Generator

The logistic map's bifurcation diagram, 2.4 to 4

A generator in the dynamics library, called 49 times across 12 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

bifurcation is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

The logistic map's bifurcation diagram, 2.4 to 4. For each parameter, the values the orbit settles into, plotted as a column of points.

Where the period doubles, and by how much the gaps shrink

Where the Rössler flow's peaks fall, as one parameter changes

The Rössler orbit at four settings of c

The Rössler flow's return map: each peak against the one before

The Rössler return map at four settings of c

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the formula counts the primitive two-colour necklaces of length 1, a necklace and its colour-swap as one ×14
the same formula counts the irreducible polynomials of degree 1 over the two-element field whose roots add to one ×12
the logistic map has exactly as many superstable parameters of period 1 as the formula says ×9
the 2-to-1 merging is inside its bracket ×4
the orbit at r = 2.8 has period 1 ×4
the lower parameter is between 0 and 4 ×2
the number of images is a whole number between 2 and 10 ×2
the parameter is between 3.57 and 4 ×2
the roots of exact period 6 are as many as the formula says ×2
the upper parameter is between 0 and 4 ×2
and so do the gaps of the cascade inside the window ×1
and so does the sine map ×1
and the windows come in the same order of periods ×1
at least three small distances below the window ×1
at the critical parameter f³ touches the diagonal ×1
below the critical parameter the channel is open ×1
chaos, then period 3, then period 6 ×1
each mean is taken over at least a hundred quiet phases ×1
each merging comes at a lower parameter than the one before ×1
one, two, four peaks before repeating, then none ×1
period 1 at c = 2.5, 2 at 3.5, 4 at 4, and a period-3 window at 5.3 ×1
rising to one side of the top, falling to the other ×1
the box holds a diagram and not an empty strip ×1
the box is a genuine rectangle inside the diagram ×1
the column count is a whole number between 40 and 1400 ×1
the diagram is drawn as marks rather than as a bitmap of elements ×1
the distance below the window is small and positive ×1
the doubling points come in order ×1
the doublings come in order ×1
the gap ratio approaches Feigenbaum's constant from this side too ×1
the hump rises as c grows ×1
the largest period included is a whole number between 4 and 7 ×1
the largest period listed is a whole number between 8 and 16 ×1
the largest period marked is a whole number between 3 and 8 ×1
the largest period searched numerically is a whole number between 5 and 10 ×1
the last ratio measured is Feigenbaum's constant to within a twentieth ×1
the main cascade's gaps shrink by Feigenbaum's constant ×1
the next superstable parameter is bracketed where the cascade predicts ×1
the number of doublings found is a whole number between 4 and 7 ×1
the number of doublings is a whole number between 3 and 5 ×1
the number of mergings is a whole number between 3 and 5 ×1
the orbit piles up beside each image of the turning point ×1
the orbit stays between the first two images of the turning point ×1
the orbit takes several steps to pass through the channel ×1
the period is a whole number between 3 and 8 ×1
the period of the window the strand belongs to is a whole number between 1 and 9 ×1
the period-three window has one superstable parameter of period three ×1
the points kept per column is a whole number between 10 and 400 ×1
the quiet phases lengthen as the inverse square root of the distance ×1
the range runs upward ×1
the range runs upward and is wide enough to draw ×1
the ratios climb towards Feigenbaum's constant ×1
the return map has its maximum inside the range — a hump, not a slope ×1
the root found has exactly the period sought ×1
the steps drawn is a whole number between 100 and 2000 ×1
the stretch drawn holds at least two quiet phases ×1
the top is rounded like a parabola, opening downwards ×1
the two maps have the same number of windows up to this period ×1
the view is one of feigenbaum, critical, histogram, merging, channel, laminar, laminarlength, zoom, cascade3, windowcount, superstable, useq, windowsmap, rossbif, rossorbits, rossreturn, rossreturns, rossdelta, rosswindow, zordhero, zordmaps, zordconv, zorddelta, zordscale, zordalpha ×1
α falls as the order of the maximum rises ×1
δ rises with the order of the maximum ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Dynamics

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.

Dynamics

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.

Dynamics

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.

Dynamics

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.

Dynamics

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.

Dynamics

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.

Dynamics

The histogram an orbit leaves

When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

Dynamics

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.

Dynamics

The same map in different coordinates

The tent map and the logistic map at four look nothing alike and are the same map, carried onto each other by a change of variable. Everything either one does the other does, and the change of variable is a sine squared.

Dynamics

The shape in every picture of itself

One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

Dynamics

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.

Dynamics

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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