Where the time goes on an attractor

A closer start buys only time: two orbits of a chaotic system that begin a millionth apart are unrelated within a few dozen steps, and no improvement in the precision of the start pushes that horizon far. Prediction of where the orbit will be is lost for good. The essays before this one measured that loss, found its mechanism in stretching and folding, and found the attractor itself to be a set with no volume and a fractional dimension.

Every average they reported quietly assumed something else: that how the orbit spends its time can still be predicted. The fraction of steps it spends in one region rather than another, the long-run average of any quantity measured along it — these are the things an experiment on a chaotic system actually reports, and they are reproducible. This essay is about the object that makes them so.

The figure below is that object for the Hénon map, the two-line recipe $x \mapsto 1 - 1.4x^2 + y$, $y \mapsto 0.3x$ that has served throughout as the model of a strange attractor. One orbit, a million steps, counted into a grid of small cells, with each cell shaded by how often the orbit visited it. The attractor’s shape is familiar. What is new is the shading: the orbit spends its time unevenly, dense along some bands and sparse along others, and half of the million steps land in fewer than a quarter of the cells it ever visits.

A distribution, not a set #

The attractor is a set of points: everywhere a typical orbit eventually comes arbitrarily close to. The shading is a different kind of object — an assignment of weight to regions, telling what fraction of a long orbit’s time falls in each. Mathematicians call such an assignment a measure, and one that the dynamics does not change from step to step — the orbit spends the same fraction of its time in a region as in that region’s image — an invariant measure.

The distinction matters because two different measures can live on the same set. A coin tossed fairly and a coin loaded two to one both take their values in the same two-element set; what differs is the weighting. The Hénon attractor carries many invariant measures, as a later section shows, and the question here is which one the hero is.

The one-dimensional version of this object has appeared before. The histogram an orbit leaves of the logistic map at full stretch is not flat — it piles up at the ends of the interval, where the map folds — and that piling is its invariant density, a function that can be written down. The Hénon attractor is one dimension up, and there is no density to write down, for a reason the section on area makes precise.

Every picture of an attractor is a picture of its measure #

There is a consequence of the distinction that applies to every drawing of an attractor anybody has made, including every one drawn here.

An attractor is drawn by following an orbit and putting a dot at each step. Where the orbit spends more time, the dots are denser; where it rarely goes, they are sparse, and a region visited once in a million steps receives no dots at all in a drawing of twenty thousand. So a dot picture of an attractor is not a picture of the set. It is a sample from the physical measure, and it shows the set only where the measure is heavy enough to be sampled.

For the Hénon attractor this means the outer tips of the folds, which the orbit visits rarely, are drawn thinner and more broken than the inner bands, although as parts of the set they are just as solid. The hero’s shading makes that bias explicit instead of hiding it in dot density, and it is why the drawings in the essay on dimension needed so many points to show the finer bands at all. Counting boxes to estimate a dimension from an orbit is likewise a count of boxes the measure reaches, and the number it produces is, strictly, a dimension of the measure rather than of the set; the two agree for these attractors, but they need not in general.

Four starts, one histogram #

The first property of the hero’s distribution is that it does not depend on where the orbit started.

Four orbits, four different starting points, four sets of 400,000 positions with nothing in common after the first few dozen steps — and the histograms of where they spent their time are the same to within the noise of counting. Sensitive dependence scrambles which bin the orbit is in at step one thousand. It does not change how many of the first million steps it spent in each bin.

The same holds for any quantity measured along the orbit.

Along each of five orbits, the average of $x^2$ over the first $N$ steps wanders for a thousand steps and then settles on $0.586$ — the same value for all five, to three decimals, by two hundred thousand steps. That is a time average converging to a number that is a property of the system, not of the start. It is the statement the ergodic theorem makes for systems that mix well: the time average of a quantity along almost every orbit equals its average with respect to one fixed measure.

The starts that disagree #

The dashed line in the averages figure is the reason for the word “almost”. The Hénon map has a fixed point, near $(0.631, 0.189)$, and an orbit that starts exactly there stays there for ever. Its average of $x^2$ is $0.631^2 = 0.399$, not $0.586$. It spends all of its time at one point.

