Astronomy & Celestial Mechanics
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Gravitation

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.

Assumes Chaos, Planetary rings and Chaos.

16 min read 6 figures Small perturbations, large consequencesMeasuring the unreachableScales that defeat intuition

A chaotic orbit forgets its initial conditions on a timescale called the Lyapunov time, and for the inner planets that time is about five million years. Where the chaos comes from is resonance overlap: when two neighbouring resonances grow wide enough, the invariant curves that separated them break, and a trajectory can wander from one to the other. On the standard map, the model that captures this in one parameter, the last curve breaks at Greene’s threshold, Kc=0.9716K_c = 0.9716.

That picture has a hole in it which the account of resonance overlap named and did not fill. Just above the threshold, a trajectory is chaotic and nothing blocks it. Yet it does not wander freely. It stays near where it started for a long time — thousands of Lyapunov times, then tens of thousands — and then crosses quickly to the next resonance, where it stays again. The figure below is a single trajectory doing exactly that: its momentum rattles around one level for thousands of iterations, jumps, and rattles around another.

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.
Fig. 1 One trajectory of the standard map at K = 1.2, a quarter above the threshold, with its momentum unwrapped and counted in whole turns. Its Lyapunov time is under four iterations, so its position is lost almost at once; yet it waits between fifteen hundred and sixteen thousand iterations near each integer resonance before moving to the next. The marks are the arrivals. The waiting, not the chaos, is what the rest of this essay measures.

What holds it there is what is left of the broken curve.

A curve with holes in it

An invariant curve on the standard map is a closed loop around the cylinder, and a trajectory started on it stays on it forever. Below KcK_c the golden-mean curve — the one whose winding is the golden ratio, the most irrational number and so the hardest to resonate with — survives, and no trajectory crosses it. Above KcK_c it no longer exists as a curve. But Serge Aubry and Ian Percival showed in the early 1980s that it does not simply disappear. It becomes a cantorus: an invariant set with the same winding, lying where the curve lay, but with gaps in it — a Cantor set rather than a line.

A cantorus is still a barrier. It is a barrier with holes, and a trajectory can pass through only by finding one. The rate at which it does is a flux: the area of phase space that passes through the gaps per iteration. Robert MacKay, James Meiss and Percival computed that flux by what they called a turnstile — a pair of lobes, bounded by the stable and unstable manifolds of a periodic orbit near the cantorus, that swap places at each iteration and carry a fixed area across. The area of the lobe is the flux, and near the threshold it grows as

Φ∝(K−Kc)3.01.\Phi \propto (K - K_c)^{3.01}.

The exponent is universal: it comes from the renormalisation that describes how the golden curve breaks, and it is the same for any smooth twist map at the breakup of a golden curve.

Why the golden curve should be the last to go is a statement about numbers rather than about orbits. A curve is destroyed by the resonances near it, and a winding close to a simple fraction is close to a strong resonance. Every irrational number can be approximated by fractions, and how well is read off its continued fraction: large terms mean a very good approximation by a small fraction. The golden ratio’s continued fraction is all ones, the smallest possible terms, so it is the number most poorly approximated by fractions of any size — the winding that keeps farthest from every resonance at once. Greene’s method finds the threshold by following the periodic orbits whose windings are the golden ratio’s successive approximations, 1/2, 2/3, 3/5, 5/8 and on, and watching when they turn unstable; the limit of that sequence is KcK_c.

It is worth setting a cantorus beside the other kind of barrier celestial mechanics knows. The zero-velocity curve of the restricted three-body problem is exact: it follows from a conserved quantity, it cannot be crossed at any time, and it never leaks. A cantorus is the opposite case — no conserved quantity forbids passage, the barrier is purely dynamical, and its strength has to be measured as a rate. Most of the confinement in the solar system is of the second kind.

The crossing time, counted

The flux prediction implies a crossing time. If a region of area AA must drain through a turnstile of area Φ\Phi per iteration, and the trajectory spreads uniformly through its region before it finds the gap, the mean time to cross is A/ΦA/\Phi, which diverges as (K−Kc)−3.01(K - K_c)^{-3.01}.

