubling bifurcation|NQDTTZ59X8]].\n\n## Explanations\n\n### L1 human\n\nPicture a recursion with a knob, such as the growth rate in a population model.\nFor most knob settings, turning it a little changes nothing important: the same number of resting states, each attracting or repelling as before.\nAt a few special settings something new happens, for example a resting state loses its pull and a two-step cycle appears.\nThose settings are the bifurcations.\nIn the logistic map $x \\mapsto r x (1 - x)$, the value $r = 3$ is one: below it solutions settle to one value, just above it they alternate between two values.\n",
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