The fixed point is not the only exception.

Every periodic orbit is an exceptional start: an orbit that returns to its own beginning after $p$ steps divides its time equally among its $p$ points, and the averages along it are averages over those $p$ points. The figure finds sixteen such orbits of the periods shown, and each gives its own average of $x^2$ — scattered around the typical value, some close to it, none equal to it. There are infinitely many periodic orbits on the attractor, established rigorously by computer for this map, each carrying its own invariant measure. And there are measures that are mixtures of these, and measures spread over sets of orbits that are neither periodic nor typical.

So the attractor carries infinitely many invariant measures, and the question is why typical starts all select the same one. The answer is that the exceptional starts, though infinitely many, occupy no area: the periodic points, and the points whose orbits are eventually drawn into them, lie on curves and countable sets. Pick a starting point at random from any small disc and, with probability one, it is not among them. The measure a start chosen this way produces is called the physical measure — the one an experiment sees, because an experiment cannot place its start exactly on a set of no area.

Why the start is forgotten #

The mechanism that makes the physical measure unique is the same one that made prediction impossible, seen from the other side.

Take a small blob of starting points and follow it. The map stretches the blob along the attractor and squeezes it across, and after a few dozen steps the blob has been drawn out into a long thin filament lying along the attractor’s bands, folded back on itself many times. Its points are then spread over the whole attractor, and spread in proportion to the physical measure — the filament has the attractor’s own distribution. From then on, whatever the blob’s original position was, its points are distributed like the measure. The start has been forgotten, and the forgetting is the stretching.

So the property that ruins forecasts — nearby starts diverge — is the property that guarantees climate. A system that did not stretch would remember its start for ever, and its long-run statistics would depend on where it began, as those of a rotation or a stable cycle do. Mixing is the name for this forgetting, and the rate at which the blob spreads is the rate at which correlations between the present and the past die away.

The analogy with a Markov chain is closer than it looks. Cut the attractor into a few regions and record only which region the orbit is in at each step: the resulting sequence of symbols behaves, statistically, much like a chain that forgets its past, and for uniformly hyperbolic attractors the correspondence can be made exact — the dynamics becomes a shift on symbol sequences, as it did for the base-β expansions whose time distribution was measured. A deterministic rule with no randomness in it produces the statistics of a random process because it forgets.

How long is long enough #

The averages figure shows five orbits agreeing to three decimals after two hundred thousand steps. That precision is itself a measurement, and it is worth reading it as one.

If successive steps were independent, the error in an average of $N$ steps would shrink like $1/\sqrt N$, with a constant set by the spread of the quantity being averaged. For $x^2$ on the Hénon attractor the spread is about $0.4$, so two hundred thousand independent steps would pin the average to about $0.001$ — which is close to what the figure shows. That the chaotic orbit does nearly as well as independent draws says its correlations die within a few steps: after a handful of iterations, where the orbit is tells almost nothing about where it will be.

For the Lorenz flow the corresponding time is longer — a trajectory spirals around one lobe several times before switching, and those spirals are correlated — so the peak histogram needs thousands of peaks to reach a few per cent of agreement. The difference between the two systems is visible in how much data each needs, before anything is known about their measures.

Ninety per cent of the time on no area at all #

The hero shaded cells of a fixed size. Refine the cells and the distribution concentrates.

With sixteen cells across the box, the cells holding ninety per cent of the orbit’s time cover a substantial fraction of the area. With five hundred and twelve across, they cover a few per cent. The share keeps falling, and it must: the Hénon map shrinks every area to $0.3$ of itself at each step, so after $n$ steps the whole trapping region has been squeezed into $0.3^n$ of its area, and the orbit lives inside that. The time is concentrated on a set of zero area.