That prediction can be checked by doing nothing more than iterating the map. Start forty trajectories in the chaotic layer at the origin, count how many iterations each takes to reach a momentum of ±2π\pm 2\pi — one full turn, which requires passing the golden cantorus — and repeat at nine values of KK.

The leak closes as the cube of the distance to the threshold. The mean number of iterations a standard-map trajectory takes to cross from the chaotic layer at p = 0 to p = ±2π, over 40 trajectories at each of 9 values of K from 1.06 to 1.7, against the distance above Greene's threshold of 0.9716, both on logarithmic axes. The measured points fall on a line of slope −3.10: the crossing time runs from 438 iterations at K = 1.7 to 258,110 at K = 1.06. The dashed line has the slope −3.01 that MacKay, Meiss and Percival derived for the flux through the gaps in the broken golden curve; the crossing time is the area to be crossed divided by that flux, so it diverges as the threshold is approached from above, with no jump at the threshold itself.
Fig. 2 Mean crossing time, in iterations of the map, against the distance above the threshold, both on logarithmic axes, over forty trajectories at each point. The measured slope is within a few per cent of the −3.01 that the turnstile flux predicts; the dashed line has that slope. At K = 1.7 a trajectory crosses in a few hundred iterations; at K = 1.06, barely above the threshold, it takes a quarter of a million. The crossing time grows continuously as the threshold is approached and has no jump at it.

The points fall on the line. The measurement is crude — forty trajectories is a small ensemble, and the spread in individual crossing times is wide — but it recovers the exponent within its own noise, from a map whose rule is two lines long. It also makes the continuity visible. Below KcK_c the crossing time is infinite because the curve is intact. Above, it is finite but diverges as the threshold is approached. Nothing dramatic happens at the threshold itself: the barrier becomes porous, and its porosity grows smoothly from zero.

This changes what the threshold means for a real system. A body in an orbit just above its own threshold is not free; it is behind a wall whose holes are small. For practical purposes — over the age of the system, say — a barrier that leaks slowly enough is indistinguishable from one that does not leak at all, and the parameter where escape becomes likely within the time available can sit far above the one where it becomes possible.

Two clocks that do not talk to each other

The Lyapunov time and the crossing time are both measured in iterations, both computed from the same trajectories, and both say something about the orbit’s future. It is natural to expect them to be related. They are not.

Two clocks on the same orbits, and no relation between them. Each point is one standard-map trajectory started in the chaotic layer: its Lyapunov time, measured from the growth of a tangent vector over its first 500 iterations, against the number of iterations it took to cross to p = ±2π, for 160 trajectories at each of K = 1.1 and 1.4, both on logarithmic axes. The Lyapunov times sit in a narrow band — medians of 4.5 iterations at K = 1.1 and 3.5 iterations at K = 1.4 — while the crossing times spread over a decade and more, with medians of 75,742 and 1,166. Within each set the correlation between the two is 0.08 and 0.06: knowing how chaotic an orbit is says nothing about when it will leave. The ratio of median crossing time to Lyapunov time is 16,865 and 338.
Fig. 3 One point per trajectory: its Lyapunov time, from a tangent vector followed over its first 500 iterations, against the number of iterations it took to cross, for 160 trajectories at each of two values of K. The Lyapunov times lie in a narrow band of a few iterations; the crossing times spread over more than a decade. Within each cloud the two are uncorrelated. At K = 1.1 the typical orbit is unpredictable after about four iterations and stays in its band for tens of thousands.

The reason is that they measure different things. The Lyapunov time is local: it is set by how strongly the flow near the trajectory shears neighbouring orbits apart, which is a property of the neighbourhood the trajectory is in, and every trajectory in the chaotic layer samples the same neighbourhood. The crossing time is global: it is set by the area of the region the trajectory has to explore and the size of the gaps it has to find, which are properties of the whole phase space. A trajectory that happens to be sheared hard in its first five hundred iterations is no more likely to be near a gap.

