A chaotic orbit clings to the edge of order

A chaotic orbit loses its predictability at a rate set by its Lyapunov exponent: two orbits that start a hair apart separate exponentially, and after a few Lyapunov times nothing about the starting point survives. It is natural to expect the orbit’s other statistics to forget as quickly — that the time it takes to come back to some region, or to leave one, should be as unpredictable as its position and as short-memoried. For the time to cross a broken barrier that is nearly right: a population behind a cantorus drains at a steady rate, like a radioactive nucleus, once its starting point is forgotten.

It fails in one place, and that place was named and set aside when the leak through cantori was counted. Most chaotic systems in astronomy are not fully chaotic. Their phase space is a sea with islands in it — regions of regular, quasi-periodic motion that chaotic orbits cannot enter — and an orbit that wanders near an island’s edge can be held there for a time with no characteristic scale. That holding is called stickiness, and the standard map, the same kicked rotor that showed where chaos comes from, can measure it directly by counting.

A clock that only counts returns #

The measurement is the simplest one available: follow one orbit for a very long time and record each moment it visits a small square of the chaotic sea, here a square 0.6 on a side round the map’s hyperbolic point. The intervals between visits are the recurrence times. Nothing else is needed — no ensemble, no choice of initial conditions beyond the orbit’s start, no Lyapunov exponent.

The mean of those intervals is fixed by a theorem. Kac’s lemma states that, for an orbit that fills a region ergodically, the mean time between visits to any part of it equals the region’s area divided by the part’s. With the square’s area of 0.36 and a torus of area four pi squared, an orbit that fills the whole torus returns every 110 iterations on average, and the orbit at K = 7 does: its 548,043 recurrences average 109.7. At K = 1.5 the mean is 78, because the orbit fills only the chaotic sea, which there covers seventy-one per cent of the torus, and the same square is a larger fraction of a smaller region.

So the mean says nothing about the dynamics beyond the size of the sea. Everything else is in the shape. At K = 7 the shape is an exponential: the probability that an absence lasts longer than t falls by a constant factor for each additional mean, so an absence of a thousand iterations, nine means, happens about three times in ten thousand, and one of ten thousand essentially never. At K = 1.5 the curve parts from its exponential after a few hundred iterations and continues on a straight line in logarithmic coordinates. Absences of ten thousand iterations, more than a hundred means, occur 150 times in the run, two in ten thousand recurrences; absences of a hundred thousand occur three times. An exponential with the same mean would put the odds of one of those at e raised to minus thirteen hundred.

Where the long absences are spent #

A recurrence time says how long an orbit stayed away, not where it went. Recording the orbit’s positions during only the longest absences shows where.

The long absences are spent at the boundaries of the islands. The orbit has wandered close to an island, entered the layer of broken curves and smaller islands that surrounds it, and circulated round the island for thousands of iterations before finding its way back into the open sea. While it is there it is not exploring the sea; it is going round, almost periodically, close to a regular orbit it cannot join.

The reason the holding has no characteristic time is the structure at the edge. Round each island is a chain of smaller islands, the remnants of resonant orbits at the island’s own rotation frequencies; round each of those is a chain of smaller ones still; and between them lie cantori, the broken invariant curves whose gaps leak at rates set by how close each is to breaking. An orbit that has worked its way into the hierarchy must work its way out again through a sequence of cantori whose leaks get slower the deeper it went. Each level adds its own time scale, and a sum of exponentials with a spread of scales that keeps going down the hierarchy is a power law.

A history dominated by rare episodes #

A power-law tail changes what an orbit’s history is made of. In an exponential world almost all of the time is spent in absences of ordinary length. In a sticky one, a handful of long episodes carry a share of the history out of proportion to their number.

At K = 1.5 one iteration in seventeen of the whole sixty million belongs to an absence longer than ten thousand iterations, and there are only a few dozen of those. If the number of absences longer than t falls as t to the −1.17, the time spent in them falls as t to the −0.17, barely falling at all across the range drawn. A summary of the orbit’s behaviour taken over any finite stretch is therefore at the mercy of whether that stretch happened to include one of the long episodes, and a long run is needed not to measure the mean, which Kac’s lemma already gives, but to sample the episodes that make the history.

Chaos that comes and goes #

The same episodes show up in the Lyapunov exponent if it is measured over short windows instead of averaged over the whole orbit.

