The last curve to go

The first thing anybody learns about chaos is that a small change in the starting conditions grows exponentially, and that growth has a rate that can be measured. It is a good first fact and it leaves behind a bad picture: that a system has a dial on it, and that turning the dial past some value converts the whole thing from predictable to unpredictable.

Nothing in that picture supports the dial. Both panels have orbits lying neatly on curves and orbits smeared over areas, and the left panel is not more orderly in any sense an individual orbit would recognise. What has changed between them is a property of the set of orbits: whether the regular ones still manage to enclose the chaotic ones.

The simplest thing that does this #

The object drawn above is Chirikov’s standard map, and it is worth a paragraph because it is not a toy.

Take a rotor — anything with an angle $\theta$ and a momentum $p$ conjugate to it — and give it a kick of strength $K \sin\theta$ once every turn. Between kicks it coasts, so the angle advances by whatever the momentum is. That gives

$$p' = p + K\sin\theta, \qquad \theta' = \theta + p'$$

with both coordinates taken modulo $2\pi$. At $K = 0$ the momentum never changes: every horizontal line is an orbit, and the whole picture is a stack of curves that nothing crosses.

The map earns its place because it is what any nearly integrable Hamiltonian system looks like near a resonance. Expand about a resonant torus, keep the resonant harmonic and its nearest neighbour, and the standard map is what comes out. So statements about it are statements about the neighbourhood of every resonance in the solar system, in a particle accelerator, and in a tokamak’s magnetic field.

The kick is what makes it a map rather than a flow, and that is a simplification rather than a distortion. A continuous system watched once per period of some driving is exactly a map, and the section through the phase space on which it is watched is where the pictures below live. Everything drawn here is a stroboscopic photograph of something that never stops moving — the same device that turns a driven oscillator’s steady response into one point rather than a curve.

What is destroyed, and in what order #

At $K = 0$ each curve is labelled by its rotation number: the average advance in $\theta$ per iteration, in turns, which is just $p/2\pi$. Turn the coupling up and the curves with rational rotation numbers are destroyed first.

The word “destroyed” is exact and is worth not softening. A curve does not become fuzzy or approximate; it stops existing, and the orbits that were on it are afterwards on something else. What replaces it is a Cantor set — the curve with a dense set of gaps punched through it — which is why the objects the subject deals in after the transition are called cantori.

The reason rationals go first is a resonance argument. An orbit whose rotation number is $p/q$ returns to the same phase of the kick every $q$ iterations, so the kicks add coherently and a tiny perturbation accumulates without limit. An orbit whose rotation number is irrational never repeats its phase, so the kicks partly cancel — and how well they cancel depends on how badly the number can be approximated by rationals.

That is the same coherence argument that makes a swing respond to being pumped at twice its own frequency, turned round: there, adding coherently is the point of the exercise; here, it is what destroys the orbit that does it.

That is the content of the KAM theorem, and it is one of the rare theorems whose statement is a number-theoretic condition on a physical object. A torus survives a small perturbation if its rotation number is sufficiently irrational — if every rational $p/q$ misses it by more than roughly $c/q^{2.5}$. The numbers hardest to approximate are the ones whose continued fractions are all ones, and the hardest of all is the golden mean.

So there is a prediction with a sharp edge to it: as the coupling rises the surviving curves are squeezed into an ever thinner set, and the very last one to go should be the golden one.

Finding the threshold by watching what escapes #

The way to measure that is not to look for the curve, which is invisible, but to look for what it stops.

Launch a set of orbits at $p = 0$, run them, and record how far in momentum they get. If any curve still spans the cylinder above them they are trapped beneath it and the excursion stays near two, less than one channel’s height. If none does, they random-walk in $p$ without limit.

The crossing of one full channel comes at $K = 1.20$ after two thousand steps, $1.11$ after twenty thousand and $1.04$ after two hundred thousand. Greene’s residue criterion — a completely different method, which asks whether the periodic orbits approximating the golden torus are stable — puts the destruction at $0.971635$.

Those numbers are converging on Greene’s from above, and the way they converge is the physics rather than a numerical embarrassment. A barrier does not vanish at the threshold; it becomes leaky. The instant the last curve breaks it leaves behind a Cantor set of what used to be the curve, with gaps of vanishing width, and an orbit can get through the gaps but takes an enormous time to find one. The flux through the remains grows from zero as a power of $K - K_c$, so any experiment of finite length reports a threshold that is too high, by an amount that shrinks as the patience grows.

That is worth stating plainly because it is the honest reading of every numerical threshold in this subject: the number a simulation reports is the coupling at which transport becomes fast enough to see, and the coupling at which transport becomes possible is smaller and has to be got at another way.

The estimate that everybody makes first #

There is a back-of-the-envelope answer to the same question, it is the first thing anyone computes, and it is wrong by a factor worth understanding.

