Period-doubling
Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
The folds that measure chaos
Apply the logistic map six times and its graph goes up and down 38 times at r = 3.5 and 64 times at r = 4. How fast that number of folds multiplies with each further step is the map's topological entropy: zero through the whole cascade of period doublings, log of the golden ratio in the window of three, log 2 at the top — and it never decreases as r rises.
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.
![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.
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.
Small Mandelbrot sets among the cubics
For most cubics Newton's method finds a root from almost every start. For some it does not: a whole region of starting points falls into a cycle and never finishes. Which cubics are the bad ones is decided by a single starting point — the one critical point that is not a root — and when the cubics are laid out in a plane and coloured by where that point goes, the bad ones form small, exact copies of the Mandelbrot set.
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.
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.
Root two, raised to itself for ever
Stack √2 to the power √2 to the power √2 without end and the tower settles at exactly 2. Solve the equation for a tower equal to 4 and √2 appears again, and that answer is false. The bases that give a limit run from e^(−e) to e^(1/e), the slope at the crossing decides everything, and a sum over labelled trees turns out to be the same number.
Named alongside it
The objects these essays reach for when they reach for this one.
BifurcationFeigenbaum constantLogistic mapPeriodic orbitUniversalityChaosRenormalisationCritical pointScalingSelf-similarityAttractorConvergence