Astronomy & Celestial Mechanics
the sky, computed
Concept

Standard map — where it appears

A two-line map of an angle and a momentum, kicked once per step by a force proportional to the sine of the angle, with a single strength parameter K. It is the simplest system showing the break-up of invariant curves and the onset of global chaos, and the model on which resonance overlap is measured.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

The last curve across the cylinder, before and after it breaks. Surfaces of section for the standard map at K = 0.5, 0.9716, 1.4, each 22 trajectories iterated 41 times from a column of starting values. Nothing here is placed: every dot is an iterate. At K = 0.5 the islands are separate and the space between them is filled with curves that run all the way round in θ — a trajectory cannot get from one island to the next, and one started beside the hyperbolic fixed point wanders 2.91 in p — 0.46 of a cylinder — and stops. At K = 0.9716, which is Greene's threshold to four figures, the last of those curves is on the point of going and the same trajectory still only reaches 4.84, which is 0.77 of a cylinder. At K = 1.4 it covers 6.1 cylinders: the barrier is gone and there is nothing left to stop it. That transition is what a chaotic zone in the asteroid belt is, drawn without any asteroids.

Where the chaos comes from

A Lyapunov time says how long a prediction lasts. It does not say what destroyed it. The mechanism is two resonances whose libration widths overlap, and the transition can be watched happening on a surface of section as one number is turned up.

gravitation · Chaos A chaotic orbit that waits, and then moves. The momentum of one trajectory of the standard map at K = 1.2, unwrapped and counted in whole turns of 2π, against the iteration number, over 39,892 iterations. The motion inside each band is chaotic from the first step — the trajectory's Lyapunov time is 3.7 iterations — but it lingers near one integer resonance for thousands of iterations before reaching the next: the waits were 3,098, 1,464, 9,461, 15,957, 6,058, 3,854 iterations. Each arrival at the next integer resonance, drawn dashed, is marked. Between the integer resonances lie the remnants of the broken invariant curves, which no longer block the trajectory but still hold it, and each passage is quick compared with the wait before it.

A barrier that leaks at a rate that can be computed

When the last invariant curve between two resonances breaks, it does not vanish. It becomes a cantorus — a curve full of gaps — and a chaotic orbit has to find the gaps to get through. The time that takes grows as the cube of how close the system is to the threshold, and it has nothing to do with the Lyapunov time, which is why an orbit can be hopelessly unpredictable and still stay put for the age of the solar system.

gravitation · Chaos How long a chaotic orbit stays away, with and without islands. The fraction of recurrence times to a small square in the chaotic sea of the standard map that exceed a given number of iterations, on logarithmic axes, counted along one orbit of 6·10⁷ iterations at K = 1.5 and K = 7 (772,165 recurrences, mean 78; 548,043 recurrences, mean 109). Dashed: exponentials with the same means. At K = 7, where the islands have all but vanished, the recurrences are exponential: the orbit forgets where it was in a few iterations. At K = 1.5 they part from the exponential after a few hundred iterations and fall instead as a power of time, with exponent 1.17 between a thousand and a hundred thousand iterations; the step at the left is the share of visits that simply stay in the square for a second iteration. A recurrence a thousand times longer than the mean is found once in 257,388 recurrences, where an exponential would give one in e to the thousand. Each mean is the area of the chaotic sea divided by the area of the square — 110 when the sea fills the torus, 78 at K = 1.5, where the islands take 29 per cent of it. The islands shorten the average absence and lengthen its tail.

A chaotic orbit clings to the edge of order

In a chaotic sea with no islands, an orbit forgets where it was within a few steps, and the time it stays away from any region falls off exponentially. Put islands of regular motion in the sea and the long absences multiply by factors of millions: the orbit reaches an island's edge, is caught in the hierarchy of smaller islands round it, and behaves as if it were regular for thousands of steps. The tail becomes a power law, and a rare few absences carry a large share of the orbit's whole history.

gravitation · Chaos Sixty futures that began a trillionth apart. The momentum of the standard map at K = 3, a kicked rotor whose phase space is almost entirely chaotic, for 60 members started 10⁻¹² apart in angle, against iteration. For 41 iterations they are one trajectory; then the separation, growing by a factor of about 2.0 each step, reaches the size of the system and they become independent random walks. Three of the 60 cross the threshold at |p| = 90 within 400 iterations (dark), the stand-in for an orbit that goes unstable. Which members cross is decided by differences far below anything that could be measured; how many cross is a property of the dynamics, and it is the only thing the ensemble can be asked for.

A probability for a system with one future

The Solar System will do one thing, and an integration of it is wrong about which thing within a hundred million years. What long integrations report instead is a probability — one per cent that Mercury's orbit goes unstable before the Sun dies — and a probability of a single event is a strange object. It is a count over an ensemble of futures that differ by less than anything measurable, which makes sense only because chaos makes those futures independent; and how many of them are needed is a binomial question with an unforgiving answer for rare events.

gravitation · Numerical integration A rare crossing reached in stages. Multilevel splitting on a kicked rotor (the standard map at K = 10), aimed at the event |p| > 450 within 400 iterations. 16 members are run until each reaches the first level or runs out of time; those that reach it are copied back up to 16, each copy restarted from its parent's crossing point with its angle nudged by a thousandth of a radian, and run on toward the next. The levels (dashed) are not fixed values of |p| but curves that rise as time runs out, so a member counts as closer to the event only if it has the time left to finish. 4 levels; the fractions passing them are 0.13, 0.06, 0.63, 0.88, and their product, 0.0043, is the estimate. Each stage is a common event; the rare one is never waited for.

A rare event reached by cloning the near misses

Counting how often an ensemble of integrations goes unstable works for an event of one in a hundred and becomes hopeless for one in a million, because a ten per cent answer needs a hundred events and so a hundred million futures. Splitting gets there by a different route: wait for the members that come closest, copy them, and run the copies on. Each stage measures a common event and the rare one is a product of common ones — provided the copies forget their parents and the stages know how much time is left.

gravitation · Numerical integration

Named alongside it

The objects these essays reach for when they reach for this one.

ChaosInvariant curveKAM theoryLyapunov timeSurface of sectionBinomial statisticsCantorusEnsembleLyapunov exponentNumerical integrationResonance overlapSolar system stability

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