A ratio that large has a name in the solar system. The asteroid 522 Helga sits in the 12:7 mean-motion resonance with Jupiter, and Andrea Milani and Anna Nobili found in 1992 that its Lyapunov time is about seven thousand years — its position along its orbit is unpredictable after a few tens of thousands of years — while its semi-major axis and eccentricity stay within narrow bounds over integrations of billions of years. They called the phenomenon stable chaos, and it was treated for a while as a paradox. On the standard map it is the ordinary case: an orbit just above the threshold is thoroughly chaotic and survives for a time that bears no relation to how chaotic it is.

Across systems the two clocks do move together

That is not the whole story, and the rest of it explains an observation that seemed to contradict stable chaos.

Predictability barely moves while survival moves by four decades. Two times measured on the same 40 trajectories at each of 8 values of K, on a logarithmic scale: the median Lyapunov time, from the growth of a tangent vector, and the mean number of iterations to cross to the next band in p. Between K = 1.06 and K = 3 the Lyapunov time changes from 4.8 to 1.2 iterations — a factor of 4.0 — while the crossing time changes from 258,110 to 31, a factor of 8,442. The first is a local property of the flow, set by how strongly neighbouring orbits are sheared apart; the second is a global property of the phase space, set by the gaps in structures the orbit has to find. Near the threshold they differ by four orders of magnitude.
Fig. 4 The two times against K, on a logarithmic scale: the median Lyapunov time and the mean crossing time, each over forty trajectories. Both fall as the perturbation grows. But the Lyapunov time falls by a factor of four between K = 1.06 and K = 3, while the crossing time falls by a factor of several thousand. They share a cause — the strength of the perturbation — and not a mechanism.

Across different values of KK both clocks shorten, because a stronger perturbation both shears harder and opens the gaps wider. So a survey of many systems, each with its own strength of perturbation, would find a correlation: systems with short Lyapunov times would tend to have short escape times. Myron Lecar, Fred Franklin and Marc Murison found exactly that in the early 1990s among asteroids in the outer belt and test particles between the giant planets — an empirical relation in which the time to a sudden change of orbit scaled roughly as a power of the Lyapunov time, with a large scatter and with Helga and its kind as outliers.

The figure shows why both findings are true at once. Across systems the two clocks share a cause, so they correlate. Within one system, at one strength, they share nothing, so an individual orbit’s Lyapunov time cannot predict its escape. And the dependence is wildly unequal: a factor of four in the Lyapunov time comes with a factor of thousands in the escape time. The escape time is dominated by how close the system is to its threshold, and the Lyapunov time barely registers that distance at all. An empirical relation fitted across the belt is therefore a relation between two quantities that each depend on a third, and it fails on exactly the bodies close to their thresholds — which are the ones where the question matters.

A leak that behaves like a decay

The crossing times at one value of KK are spread over a decade. The shape of that spread says something about the mechanism.

The fraction still trapped, at three strengths of the chaos. The fraction of 240 standard-map trajectories still inside their starting band after a given number of iterations, on a logarithmic time axis, for K = 1.1, 1.2, 1.4. Each curve has the same shape, shifted along the axis: mean residence times of 90,583, 12,297, 2,004 iterations. The dashed curves are exponential decays with the same means. Early on the measured curves lie above them, because every trajectory starts at the same point and must first spread through its band before any can reach the gaps; past the mean they fall as the exponentials do, and at twice the mean 9%, 12%, 11% remain, against the exponential's 14%. Once the starting point is forgotten, escape through a cantorus behaves like a decay with a rate rather than a clock with a deadline.
Fig. 5 The fraction of 240 trajectories still in their starting band after a given number of iterations, on a logarithmic time axis, at three values of K, with exponential decays of the same mean drawn dashed. The three measured curves have the same shape, shifted by decades. For the first part of each, fewer trajectories have left than an exponential would allow; beyond the mean, they drain as the exponentials do.

The early deficit is the trajectories spreading out. They all begin at one point, and none can leave until some have reached the region near the gaps, so there is a dead time before the first escapes. After that, the population drains at a steady rate: at each iteration, a roughly fixed fraction of what remains finds a gap. That is the behaviour the turnstile predicts. Each lobe carries the same area across at every iteration, and if the region behind the cantorus is well mixed — which chaos ensures — the probability of being in the lobe is the same at every iteration, independent of how long the trajectory has already waited.

