An orbit stuck to an island

Doubling without an attractor followed period doubling into maps that keep area. There, nothing attracts: an orbit never settles, and every cycle in the cascade is surrounded by closed curves and chains of little islands. The essay ended on a question about how much of the plane near such an orbit is islands and how much is chaotic sea. This essay asks a question that turns out to be more interesting. What does the existence of those islands do to the chaos around them?

The expected answer is that it does nothing. A chaotic orbit should forget its past quickly, and the island’s only role would be to take up room. That picture is correct for maps without islands, such as Arnold’s cat map, whose statistics become uniform within a few dozen steps. It is wrong for maps with islands, in an instantly visible way. A sea orbit can go for tens of thousands of steps behaving as if it were regular, and the frequency of such stays decays only as a power of their length. The orbit is sticky. That feature changes every long-run average of the system, it has been studied in the motion of asteroids and of charged particles in magnetic fields, and it has resisted a complete analysis for fifty years.

The sea and its holes #

The test subject is the standard map, which the last circle to break has already studied for what happens at the golden number:

$$p' = p + \frac{K}{2\pi} \sin 2\pi x, \qquad x' = x + p',$$

both modulo 1. It keeps area exactly, so nothing can be attracting. At small $K$ almost every orbit lies on a smooth closed curve. At $K = 2$ the picture is mixed. A single orbit started next to the hyperbolic fixed point wanders all over a connected region, the sea. Inside the sea are holes that it never enters, and when an orbit is started inside one of them it stays there, tracing out a closed curve or a chain.

That the two kinds of motion share the plane is a theorem of sorts. Kolmogorov, Arnold and Moser’s theory guarantees that closed curves survive perturbation near an elliptic point, while the sea arises where they have broken up. A generic area-preserving map is thought to be always mixed: an elliptic point with a chain of islands around it is a robust structure, created and destroyed as parameters change, and it is believed that islands exist at nearly all values of $K$, though this has not been proved. Lyapunov estimates sort the orbits, with the regular ones below 0.01 in a run of 3,000 steps, and that is how the blue seeds of the figure were picked.

Waiting for a return #

Choose a small box in the sea, here of side 0.08 around the hyperbolic point, and follow one long orbit, recording the time between each visit to the box and the next. Then ask what share of those gaps exceed a given length. For a uniformly chaotic system, the answer is a theorem: visits to a small box behave like a Poisson process, so the gaps’ tail decays exponentially, at a rate set by the size of the box.

The cat map does just that. Over 4 million steps its longest gap is 2,896 and the curve drops vertically. The standard map starts out the same way, since most returns are quick, and then the curve bends over and runs in a nearly straight line across five decades. A power law with exponent 1 or 2 is enormously heavier than an exponential: in the run of 400 million steps there is a gap of more than half a million. A fit between 1,000 and a million gives a slope of $-1.65$.

Boris Chirikov and Dima Shepelyansky reported in 1984 that the tail of these return times in an area-preserving map falls like a power with an exponent close to 1.5, a figure that was refined over decades by larger computations. Giampaolo Cristadoro and Roland Ketzmerick in 2008 argued for a universal value of about 1.57, using longer runs than anybody had managed. The present measurement of 1.65 is not a precise check on that. It is one run at one parameter, and a fit over a range on which the curve visibly wobbles. Such wobbles are real, and they have been explained as the signature of successive levels of the island hierarchy. So a slope measured over three decades can be off by a tenth, and the figure shows the shape and does not determine the number.

The consequences of a power-law tail are stark. Its mean exists, since the mean return time is fixed exactly at the inverse of the box’s area by Kac’s lemma, but its variance is infinite at an exponent below 2. That has a blunt practical meaning. A time average over a chaotic orbit converges, as the ergodic theorem requires, but its fluctuations do not obey the central limit theorem in the usual way, and the error is dominated by the single longest stay.

One orbit caught and released #

The stickiness can be seen in a single run, by following the stretching rate over short windows.