At K = 7 the rate at which neighbouring orbits separate is the same in every window to within a few per cent; the map is uniformly chaotic, and a measurement of the exponent over any stretch gives the same answer. At K = 1.5 a tenth of the windows show a separation rate less than a quarter of the average, and some show almost none. Those are the sticky windows: the orbit is circulating at an island’s edge, where neighbouring orbits move nearly together, and over five hundred iterations it would pass most tests for regularity.

That bears directly on how chaos is diagnosed. An orbit integrated for a fixed time and found to have a small Lyapunov exponent is not shown to be regular; it may be sticky. An orbit found to have a large one is chaotic, but the time it then takes to leave its neighbourhood is not set by that exponent — the two clocks do not talk to each other — and in a phase space with islands it can be far longer than any exponential estimate. A chaotic orbit’s chaos is not a property of the orbit alone. It comes and goes with where the orbit is.

Islands measured with a clock #

Kac’s lemma also works in reverse. If the mean recurrence time is the sea’s area divided by the square’s, then the sea’s area is the mean recurrence time multiplied by the square’s — and the area of the islands, which the orbit never enters, follows without ever drawing one.

The islands take four-tenths of the torus at K = 1.2, just above the value at which the last curve spanning the cylinder breaks, and shrink as the kicks strengthen. They do not shrink smoothly: between K = 2.6 and 3.5 the sea’s share stalls, because new islands are born round periodic orbits that become stable over a range of K even as the old ones lose ground. Above K = 4.5 they hold only a couple of per cent.

That is a measurement made purely by timing, and it is worth noticing what it does not need: no picture of the section, no search for invariant curves, no decision about which points belong to which island. It needs an orbit long enough for its mean recurrence time to converge, and a region known to lie in the sea.

The excess against the kick #

The size of the stickiness follows the islands.

The excess of long absences is astronomically large while the islands are large, and it falls by six orders of magnitude between K = 1.2 and K = 4.5, without falling smoothly, just as the islands’ area does not fall smoothly. At K = 7 no absence of two thousand iterations occurred in sixty million, which an exponential with mean 110 would also predict. The stickiness is a property of the islands, not of the chaos round them. A map can be very chaotic and very sticky at once, provided it has islands for its orbits to cling to.

Resonance sticking in the solar system #

Real dynamical systems in astronomy are of the sticky kind. The asteroid belt, the Kuiper belt and the population of comets crossing the planets’ orbits all occupy phase spaces in which resonances with Jupiter or Neptune create islands of regular motion surrounded by chaotic layers, the overlapping resonances that make the chaos in the first place.

In the scattered disc beyond Neptune, bodies whose orbits are chaotic, driven about by encounters with the planet, are repeatedly captured for long spells into mean-motion resonances, where they circulate quasi-regularly before escaping again. Simulations of the population find that much of its lifetime is spent in such captures, and that the distribution of survival times has a long tail; the effect is called resonance sticking, and it is the same phenomenon as the blue points pressed against the islands’ edges. For comets on planet-crossing orbits, likewise, the fraction still present after a given time falls more slowly than an exponential, because some of them spend long periods near resonances. The comets on their first passage are those that have not yet been caught and held by any of those resonances.

Among the asteroids it has a quieter consequence. The bodies with short Lyapunov times that nevertheless keep their orbits for the age of the solar system — the stable chaos found in the outer belt, where a Lyapunov time of a few thousand years sits beside a lifetime of billions — are orbits trapped in thin chaotic layers bounded by cantori and islands, wandering without going anywhere. A population drawn from a sticky phase space contains such bodies as a matter of course; their presence is not a paradox but a tail.

The gaps that are not quite empty #

The asteroid belt shows the same structure in its counts. Where Jupiter’s resonances overlap, the belt has gaps that the resonances cleared, and the largest of them are chaotic zones from which an asteroid is removed — pumped to an eccentricity that crosses Mars or the Earth — within a few million years. The removal is not uniform across a gap. Near its edges, where the chaotic zone meets the regular orbits outside it, removal times stretch to hundreds of millions of years, and small populations survive there that a uniform rate would long since have emptied.

Inside some resonances the islands are large enough to hold populations of their own. Jupiter’s 3:2 resonance contains a broad stable island that holds the Hilda asteroids in quasi-regular orbits, and the 2:1 resonance contains smaller ones; between such islands and the chaotic zones around them lie the boundary layers where stickiness acts, and asteroids found there have lifetimes far longer than the chaos surrounding them would suggest. An ancient sweep of a secular resonance through the inner belt is a different process, a resonance moving through a population rather than a population leaking out of a resonance, but both leave counts that depend on how long a chaotic orbit lingers near order before it goes.