So a chaotic orbit behind a cantorus is best described as a radioactive nucleus rather than as a clock running down. There is no deadline. A body that has stayed in a resonance for a billion years is not overdue to leave; its chance of leaving in the next million is what it always was. That is a statement about populations, and it is how the question has to be asked in the solar system: not when will this asteroid leave the resonance, which is unanswerable beyond a few Lyapunov times, but what fraction of the population in this resonance leaves per million years, which the flux answers.

The exponential tail is not universal, and it is worth being exact about when it fails. Near the boundaries of the regular islands the trajectory can become stuck for very long times, because a hierarchy of smaller islands and cantori surrounds each one, and escape from that hierarchy is slower than any exponential. In long simulations and in large ensembles, residence times in mixed phase spaces generally acquire a power-law tail from this stickiness. The ensembles drawn here are too small and too short to reach it; the trajectories start in the separatrix layer, away from the islands, and drain before they find them.

Where the orbit actually spends its time

The same trajectories can be followed on the surface of section, which shows where the cantorus is and what the trajectory does near it.

Where a leaking orbit spends its time. One standard-map trajectory at K = 1.1, plotted on the section at every 12th of its 50,617 iterations before its momentum first reached ±2π. The dashed lines mark the mean momenta of the golden-mean invariant curves, at 0.382 and 0.618 of a turn and their mirrors — the last curves to survive as K rises, and broken here into cantori. The trajectory spent 48% of its iterations inside the inner pair, around the integer resonance where it started, and the rest wandering the band between the cantori and their mirrors without yet reaching a full turn. The island around θ = π, p = 0 is empty in all 50,617 iterations: it is filled with regular orbits the chaotic one cannot enter.
Fig. 6 One trajectory at K = 1.1, drawn on the section at every twelfth iteration until it first reaches a full turn of momentum. The dashed lines are the mean momenta of the golden curves that no longer exist, at 0.382 and 0.618 of a turn and their mirrors. About half of the trajectory’s iterations fall inside the inner pair of lines, around the integer resonance where it started; the rest are spread through the band beyond, among the remnants of the broken curves, without yet reaching a full turn. The empty region around θ = π is the island of the integer resonance, which the chaotic orbit never enters in fifty thousand iterations.

The picture is the transport problem in one frame. The trajectory is confined not by a wall but by a region it has not yet found the way out of, and the points crowd along bands of the section where fragments of the broken curves turn it back again and again. The gaps in the cantorus are too small to see at this resolution — their total area per iteration is a small fraction of the plotted region — and that is exactly why the crossing takes so long.

What this means for asteroids and comets

The standard map is a model, not a solar system. It has one degree of freedom and one resonance chain, and its cantori are a clean case. The ideas transfer, with two amendments.

The first is that a real resonance has more than one boundary and more than one kind of leak. An asteroid in a mean-motion resonance with Jupiter sits in a region bounded by cantori of several windings, and its eccentricity is also driven slowly by secular effects and — for small bodies — by the Yarkovsky drift in its semi-major axis, which move it toward or away from the chaotic layer. The turnstile flux is the mechanism of escape once a body is in the chaotic zone; the drift decides how it got there. The Kirkwood gaps are what the whole process leaves: resonances whose chaotic zones connect to planet-crossing orbits, drained over the age of the solar system, next to resonances whose zones do not, and which remain populated.

The second amendment is dimensional. With two or more degrees of freedom, an invariant torus no longer divides the phase space in two, so it cannot be a barrier even when intact; a trajectory can go around it. The transport that results, Arnold diffusion, is slow in a different and much more extreme way, and it has never been unambiguously demonstrated in a real planetary system.

