The road paved with doublings

One map, one dial. For each setting of the dial, run the map until it has forgotten where it started, then record where it ends up. Plot those records in a column above the dial’s setting, and move along.

That is the most reproduced picture in this subject, and everything in it can be read.

Reading a column #

A column is not a picture of an orbit. It is a picture of the set of values an orbit eventually takes, with the journey there thrown away.

One mark means the orbit settles on a single value: a fixed point. Two marks mean it alternates between two. Four means a cycle of four. A column that is a solid smear means the orbit never repeats, and the smear is the set of values it visits.

Reading left to right, then: a single curve, which splits into two, which each split into two, and so on — until the splitting stops being resolvable and the diagram goes dark.

The first split, and why it is where it is #

The single curve on the left is the fixed point $1 - 1/r$, and it ends at $r = 3$.

That is not an accident of the drawing. The slope test says the fixed point attracts while $|2 - r| < 1$, so it stops attracting exactly when $r = 3$ — and what appears instead is a two-cycle. The picture’s first fork is an inequality becoming an equality.

The same reasoning gives the second fork. A two-cycle is a fixed point of the map applied twice, its slope is the product of the two slopes, and when that product passes $-1$ the two-cycle stops attracting and a four-cycle appears. There is nothing new in the mechanism; it is the same bifurcation happening to a more complicated object.

The second fork, computed #

The paragraph above says the second fork works the same way as the first. It is worth doing, because it is the last one that can be done, and knowing where the closed forms stop is most of why the next rung exists.

A two-cycle is a pair $p, q$ with $f(p) = q$ and $f(q) = p$. Both are fixed points of $f$ applied twice, and dividing that quartic by the two ordinary fixed points already known leaves a quadratic. Its roots satisfy

$$p + q = \frac{r+1}{r}, \qquad pq = \frac{r+1}{r^{2}},$$

which is all that is needed — the individual values never appear again.

The cycle attracts while the slope of the twice-applied map at either of its points has size below one, and by the chain rule that slope is the product $f'(p)f'(q)$, which for the logistic map is $r^2(1-2p)(1-2q)$. Expanding and substituting the two symmetric quantities above collapses it to a polynomial in $r$ alone:

$$-r^{2} + 2r + 4.$$

Now read both ends off it. Setting the multiplier to $+1$ gives $r^2 - 2r - 3 = 0$, whose positive root is three — the parameter at which the two-cycle is born, with the fixed point handing over its stability exactly as it loses it. Setting the multiplier to $-1$ gives $r^2 - 2r - 5 = 0$, whose positive root is

$$1 + \sqrt6 = 3.449489\ldots,$$

and there the two-cycle stops attracting and the four-cycle appears. The first two forks are a root of a quadratic each, and both are visible in the figure at exactly those places.

The third fork is where this stops. A four-cycle’s stability condition is a polynomial of degree twelve in $r$, its relevant root is near $3.544090$, and it has no expression in radicals of any use. Every fork after that is worse, and none of the accumulated thresholds has a closed form.

That is the reason the subject changed the question. With the exact thresholds out of reach, what remains measurable is how they space — and the first three already say something. The gap from the first fork to the second is about $0.4495$; from the second to the third, about $0.0946$. Their ratio is a little over four and three-quarters.

Run the same division on later gaps and the ratio drops and settles, and what it settles on is neither an accident of this map nor a number anyone chose: it is the constant the next rung is about, and it is the same constant for the tent map’s smooth cousins and for every other hump. The exact positions of the forks are a property of the logistic map; the rate at which they close in is not.

The cascade, and why it accelerates #

What is new is the rate. The forks come faster and faster, and by the fifth one they are too close together to separate by eye.

The reason is the chain rule. A cycle of length $n$ has slope equal to the product of the map’s slopes at its $n$ points, and a product of many numbers moves faster than a product of few as the parameter changes. Each doubling therefore needs less room than the one before, by a factor that settles down to something specific.

Because the intervals shrink by a roughly constant factor, they form something close to a geometric series — and a geometric series with a ratio bigger than one converges. Infinitely many doublings fit into a finite stretch of the dial. They accumulate at

$$r_\infty = 3.569945672\ldots$$

which is where the cascade ends and the picture goes dark.

What is past the end #

To the right of the accumulation point the orbit is no longer periodic. It fills out bands, and the bands merge as $r$ rises, until at $r = 4$ the orbit visits the whole interval.

