The staircase that shows the whole orbit

A rule that turns one number into another can be applied to its own output. Take $x$, compute $f(x)$, then compute $f(f(x))$, and keep going. The sequence that comes out is called the orbit of the starting point, and almost every question in this field is about what it does in the long run.

The difficulty is that an orbit is a list of numbers, and a list of numbers is the least legible thing in mathematics. There is a way of drawing it that fixes this completely.

The construction is two moves, repeated. From a point on the diagonal, go vertically to the curve — that is applying the map, since the curve’s height above $x$ is $f(x)$. Then go horizontally back to the diagonal — that is taking the answer and treating it as the next input, since the diagonal’s height above $y$ is $y$.

Every corner of the resulting staircase is a step of the orbit. The picture is not an illustration of iteration; it is iteration, with the arithmetic replaced by two rulers.

What the shape says #

The value of the drawing is that the orbit’s fate is a property of the staircase’s shape, and shapes are what eyes are for.

A crossing of the curve and the diagonal is a point with $f(x) = x$ — a fixed point, a value the map leaves alone. The staircase above walks into one and stays. That is the tamest thing an orbit can do and it is the behaviour most rules have most of the time.

A closed rectangle is a two-cycle: $f(a) = b$ and $f(b) = a$. The orbit alternates forever, and the fixed point is still there — the curve still crosses the diagonal — but the staircase no longer goes to it.

Three parameters of one map, three completely different long-run behaviours, and the difference is visible before any of it is calculated. That is what the drawing is for.

It is worth noticing how little separates them. The three figures differ in one coefficient, by amounts a reader would call negligible if the number appeared in a measurement — $2.8$, $3.2$, $3.9$ — and the behaviour they produce is not a graded change but three different kinds of thing. A system that settles, a system that alternates and a system that never repeats are not points on a scale, and nothing in the rule announces which one a given coefficient will give.

The same orbit, plotted against time #

A cobweb plots the orbit against itself. The obvious alternative is to plot it against the step number, which is what a reader who had never seen a cobweb would draw.

The comparison is the argument for the cobweb, and it generalises past this field. A time series shows what happened; a plot of the orbit against itself shows the rule that made it happen. The second is what a reader needs in order to predict anything, and the first is what the data actually is.

That is why this collection reaches for the cobweb even when the sequence is short: it puts the mechanism and the outcome in one picture, and the mechanism is the part that generalises.

One picture, three fields #

The cobweb is worth recognising because it is not confined to this subject. Any process that produces its own next input has one, and three that this collection has already built are exactly that.

Euclid’s algorithm takes a pair and returns a pair, and the sequence of leftovers is an orbit; the reason it terminates is that the map is a contraction on the whole numbers, which is the same statement as a staircase walking into a fixed point. Newton’s method is a map from a guess to a better guess, and its whole theory is the question of which starting points have orbits that converge. Even the continued fraction is an orbit: the Gauss map $x \mapsto \{1/x\}$ applied over and over produces the quotients, and the reason the golden ratio’s expansion is all ones is that its orbit is a fixed point of that map.

So a reader who learns to read a staircase gets three earlier essays back with something added, which is the argument for the picture rather than for the subject. What is new here is only the willingness to ask what the orbit does forever, rather than treating it as a means to an answer.

The transient, and why it is discarded #

The first few steps of a staircase usually go somewhere the rest of it does not. An orbit started at $0.2$ takes a while to reach the two-cycle, and those early steps are called the transient.

Every figure in this field that claims to show “what the map does” has thrown a transient away first, and the number of steps discarded is a judgement rather than a computation. Discard too few and the picture shows a starting point rather than a map; discard too many and a slow approach is reported as an exact arrival.

That is not a small caveat. Some maps have transients thousands of steps long, and a system can spend a very long time doing something it will eventually stop doing — transient chaos, where an orbit wanders convincingly for a thousand steps and then drops into a two-cycle. Nothing in a picture of the first hundred steps distinguishes that from chaos that lasts forever.

Different maps, same construction #

Nothing in the cobweb is about the logistic map. Any rule from an interval to itself can be drawn this way, and swapping the curve swaps the subject.

The doubling map is the simplest chaotic system there is, and it is chaotic for a reason a reader can hold: in binary, multiplying by two shifts the digits left and dropping the whole part discards the leading one. So the orbit of $x$ is the sequence of tails of its binary expansion, and asking where the orbit goes is asking what the digits of $x$ do — which for almost every $x$ is nothing describable at all. That is the same “almost every” that governs the continued fractions of real numbers: a complete description of the typical case and nothing about any case anybody can name.

The tent map is worth having beside the logistic one because it removes a possible explanation. Whatever produces the wandering, it is not the curvature of the parabola: a rule made of two straight lines does the same thing. What the two have in common is a hump — they fold the interval over itself — and folding turns out to be the whole story. A rule made of two straight lines producing behaviour nobody can summarise is the same surprise that a triangle of parities produces, and for a related reason: simple local rules are under no obligation to have simple global consequences.

Why the hump is compulsory #

A rule with no hump cannot do anything interesting, and the reason is worth one paragraph because it bounds the whole subject.

Suppose $f$ is increasing everywhere on the interval. Then the orbit of any point is monotone: if $f(x) > x$ the next step is larger still, and the sequence rises until it meets a fixed point. It cannot oscillate, because oscillating requires overshooting, and an increasing map never sends a smaller input to a larger output than it sends a larger one. So every orbit of every increasing map converges to a fixed point, and the entire long-run behaviour is decided by where the curve crosses the diagonal.

