Period-doubling
Named by 4 essays across one field — 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Logistic mapBifurcationChaosAttractorFeigenbaum constantPeriodic orbitSelf-similarityConvergenceCritical pointGolden ratioInvariant densityLyapunov exponent