Calling this “chaos” is standard and slightly unhelpful, because the word does most of the work in the sentence. What is actually true is three things at once: the orbit is completely determined by its starting point, it never repeats, and nearby starting points end up unrelated. The third of those is what makes it unpredictable, and it is the only one of the three that has anything to do with the ordinary meaning of the word.

The windows #

The dark region is not uniformly dark. Look at it and there are gaps — vertical bands where the smear collapses back to a small number of marks.

Every window contains its own complete period-doubling cascade — three to six to twelve to twenty-four — and its own accumulation point, and its own windows inside that. The diagram contains copies of itself at every scale, which is what makes it a fractal rather than merely a complicated curve.

That is worth pausing on. The picture was produced by a rule with one parameter and no self-reference in it whatever. The self-similarity is a consequence, and it is the same consequence for every map with a hump.

Where the diagram came from #

The picture has a date and a discipline, and both are surprising.

The logistic map is a model of a population that breeds and then runs out of food: this year’s numbers multiplied by a growth rate, multiplied again by whatever fraction of the habitat is still free. Biologists had been writing it down since the 1840s. What nobody had done was turn the growth rate up and watch, because a model whose answer refused to settle was assumed to be a model somebody had got wrong.

Robert May, an ecologist, did it in 1976 and published the result in Nature under a title that reads as a warning rather than a finding: Simple mathematical models with very complicated dynamics. His point was directed at his own field. A population fluctuating wildly year on year had always been read as evidence of a complicated environment — weather, predators, disease — and here was the same fluctuation coming out of one equation with one parameter and nothing external at all.

That is the transferable claim, and it is why the diagram escaped biology within a decade. Irregular data is not evidence of a complicated cause. Whatever else this picture is, it is a standing objection to the reflex of explaining variation by finding more variables.

The order the periods arrive in #

The windows are not in an arbitrary order, and the theorem that says so is one of the strangest in the subject.

Sharkovskii, in 1964, showed that the periods of a continuous map of an interval arrive in a fixed order — and that the order begins

$$3,\ 5,\ 7,\ 9,\ \ldots,\ 2\cdot3,\ 2\cdot5,\ \ldots,\ 8,\ 4,\ 2,\ 1$$

with the odd numbers first, then twice the odds, then four times the odds, and the pure powers of two at the very end. If a map has a cycle of some period, it has cycles of every period later in that order.

The famous corollary is the first entry. A map with a cycle of period three has cycles of every period whatever — the result Li and Yorke published in 1975 under the title Period Three Implies Chaos, unaware that Sharkovskii had proved the stronger statement eleven years earlier in Ukrainian.

So the period-three window is not one window among many. It is the last kind to appear as the parameter rises, and its appearance certifies that everything else is already there.

The theorem’s shape is worth admiring separately from its content. It is a statement about every continuous map of an interval — no smoothness, no formula, no parameter — and it says that an ordering exists which all of them obey. Nothing about a map is assumed except that it does not tear the interval, and the conclusion is a rigid combinatorial fact. That is closer in spirit to Brouwer’s theorem than to anything else in this field: a topological hypothesis, and a conclusion nobody would guess from it.

The rediscovery is worth a line too, because it is a recurring failure rather than an anecdote. Li and Yorke’s paper was written in English in a widely read journal; Sharkovskii’s was in Ukrainian in a journal almost nobody outside the Soviet Union saw. Eleven years and a stronger theorem separated them, and the name that stuck to the field — chaos — comes from the later, weaker paper’s title.

Counting what is in there #

A question the diagram invites and does not answer: how many cycles does the map have at a given parameter?

The count is exact and it grows explosively. A cycle of period $n$ is a fixed point of the map applied $n$ times, and the graph of that iterate has about $2^{n-1}$ humps, each of which can cross the diagonal twice. Past the accumulation point the logistic map really does have about $2^n/n$ cycles of period $n$, for every $n$ — thousands of them by period twenty, and infinitely many altogether.

All but at most one of them are repelling. That is the resolution of what looks like a contradiction between this essay and the last one: the diagram shows a single attractor above each parameter because an attractor is what an orbit finds, and the enormous population of cycles it does not find leaves no mark on the picture at all.

