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  "body": "\n## Summary\n\nA period-doubling bifurcation, or flip bifurcation, happens in a family of maps when the derivative at a fixed point crosses $-1$.\nThe fixed point changes stability, and an orbit of period $2$ is born from it.\nFor flows, the same happens to a periodic orbit whose Floquet multiplier crosses $-1$: a new periodic orbit of about twice the period appears.\nA sequence of such bifurcations, at parameters that accumulate geometrically, is one of the main routes to chaos, governed by the [[Feigenbaum constants|F1EHT2CPHC]].\n\n## Explanations\n\n### L1 human\n\nPicture a population that settles to the same size each year.\nWhen a parameter such as the growth rate is increased, this steady state can lose stability by overshooting: too large one year, too small the next.\nThe population then settles into an alternating pattern, high and low, of period two.\nThis is a period-doubling bifurcation; in the logistic map $x \\mapsto r x (1 - x)$ it happens at $r = 3$.\n",
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