Lifetimes quoted as half-lives #

Planetary scientists routinely quote dynamical lifetimes: how long, typically, a near-Earth asteroid survives before hitting the Sun or a planet or being thrown out; how long a comet stays among the giant planets. For an exponential process a single number describes the whole distribution, and half-life and mean are interchangeable. For a sticky one they are not. The median lifetime of a population is set by its ordinary members, the mean by its stragglers, and the fraction still present after many medians by the power law rather than by any exponential extrapolated from the early decay.

That matters whenever an observed population is compared with a source. A steady population is the source’s rate multiplied by the mean lifetime, so if the mean is dominated by stragglers caught in resonances, the inferred source rate depends on modelling those stragglers correctly. And it matters for transport between regions, as in the captures of comets that pass through the narrow necks near the Lagrange points into temporary orbits round Jupiter, where how long a captured comet stays is a recurrence problem of exactly this kind.

Why the tail is a power #

The power law is not an accident of this map. Models of transport near island boundaries treat the hierarchy of islands and cantori as a tree of states, each level with its own escape rate, and the orbit’s descent and climb through the tree as a random walk on it; with escape rates that scale geometrically from level to level, the time to leave is a sum over the tree whose distribution has a power-law tail. Long simulations of area-preserving maps have measured exponents near 1.5 for the recurrence tail, the value those models suggest, with variations from system to system and over the range of times sampled — the 1.17 measured here, between a thousand and a hundred thousand iterations, is in the range found for intermediate times before the asymptotic slope is reached. Whether there is a single universal exponent has been argued for forty years without settlement, partly because the asymptotic regime is reached only after times that no simulation easily samples.

What is not in doubt is the qualitative point: any phase space with islands surrounded by a fractal hierarchy of smaller islands and cantori produces recurrence times with no characteristic scale, and the mean, though fixed by Kac’s lemma, is reached by a distribution whose long absences carry much of the time.

The same map outside astronomy #

The standard map was not invented for planets, and the stickiness it shows turns up wherever motion is described by an area-preserving map. A charged particle circulating in a storage ring receives a kick from each focusing magnet once a turn, and its motion in the plane of position and momentum is a map of exactly this kind; particles that wander near the edges of the stable region round the design orbit linger there before being lost, and the time a beam survives is governed by the same tails. The magnetic field lines in a fusion device, followed once round the torus at a time, also obey an area-preserving map, with magnetic islands where the lines close on themselves and chaotic regions where they wander. A field line near an island’s edge follows it for many turns before leaving, and because heat flows along field lines, the stickiness slows the escape of heat from the plasma’s edge.

Cold atoms in an optical lattice switched on in brief pulses realise the kicked rotor itself, with laser light supplying the kicks. There the motion is eventually governed by quantum mechanics, which suppresses the classical diffusion altogether after a time, but over the early kicks the classical structure, islands and stickiness included, is visible in the measured distributions of momentum. The arithmetic of an orbit clinging to the edge of order is the same in a ring of magnets, a torus of plasma and a belt of asteroids, because in each the motion conserves phase-space area and has islands to cling to.

What the counting leaves out #

Every number here comes from one orbit per value of K, run for sixty million iterations, with recurrences to one square in double-precision arithmetic. One orbit is the right sample for the time averages Kac’s lemma speaks of, but the tail beyond a hundred thousand iterations is sampled by a handful of events, and its slope there is uncertain; rounding error, which perturbs an orbit by a part in ten to the sixteen per iteration, can in principle let an orbit cross a barrier it should not, though not at a rate that would affect absences of this length. The square was chosen inside the sea for every K drawn; a square that touched an island would count an orbit that never returns. And the standard map has one degree of freedom, a two-dimensional phase space in which invariant curves divide the plane. In more degrees of freedom they cannot, and stickiness becomes a different and slower question.

Still open: stickiness when the barriers do not divide #

In a system with two or more degrees of freedom, the regular tori are too low in dimension to enclose anything, and chaotic orbits can move along the web of resonances that threads between them — Arnold diffusion. Islands still exist and still hold orbits near their edges, but an orbit stuck near one is no longer confined to circulate round it; it can drift slowly along the resonance it is in. Whether stickiness in that setting still produces power-law tails, with what exponents, and whether the long-lived chaotic bodies of the solar system are held mainly by stickiness or by the slowness of diffusion along resonances, are questions that the planetary systems ask and that simple maps cannot answer.