That is what it means for the physical measure to be singular: it gives all its weight to a set that ordinary area ignores. It has no density — no function whose integral over a region gives the time spent there — because a density would give zero weight to a set of zero area. The logistic map’s histogram converged to a curve; the Hénon map’s histograms, refined, converge to nothing that can be drawn as a surface over the plane. Along the unstable direction, where the attractor stretches, the measure does have a density; across it, where the attractor is a Cantor dust of strands, it does not.

This is the property that makes the physical measure hard to construct. Showing that time averages converge requires controlling the orbit on a set that ordinary area cannot see, and the tools that work for smooth densities do not apply.

The same measure on the Lorenz flow #

The Lorenz flow, the other standard model, has a physical measure too, and its return map is the natural place to see it.

Each peak of $z$ is one point of the return map, and counting how often the peaks fall at each height gives the physical measure of the flow seen on that section. Two trajectories started far apart give the same distribution. Its shape — a single smooth hump spread across the whole range — is also a statement about predictability: knowing the physical measure, the probability that the next peak exceeds forty can be read off as the area of the hump beyond forty, even though which peak will exceed it cannot be predicted at all beyond a few turns.

That is the practical content of the whole essay. Weather is a chaotic flow; the weather on a particular day a month ahead is unpredictable, and the climate — the distribution of weather over many days — is predictable. The physical measure is the mathematical object that the word climate points at.

Sinai, Ruelle and Bowen #

The theory of these measures was built in the 1970s, for attractors with a property the Hénon and Lorenz attractors only approximately have. Yakov Sinai, David Ruelle and Rufus Bowen studied uniformly hyperbolic attractors, on which every point has a direction that is stretched and a direction that is squeezed, at rates bounded away from one, everywhere. For those, they proved that a physical measure exists, is unique, and has a density along the stretched directions — which is exactly the combination the singular-area section found numerically. Such measures are called SRB measures after the three of them.

The Hénon attractor is not uniformly hyperbolic: its folds create tangencies between stretched and squeezed directions, where the uniform rates fail. For the Hénon map, Michael Benedicks and Lai-Sang Young proved in 1993 that an SRB measure exists — but for a set of parameters near a different corner of the family, where the fold is sharp and the squeezing strong, and their set of parameters is large in measure but full of holes. For the classical parameters $a = 1.4$, $b = 0.3$, the existence of a physical measure has not been proved. Every figure on this page is consistent with one, and the reason no proof exists is the subject of a region the orbit cannot leave.

For the Lorenz flow the situation is better. Warwick Tucker’s computer-assisted proof of 1999 showed that the Lorenz equations at the classical parameters have a robust strange attractor of the geometric type studied in the 1970s, and those were already known to carry SRB measures. So the histogram of peaks above is a picture of an object known to exist.

What a million steps cannot show #

Every distribution here is a finite sample of an orbit computed in floating-point arithmetic. The computed orbit is not an exact orbit of the map — rounding errors are amplified exactly as differences in the start are — and whether its statistics are those of some true orbit is a question about shadowing, which for non-hyperbolic maps like Hénon’s is not settled in general. The agreement between starts, the convergence of averages and the concentration on small area are all measured on computed orbits, and they are evidence about true ones rather than proof.

The finite length matters too. Averages converge like the inverse square root of the number of steps when correlations decay fast, and the figures show the three-decimal agreement that a few hundred thousand steps can deliver. They cannot show whether the true physical measure is exactly what the histograms suggest in regions the orbit rarely visits.

And the grids have a finest scale. The singular-area figure stops at 512 cells across, where the share holding ninety per cent of the time is a few per cent. That it tends to zero is an argument from the map’s constant area contraction, not a measurement; the measurement is that it falls at every step of refinement drawn.

Still open: whether the classical attractor exists at all #

The next question is sharper than it looks. Everything above assumed that the Hénon map at $a = 1.4$, $b = 0.3$ has a strange attractor — a chaotic set on which typical orbits wander for ever. It might instead have a stable periodic orbit of very long period, which every typical orbit eventually settles into after an arbitrarily long chaotic-looking transient. The figures could not tell the difference, and neither, so far, can any proof. How much of the attractor’s existence can be proved, and by what kind of argument, is a region the orbit cannot leave.