So the dark region is not a region where structure has broken down. It is a region so full of structure that none of it is visible, because a drawing made by running orbits can only ever show the parts that attract. A picture of the outcome is not a picture of the system, and this is the clearest case of that in the collection.

How the picture is actually made #

Every column here discards a transient before recording anything, and the number discarded is a judgement rather than a measurement.

Near a bifurcation the discarding matters most, because convergence there is slow: at exactly $r = 3$ an orbit takes thousands of steps to approach the fixed point, so a column computed with a short transient shows a smear where the true attractor is a point. Every published version of this diagram has that artefact somewhere, and it is invisible to a reader who does not know to look for it.

The figures here settle four hundred steps before recording sixty, and the check that keeps them honest is not the transient count but the period: at four parameters the generator counts the period off the settled orbit and compares it with what the cascade requires. A transient still running would give the wrong period and the build would stop.

What the picture cannot show #

The horizontal axis is a continuum and the drawing has four hundred and sixty columns. Between any two of them is an interval containing infinitely many windows, and no drawing resolves them.

That is not a defect of resolution but of kind. The set of parameters at which the map is chaotic and the set at which it is periodic are interleaved everywhere: in any interval, however small, there are parameters of both kinds. The windows have positive total width, and so does the chaotic set, and neither contains an interval free of the other.

So a reader looking at a dark region and concluding “the map is chaotic here” is making a statement that is false at densely many parameters inside it — including, quite possibly, the exact one they care about. What is true is a statement about measure: most parameters in the dark region, in the sense of area, are chaotic. Jakobson proved that in 1981 and it is much harder than the picture makes it look.

There is a second thing the picture hides, and it is the one that costs money. A physical system sitting in a narrow window behaves periodically; nudge its parameter by a thousandth and it is chaotic. Nothing about the system announces which side of a window edge it is on.

And a third, which is about the vertical axis rather than the horizontal. A column is drawn from a finite sample of a set that is usually infinite, so the density of marks in it carries no information — a value visited a thousand times more often than another is drawn at the same weight. The genuine object above each chaotic parameter is a probability distribution rather than a set, and this diagram shows its support and discards its shape. For the logistic map at $r = 4$ that distribution is known exactly, and it is not uniform: it piles up at both ends, so an orbit spends far more of its time near $0$ and $1$ than the even smear in the picture suggests.

The same picture, from a different map #

The last thing to say about this diagram is that it is not really about the logistic map.

Run the same procedure on the tent map, on $r\sin(\pi x)$, on any map of an interval with a single smooth hump, and the same cascade appears: doublings accelerating to an accumulation point, then bands, then windows in the same Sharkovskii order. The parameters differ. The shape does not.

The word for that is universality, and it is the most surprising thing in this field — a word this collection has used before, for the bell curve arriving from any starting distribution, and meaning the same thing here: not that a simple rule produces complicated behaviour, but that different simple rules produce the same complicated behaviour, with the same numbers attached.

Doubling in the other direction #

One more feature of the diagram is worth naming because it is visible and counter-intuitive: the bands past the accumulation point merge as the parameter rises, in a mirror image of the way the branches split before it.

Just past $r_\infty$ the orbit is confined to $2^k$ narrow bands and visits them in a fixed cyclic order — so it is still, in a weak sense, periodic: which band it will be in is completely predictable, and only where in the band is not. As $r$ rises the bands merge in pairs, halving the count, until at $r = 4$ there is one band and the orbit ranges over the whole interval.

So the picture has two cascades running towards each other, one of periods doubling and one of bands halving, meeting at the accumulation point. Reading it in that direction makes $r_\infty$ look less like an ending and more like a boundary between two ways of counting the same thing.

The band structure is also the honest answer to “is this system predictable”. Between $r_\infty$ and about $3.68$, a forecaster who is asked which band the orbit will be in a hundred steps hence is right every time; one asked for the value is wrong immediately. Which of those counts as prediction depends entirely on the question, and this collection’s other unpredictable system has exactly the same double character.

Where the ladder goes next #

Those numbers are the next rung. The intervals between doublings shrink by a factor that approaches something specific, and that something is the same for every map with a smooth hump — a constant that belongs to the cascade rather than to any map in it.

The other direction is what the dark half of the diagram means for prediction. Every column in it is a set of values an orbit visits without repeating, which means two orbits started next to each other end up in different parts of that set: a difference too small to draw becomes the whole difference, and it does so in a number of steps that can be counted.