Chirikov’s criterion says: find the two largest resonances, work out how wide each is, and declare global chaos when they touch. Each primary resonance of this map is a pendulum with a separatrix reaching $2\sqrt{K}$ in momentum, and they sit $2\pi$ apart, so they touch when $4\sqrt{K} = 2\pi$, giving $K = \pi^2/4 = 2.467$.

Against a measured $0.971635$, the estimate is too large by 2.54. It is too large for a reason the picture makes visible: between the two big islands there are islands at every rational rotation number, and by the time the two largest have grown enough to touch, the small ones have already destroyed most of what lay between them. The criterion counts two of an infinite family and ignores the rest.

What it gets right is the scaling. The widths really do go as $\sqrt{K}$; the criterion’s structure is correct and only its coefficient is optimistic. That is the usual state of an overlap estimate, and it is why the criterion is still the first calculation to do and never the last word. A factor of two and a half in a threshold is a serious error in an accelerator design and an entirely acceptable one when the question is whether a system is anywhere near trouble.

The same distinction runs through the whole of this collection: a stationary path is not always a minimum, and a criterion that identifies the right structure can be off by a constant that only a calculation supplies.

What survives, measured against a closed form #

The treads of the staircase are worth one more look, because one of them can be checked against arithmetic done on paper.

The tread at zero rotation number is the primary island, and treating that one resonance as a pendulum gives its boundary exactly. Two corrections separate the prediction from the $\sqrt{K}/\pi$ that a first attempt writes down, and both matter. The elliptic fixed point of this map is at $\theta = \pi$, not at $\theta = 0$, so the separatrix reaches its full height only there and the probe line cuts a chord of it. And the map’s momentum is not the pendulum’s: the kick precedes the drift, so the momentum symmetric in time is $p + (K/2)\sin\theta$, and the island’s upper edge sits that much lower on the axis the staircase is drawn against.

With both corrections the pendulum predicts 0.114 of a channel at $K = 0.35$ and the staircase measures 0.114. The agreement is not the point; the point is that the agreement had to be earned, and that the version of the formula everybody remembers was out by two-thirds.

Chaotic does not mean random #

Once the curves are gone the momentum wanders, and the obvious next question is how fast. The obvious answer is to treat each kick as independent: the mean square change in $p$ per iteration is then $\langle K^2\sin^2\theta\rangle = K^2/2$, which is called the quasilinear estimate and is what almost every application uses.

It is not one anywhere. Just above the threshold the ratio is 0.07 — the curves are gone but the gaps left behind are narrow, so most orbits are still stuck for most of the run. It climbs through one as the coupling grows, and near $K = 2\pi$ it reaches the hundreds, which is not a diffusion coefficient at all. At that coupling the map has accelerator modes: orbits that receive very nearly the same kick every iteration, so their momentum grows linearly rather than as a square root. A handful of them in an ensemble dominates the mean square completely.

This is the correction the first rung of this ladder needs. A positive Lyapunov exponent says that two neighbouring orbits separate exponentially. It says nothing whatever about whether one orbit’s successive steps are statistically independent, and the two are routinely confused. A system can be as chaotic as anything and still remember enough about its own past to spread at fifty times — or a fiftieth of — what a coin-flipping model predicts.

Tracking the energy of a chaotic trajectory through an integration is what makes the exponent measured from that run trustworthy. The individual path is meaningless after a few Lyapunov times — it is the accumulated rounding error amplified — but a conserved quantity that stays conserved shows the integrator is following some trajectory of the true system, and the statistics of that trajectory are the answer being sought.

Where the mechanism shows up #

The map’s structure is not an artefact of its own simplicity. Three places where the same arithmetic decides something:

The asteroid belt has gaps at the rational periods. The Kirkwood gaps sit where an orbital period is a simple fraction of Jupiter’s, and the mechanism is the one above: at those rotation numbers the perturbation adds coherently, the resonance is wide, and material is removed. The gaps are treads on a staircase.

Particle accelerators are designed around the tune. The number of betatron oscillations per turn is a rotation number, the machine is deliberately tuned away from low-order rationals, and the design margin is an overlap calculation. A tune that drifts onto a resonance loses the beam.

And a tokamak’s magnetic surfaces are invariant tori. Field lines wind round the torus with a rotation number, the surfaces confine the plasma exactly as the curves above confine the orbits, and a perturbation that destroys them lets heat out. A field line in a conducting fluid is already a curve nothing crosses; what this argument adds is that the surface those lines lie on can be destroyed while every individual line is still perfectly well behaved, and the topology of what is left is not something a local measurement can report. The transition studied here is the one that decides whether a machine confines or leaks.

