Period-doubling — the series
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The road paved with doublings
Turn one dial slowly and watch what a map settles into. It settles on a point, then on two points, then four, then eight — faster and faster, and the doublings run out at a parameter that is finite.
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A constant that does not care which map
The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.
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The dark lines are one point's orbit
Past the end of the period-doubling cascade the bifurcation diagram turns into grey bands crossed by darker curves. Every one of those curves is the orbit of a single point — the top of the hump — and the places where they meet are exactly where the bands merge, in a second cascade that runs backwards at the same rate.
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The window that opens with a stutter
The period-three window does not fade in. At r = 1 + √8 a cycle of three appears out of nothing, and just before it does, the chaotic orbit keeps imitating the cycle that is not there yet — for twenty steps, then fifty, then hundreds, in quiet stretches whose length grows as one over the square root of the distance to the window.
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![A strand of the logistic map's diagram between r = 3.8284 and 3.8572, upside down. The bifurcation diagram of the logistic map cropped to r in [3.8284, 3.8572] and x in [0.44, 0.565], with the value axis inverted.](/f/a-strand-of-the-logistic-map-s-diagram-between-r-3-8284-36b84e6c.svg)
Every window is the whole diagram again
Zoom into one strand of the period-three window of the logistic map and the whole bifurcation diagram is there again — a single value, a fork, a cascade of doublings, chaos, bands and windows — upside down and ninety-seven times narrower. The doublings inside it shrink by Feigenbaum's 4.669, exactly as the originals do, because near the top of its hump the map's third iterate is itself a one-humped map, turned over.
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Windows counted like necklaces
The logistic map has one window of period three, two of period four, three of period five, five of period six, nine of period seven — and the sine map, which shares no algebra with it, has exactly the same numbers, in exactly the same order along the parameter. The counts are 1, 1, 1, 2, 3, 5, 9, 16, 28, 51, and they are the number of ways to thread a necklace of beads in two colours.
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A flow that doubles like a parabola
Otto Rössler's three equations have one nonlinear term and a single knob. Turn it and the orbit closes after one loop, then two, then four, then never — the logistic map's cascade, in a flow in three dimensions. Record each peak of the orbit against the one before and the reason appears: the points lie on a single rounded hump. The gaps between doublings shrink by ratios 3.49, 4.54, 4.59, climbing towards Feigenbaum's 4.669, and the chaos contains a period-3 window that repeats the whole cascade.
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One constant for each shape of top
Feigenbaum's 4.669 is the same for every smooth one-humped map — provided the hump is rounded like a parabola. Make the top flatter, like the graph of x to the fourth, and the doublings crowd together at 7.285; make it sharper, like x to the power 1.25, and they crowd at 3.260. Measured across twenty shapes of top, the constant is a smooth rising curve with 4.669 as one point on it, and the spatial constant falls as it rises.
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Doubling without an attractor
Feigenbaum's 4.669 holds for every map that folds and loses area. A map that keeps area — a frictionless pendulum sampled once a swing — has no attractors at all, yet its periodic orbits double just the same: a period-1 orbit at a = 3, period 2 at a = 4, then 4.1205, 4.1344, 4.1360 and on. The intervals shrink by 8.721 each time, not 4.669, and the orbits by 4.018, not 2.503. Let the map leak a little area and the cascade starts with one constant and finishes with the other.
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An orbit stuck to an island
A chaotic orbit in the standard map stretches at 0.44 a step, and then for 19,000 steps it almost stops stretching at all. It has wandered into the fringe of an island of regular motion and been caught. So gaps between visits to a box do not die away exponentially as they would in a uniformly chaotic map; they fall as a power, and an orbit of 400 million steps still has not seen the longest.