Bifurcation
Named by 14 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.
How fast two orbits part
The word "sensitive" is an adjective. Averaging the logarithm of one derivative along an orbit turns it into a number — one that says how many steps of prediction the map allows, and whose sign says whether it allows any.
Two lobes and no cycle
Three equations, three variables, and a trajectory that never crosses itself, never repeats, and never leaves a region of zero volume. The set it settles onto is not a point, not a loop, and not a surface.
The staircase that is flat almost everywhere
A map of the circle advances by an average amount each step. Plot that average against the parameter driving it and the graph is flat over an interval at every rational, rises only on a set of measure zero, and still climbs from nothing to one.
How a lock comes apart
Inside a plateau the map has two periodic orbits, one attracting and one pushing away. Track them to the plateau's edge and they run into each other and vanish together — so the edge of a tongue is a collision, and the width of the plateau is how far the two can be pulled apart.
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.
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.
A whole interval of speeds
Below the critical line every orbit of the circle map goes round at the same average speed. Above it the map folds back on itself, and the speed depends on where the orbit starts — not a few different values but a whole interval of them, every fraction in it the speed of some periodic orbit, and almost none of them ever seen by an orbit started at random.
![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.
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.
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.
A crowd of clocks that falls into step
Give a thousand oscillators a thousand different natural rhythms and let each be pulled towards the average phase. Below a definite pull nothing happens; above it a cluster forms and grows, and Kuramoto's 1975 calculation says exactly where: at twice the spread of the frequencies, with the order growing as the square root of one minus the threshold over the pull. A simulation lands on that curve to the third decimal.
Stable cycles a hair from Hénon's chaos
Nobody knows whether the Hénon map at a = 1.4 is chaotic, because a stable cycle in a window too thin to see would look exactly the same. So hunt the windows: a stable orbit of period 19 at a = 1.4001616, proved to exist and to attract by interval arithmetic, in a window 15 hundred-millionths wide and 0.00016 from the classical value — and nothing nearer, down to a spacing of a ten-billionth.
Named alongside it
The objects these essays reach for when they reach for this one.
Periodic orbitLogistic mapChaosOrbitPeriod-doublingSelf-similarityStabilityAttractorUniversalityCircle mapFeigenbaum constantRotation number