For most of the time, the local exponent bounces around its mean of 0.44. Then, at about 30,000 steps into this segment, it falls to almost nothing and remains there for 19,000 steps. For that stretch the orbit is doing nothing that distinguishes it from a regular one, with an exponent below 0.06 in every window. Then it leaves, as abruptly as it came, and resumes its chaotic behaviour. There is no external event, and no parameter changed. Over longer runs, longer stays occur.

This matters to anything estimated from a simulation. A Lyapunov exponent calculated from a run of a few thousand steps depends strongly on where the run starts. If it started during this episode, the estimate would be near zero and the orbit would be declared regular. And it says that the very idea of a typical chaotic orbit over any fixed length of time is misleading here, in a way that it is not for a map without islands. The picture of chaos as something that forgets its past quickly applies only to a part of the sea at a time.

The contrast with the theorem that almost every orbit is fair is worth stating exactly, since nothing here contradicts it. Within the sea, time averages do converge to space averages, so far as anyone believes the sea to be one ergodic component. What fails is the speed. In the cat map, which always comes back, a time average over $n$ steps has an error of order $1/\sqrt n$, as for independent samples. With a heavy-tailed stay, the error decreases more slowly, so that a computed average may look settled for a long time and then shift, when a long stay happens. A run that has seen no long stay is not one that has measured the right answer. It is one whose sample happens not to have reached the rare events that dominate its error.

Where the delay comes from #

Why should an orbit stay? The answer lies in what surrounds an island, which turns out to have structure at all scales.

The outermost closed curve of an island is not where the regular motion stops. Beyond it are the remains of curves that have broken up. As the last circle to break explained, a circle at an irrational rotation number does not simply vanish as $K$ passes its breakup value. It becomes a cantorus, a Cantor set that is invariant and has rotation number the same, with gaps through which an orbit can leak. Ian Percival described these sets around 1980, and Robert MacKay, James Meiss and Percival showed in the 1980s that the flux through one can be extremely small. A cantorus just above its breakup value behaves as a nearly impenetrable wall. An area of phase space between two cantori is entered and left slowly, like a room with a tiny door.

Around the island, then, is a nested set of partial walls. A sea orbit that gets through the outermost finds itself in a chamber whose door to the next chamber is smaller still. It can go further in, but the probability of doing so at each level is small. And each island chain has its own satellite islands, each surrounded by a similar structure. This is the “islands around islands” picture, and the theory of how the time spent scales from one level to the next, due to Meiss and Edward Ott in 1985 among others, treats it as a Markov tree, in which an orbit moves up and down a hierarchy with given rates. The tree gives a power law with a universal exponent, in a suitable limit. That exponent has been the subject of a long debate. Its value in the model of Cristadoro and Ketzmerick is the 1.57 quoted above, whereas earlier models gave somewhat different numbers, and it is not known to what extent a real map follows any such tree.

There is a clean analogy with the Sinai billiard and the stadium, where no islands exist yet recurrences follow a power law. The stadium’s trajectories can bounce between its two parallel walls for a long time. Its decay is polynomial for a simple geometric reason, not a hierarchical one. So a power law on its own does not prove the presence of islands. The explanation differs from one system to another.

What stickiness does to transport #

The map above wraps $p$ round a circle. Unwrap it, so that $p$ can grow without bound, and a chaotic orbit’s momentum performs something very like a random walk. Each kick adds an amount set by the position, and for large $K$ those kicks are nearly independent, so the mean square of $p$ grows linearly in time, with the diffusion rate that a walk’s own central limit theorem would predict. Chirikov computed that rate, and there are oscillating corrections to it as $K$ varies.

But at special values, $K$ near $2\pi$ and its multiples, a small set of islands has a different character. Inside them the momentum grows by a whole number each step, steadily and for ever, and they are known as accelerator modes. An orbit from the sea that sticks near one of those islands is carried along with it, for a stay of random length, covering a distance proportional to the stay. The result is a flight, whose length has a power-law distribution, and the sum of such flights is not a normal diffusion at all. Its spread grows faster than the square root of time. George Zaslavsky made much of this, describing a system’s transport as a fractional kinetic process. So a tiny island, perhaps invisible on a grid, can dominate how a whole phase space’s momentum spreads out. Its effect is out of all proportion to its area.