Almost every exponent of a force law gives an apsidal angle that is not a rational fraction of a turn, so almost every orbit fails to close and fills an annulus instead. That is where the mechanism shows up in the oldest problem there is: closure requires a resonance between two frequencies, the inverse square and the linear spring are the only laws that supply one, and everything else is quasi-periodic — which is the same condition the tori in this essay are destroyed by.

The prize essay that had to be recalled #

The structure this page draws was first seen in the problem the applications section names, and the circumstances are worth recording.

In 1889 Poincaré won a prize offered by the King of Sweden for work on the stability of the solar system, with a memoir on the three-body problem that concluded, essentially, that the motion was regular. While the volume was being printed an editor raised a query about one passage. Poincaré looked again and found that he had assumed something he had no right to: that two families of trajectories emerging from an unstable periodic orbit either coincide or do not meet at all.

They do meet, and once they meet at one point they must meet at infinitely many, in a tangle whose complexity he described as something he would not even attempt to draw. That tangle is chaos, and it was found by a man checking his own prize-winning argument.

He had the printed copies withdrawn and paid for the reprinting himself — more than the prize was worth — and the corrected memoir is where the subject begins. What makes the episode worth more than an anecdote is what the error was: not a slip in arithmetic but an unexamined assumption of regularity, of exactly the kind the phase portraits at the top of this page are drawn to expose.

The solar system, measured against its own Lyapunov time #

The question Poincaré was answering is now answered numerically, and the answer is that the solar system is chaotic.

Integrations of the planets’ motions give a Lyapunov time of about five million years for the inner planets: an uncertainty in a planet’s position grows by a factor of $e$ over that interval. Fifteen metres of error in the Earth’s position today becomes about a hundred and fifty metres after ten million years and, continuing, a distance comparable with the orbit itself after a hundred million.

So the solar system’s positions are unpredictable on timescales short compared with its age, and the question “where will the Earth be in two hundred million years” has no answer that any improvement in measurement or computation could supply. That is a strong statement about a system usually offered as the model of clockwork regularity.

What remains predictable is statistical. Long integrations run many times with slightly different starting conditions give a distribution of outcomes, and the great majority of them look much like the present arrangement for the remaining lifetime of the Sun. A small fraction — around one in a hundred — do not: Mercury’s eccentricity, driven by a resonance with Jupiter, grows large enough to destabilise the inner system, with collisions among the possible endings.

That is the honest form of an answer about a chaotic system, and it is the form this essay’s diffusion figure warns about. There is no trajectory to quote, there is a distribution over trajectories, and the useful question is what fraction of it does something interesting.

What the picture cannot show #

The curves are invisible. A phase portrait shows orbits, and an invariant curve is drawn only if an orbit is launched exactly on it, which cannot be arranged for an irrational rotation number. Every curve in the panels above is a nearby orbit standing in for one, and the distinction matters at exactly the moment of destruction: an orbit on a barely surviving curve and an orbit in a barely open gap look identical for a very long time.

Nothing here is a proof of anything. The KAM theorem’s hypotheses are much stronger than “the perturbation is small”, the constants it produces are enormously conservative, and the numbers computed here are outside every range in which it has been proved. Greene’s criterion, which supplies the 0.971635, is itself a conjecture supported by extremely good numerics rather than a theorem.

The measured thresholds are upper bounds. As the barrier figure shows directly, watching longer lowers the number. Nothing in a finite computation can establish that a curve exists — only that nothing got past during the run.

The map is area-preserving and most things are not. Every statement above depends on phase-space area being conserved, which is what makes an invariant curve a permanent barrier rather than something an orbit spirals across. Add the smallest amount of friction and the whole structure is replaced by attractors — the curves stop being barriers because there is no longer anything to conserve — and the analysis has to start again. The three regimes of a damped oscillator are the beginning of that other story.

And two dimensions is a special case. In a map of this kind an invariant curve is a closed loop that divides the phase space into an inside and an outside, so a surviving curve is a genuine wall. With more degrees of freedom the invariant surfaces have too low a dimension to divide anything, and an orbit can travel around them. That is Arnold diffusion, it happens at every coupling however small, and it is why the sharp threshold this essay is about does not exist in the systems most of the applications above actually are.

The ladder from here #

Later rungs on this anchor: Arnold diffusion, and why the wall this essay is about disappears the moment there are three degrees of freedom; the renormalisation argument behind Greene’s criterion, which explains why the golden mean and produces the critical exponents of the transition; the stickiness of island boundaries, which gives chaotic orbits a heavy-tailed distribution of escape times and breaks the diffusion picture in a second way; strange attractors, which are what the same phenomena look like when dissipation is added and phase-space area is no longer conserved; and the transition to turbulence, where the sequence of destroyed resonances is what a physical fluid actually does.

The neighbouring ladders are the exponential separation of nearby trajectories, which is the rung this one stands on, and the orbits that do not close, whose rosettes are the same statement about rational and irrational winding made about a central force rather than a kicked rotor.