That is a complete theory of a whole class of maps, obtained by looking at a picture and noticing that the staircase has nowhere else to go. What it means is that everything below — cycles, chaos, the bifurcation diagram, the Feigenbaum constant — needs a map that turns around somewhere. One hump is enough, and one hump is what every map in this field has.

Stretch and fold #

The mechanism underneath every picture on this page has two moves, and they are the two moves a baker uses on dough.

Stretch. Near the top of the hump the map takes a short interval and returns a longer one. Two nearby points move apart, and repeating that separates them exponentially.

Fold. The interval has nowhere to go — the map sends $[0,1]$ into $[0,1]$ — so the stretching has to be paid for by folding the interval back over itself. Points that were far apart are brought together.

Stretching alone gives escape to infinity, which is boring. Folding alone gives contraction to a point, which is also boring. Doing both, forever, gives an orbit that never settles and never leaves: nearby points separate, distant points are brought together, and no amount of running the rule produces a repeat.

That is the whole of what “chaotic” means here, and it is visible in the cobweb: the curve has to rise above the diagonal and come back down.

Counting the crossings #

A question the picture answers immediately, and which is fiddly to settle any other way: how many cycles does a map have?

A fixed point is a crossing of the curve with the diagonal. A two-cycle is a fixed point of $f$ applied twice — so it is a crossing of the curve of $f \circ f$ with the diagonal, and $f \circ f$ is a curve with two humps. Three-cycles are crossings of a four-humped curve, and in general the graph of $f$ applied $n$ times has about $2^{n-1}$ humps for a map with one.

Each hump can cross the diagonal twice, so the number of periodic points grows exponentially with the period. That is a counting argument of exactly the kind the number field runs on, made here about intersections rather than lattice points, and it says something a picture of one orbit cannot: a chaotic map has infinitely many periodic orbits, densely packed, and an orbit chosen at random hits none of them.

Both halves of that are worth holding together, because they sound contradictory and are not. Periodic points are everywhere; the set of them has measure zero; so a starting point picked by throwing a dart misses every one of them with probability one. A dense set that a dart misses with certainty is not a paradox, and the rationals on a number line are the same situation in a setting with no dynamics in it at all. The picture at $r = 3.9$ is wandering because the exact periodic orbits it is passing near are all repelling.

Where the diagonal earns its place #

The diagonal looks like scaffolding and is the load-bearing part of the drawing, which is worth saying because a reader can be forgiven for treating it as a guide line.

Three separate things are the diagonal. It is the set of fixed points — every crossing with the curve is one. It is the return path: the horizontal move that turns an output into the next input only works because the diagonal’s height above $y$ is $y$. And its slope of one is the reference the curve’s slope is compared against, which is what decides whether a crossing attracts.

Take the diagonal away and none of the three survives. There is no way to feed the output back in, no way to see a fixed point, and nothing to compare a slope with. This is why the cobweb has no useful variant without it, and why every drawing of iteration ever made has one.

The same construction on a map from a plane to itself has no diagonal, because the graph of such a map lives in four dimensions. That is the reason this field’s elementary pictures are all one-dimensional, and the reason the Lorenz system has to be drawn as a trajectory rather than as a staircase: the trick does not survive the extra dimension, and what replaces it is a picture of the orbit alone with the rule left out.

What the picture cannot show #

The staircase is drawn for one starting point, and the interesting statements are about all of them.

Worse, the drawing is finite in a way the subject is not. Sixty steps of a chaotic orbit look like sixty steps of a random sequence, and no number of steps anybody can draw distinguishes them. The distinction is that the orbit is determined — the same start gives the same sequence every time — and determinism is invisible in a picture of an outcome. A reader looking at the $r = 3.9$ figure is looking at a completely predictable object and cannot tell.

There is also a limit that is arithmetic rather than graphical. A computed orbit of a chaotic map is wrong almost immediately: rounding at the sixteenth decimal place is a perturbation, perturbations grow exponentially, and after fifty steps the drawn orbit shares nothing with the true orbit of the stated starting point. Every chaotic figure on this site has that property.

What rescues it is a theorem rather than a hope. The shadowing lemma says that for these maps a computed orbit, wrong as it is, stays close to the true orbit of some other nearby starting point. So the picture is an honest picture of the system, and a dishonest picture of the number in its caption. Which of those a reader needs depends on the question, and it is worth saying which is being offered.

Where it goes next #

The three figures at the top of this essay differ in one number, and the difference between settling, cycling and wandering is a matter of where that number sits. Following it across its whole range gives the bifurcation diagram, which is the most reproduced picture in the subject and the next rung of this ladder.

Before that, the crossing itself is worth understanding. Both the settling staircase and the escaping one have a fixed point; one walks in and the other walks out, and what decides which is a single number — the slope of the curve where it crosses — that can be read off the picture. It also connects this field to one already built here: Brouwer’s theorem guarantees a fixed point exists for any continuous map of a disc, and says nothing whatever about whether anything will go there.

The other direction is the one the last figure opened. A map that stretches and folds separates nearby starting points, and what that does to prediction is the subject this field is best known for. The doubling map is the cleanest case, because its orbit is literally the digits of its starting point — so asking what the system will do in fifty steps is asking for the fiftieth binary digit of a number nobody measured to fifty places.