Feigenbaum constant
Named by 5 essays across one field — each of them below, with the objects they name alongside it.
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 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 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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Period-doublingUniversalityBifurcationRenormalisationCritical pointLogistic mapScalingArea-preserving mapAttractorChaosConvergenceDifferential equation