How much of the plane the sea takes #

At $K = 0.5$ a single orbit reaches about 3% of the grid, because closed invariant circles run across the whole space and confine it to a thin layer. The last of those circles breaks near $K = 0.9716$, and past this the share climbs rapidly: 0.408 at $K = 1$, 0.772 at $K = 2$, 0.892 at $K = 2.5$. Between 2.5 and 3.5 it stands still, because the main island persists and then new ones have appeared. It reaches 0.982 by $K = 4.5$, and after that it is essentially one on a grid of this size.

The picture is a lower bound on what is chaotic, and at high $K$ it does not say what it appears to say. Each cell is marked if the orbit ever enters it, so a cell containing a small island and a bit of sea still counts. And the area of islands at high $K$ is too small for this grid to see, but stable periodic orbits are known to exist at arbitrarily high $K$: Pedro Duarte’s theorem of 1994 shows that the set of parameters with elliptic islands is residual in a suitable sense. So the sea does not win completely on any known interval of parameters.

The limit of the figures #

No simulation can see a power law to its end. Every curve here stops at the length that the sample could reach, and a true cut-off could be just past that. It is also possible that it is a slightly different exponent that would appear at much longer times. For this reason the debate on the exponent has continued, in spite of runs of $10^{12}$ and more steps being done by the specialists. There is a further issue with precision. A computed orbit in double precision is not a true orbit, and a long stay very near a regular region is exactly where the difference matters. Numerical orbits can get into places that a true orbit cannot, and cannot get into some places that it can. Careful studies have looked into this, but none was attempted here.

Moreover, these figures are for one value of $K$ and one box. The cat map is a fair comparison for the exponential cut-off, and it has been proved to have one. For the standard map at $K = 2$ almost nothing about the tail has been proved.

What was computed and what was quoted #

One orbit of 400 million steps was run for the standard map’s tail and 4 million for the cat map’s, recording gaps between visits in logarithmically spaced bins, and the slope was fitted by least squares on the logarithms of the tail between 1,000 and one million. The stay was found in a run of six million steps, with windows of 500. Grid shares came from 3 million steps at each $K$. Each regular seed was identified by a local exponent below a threshold over a finite run, which is not a proof that it is regular.

Everything about the exponents in the literature, the theory of cantori, the Markov tree models, the behaviour at high $K$ and the breakup value 0.9716 are quoted. The values measured here should be read as a demonstration of shape, since the sample is modest by the standards of the field.

Still open: the exponent, and the area of the sea #

Does the recurrence tail of a generic area-preserving map have a universal exponent, and what is it? Long computations suggest something near 1.57, with log-periodic oscillations that make a clean measurement very hard. Nothing has been proved, and the existence of a limiting exponent at all is an open problem.

Even more basic is a question that has been open since the 1960s. Does the chaotic sea of the standard map have positive area? Every picture says yes. Yet no proof exists, for any $K$, that the set of points with a positive Lyapunov exponent has positive measure. It is among the most famous open problems in dynamics, sometimes known as Sinai’s conjecture, and it stands in a remarkable contrast to the fact that the pictures are unmistakable. For all that has been proved, the entire rust cloud in the first figure might be a set of measure nought, with long-lived illusions of chaos in between a dense set of islands. Michael Benedicks and Lennart Carleson needed all the machinery of their proof to get this for Hénon’s map, which loses area, and in the area-preserving case the elliptic islands remove the leverage they used.

A sea with a memory #

An orbit in a chaotic sea with islands is caught and held near their fringes for stays whose lengths follow a power law, and this makes gaps between visits heavy-tailed where a map without islands has an exponential tail. The stretching rate of a single orbit drops to almost nothing for 19,000 steps and then comes back, and over 400 million steps the tail’s slope is about 1.65.

So islands do not only take up room. They reach into the chaos around them and change its statistics at every time scale, through a fringe of partial walls that no one has fully described. It is a warning about what a finite computation can reveal. The rate of chaos measured on an orbit can depend on when it is measured, and a single stay can be as long as the entire sample. The questions that seem most basic, whether the sea has positive area and how its tail behaves, have been open for as long as anybody has been able to draw that first picture.