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  "body": "\n## Summary\n\nA period-doubling cascade is an infinite sequence of [[period-doubling bifurcations|NQDTTZ59X8]] in a [[family of maps|EE9X3NZW81]].\nIn it, stable cycles of periods $1, 2, 4, 8, \\dots$ appear one after another.\nThe parameter values of the bifurcations increase to a finite limit, beyond which the dynamics can be chaotic.\nFor the logistic map $x \\mapsto r x (1 - x)$ the cascade starts at $r = 3$ and accumulates near $r \\approx 3.5699$.\nFor typical smooth unimodal maps, the gaps between successive bifurcation values shrink at a universal geometric rate.\nThis rate is the [[Feigenbaum constant|F1EHT2CPHC]] $\\delta \\approx 4.6692$.\n\n## Explanations\n\n### L1 human\n\nTurn up the growth rate in a simple population model.\nFirst the population settles to one value, then it alternates between two values, then it cycles through four, then eight, and so on.\nEach doubling happens sooner than the last, and the doublings pile up at a finite setting.\nPast that point the population can jump around in a way that never repeats.\nThis road from order to chaos is seen in many models and in experiments with fluids and electronic circuits.\n",
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