Comets offer the most direct illustration of residence behind partial barriers. Short-period comets that pass close to Jupiter are temporarily captured into orbits around the planet, or into resonances with it, for periods that range from a few orbits to thousands. The distribution of capture durations follows statistics of the same general kind as the crossing times here: a short dead time and then a decay, with the rate set by the geometry of the tubes and turnstiles that connect the regions near Jupiter’s Lagrange points. The same geometry is what spacecraft trajectory designers exploit to move between those regions at almost no cost, by finding the gaps deliberately.

How the numbers here were measured, and what they omit

Every number in the figures is counted off iterations of the standard map, p′=p+Ksin⁡θp' = p + K \sin\theta, θ′=θ+p′\theta' = \theta + p', in double precision, with no fitted parameters except the slope in the crossing plot. The Lyapunov times come from a tangent vector renormalised at every step over the first five hundred iterations. The chaotic trajectories start near the hyperbolic fixed point at the origin, spread over a twentieth of a radian in angle, which places them in the separatrix layer and away from the islands.

Floating-point error is not a concern for the statistics, though it would be for any individual trajectory: after a few tens of Lyapunov times, the computed orbit is not the true orbit from its initial condition, but by the shadowing property of hyperbolic systems it stays close to some true orbit of the map, and the ensemble statistics are those of true orbits. The crossing times are ensemble properties and are safe. The exact sequence of waits in the hero figure is not reproducible on a different machine, and does not need to be.

What the figures omit is the long tail. With forty to 240 trajectories per value of KK and at most a few million iterations each, the ensembles see the bulk of the distribution and not the sticky trajectories trapped near islands for a hundred times the mean. Establishing the tail’s exponent requires ensembles thousands of times larger, and the literature’s estimates of it are still debated.

What the barrier teaches about stability

The account of resonance overlap established that a Lyapunov time measures predictability and not survival. This one shows why, and by how much. Survival is controlled by the flux through the gaps in the broken curves, which depends steeply on the distance from the threshold — as its cube, near the golden curve’s breakup — while predictability depends on it only weakly. A system just past its threshold is fully chaotic, entirely unpredictable within a few Lyapunov times, and in practice as confined as if the curve had never broken.

That is the resolution of stable chaos, and it is quantitative. An orbit that is chaotic and stable is an orbit behind a cantorus with small gaps. How stable it is follows from the size of the gaps, and the size of the gaps follows from how far past its threshold the system sits.

Still open: transport with more freedom, and the sticky tail

The next question is the dimensional amendment the asteroid section named and set aside: what happens when there are two or more degrees of freedom and a torus cannot enclose anything, so that transport proceeds along the web of resonances rather than across barriers. Arnold diffusion is the name for it, Nekhoroshev’s theorem bounds how slow it must be, and whether it has ever been observed in a planetary system — rather than in a model built to show it — is the open question any account of it has to answer honestly. The other direction is the tail: stickiness near the islands, the power law it gives the residence times, and whether the exponent is universal.

Where this goes next

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. A separation of 10⁻¹⁰ becomes 0.0285. The distance between two copies of Burrau's problem, started 10⁻¹⁰ apart in one coordinate of one body and then stepped in lockstep — one loop, one step size, both states advanced by it — for 60 time units. The vertical axis is logarithmic, so the straight stretch is exponential growth, and its slope is the Lyapunov exponent: 0.3413 per time unit, fitted over the 604 samples lying between ten times the starting offset and a tenth of the system's own size. That is an e-folding every 2.93 time units, so the separation multiplies by ten every 6.75 and by 7.8·10⁸ across the whole run. It has not saturated within the drawn interval: the growth rate over the last tenth of the run is still 0.6002 per time unit, 176% of the fitted exponent, and the separation has reached only 0.90% of the system's own size. The exponent is what sets a prediction horizon, and this run is drawn short of it on purpose: the straight line is the measurement, and the flat part that follows is arithmetic about how far apart two bounded systems can get. A prediction with an expiry date The inner solar system's Lyapunov time is about five million years, so a centimetre of error becomes an orbit in a hundred million. The ephemeris dies while the system survives — because the elements stay bounded when the phase does not, and only one of those is what stability means. The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye. Resonance clears a gap in one place and locks a moon in another When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

CantorusChaosChaotic diffusionInvariant curveKAM theoryLyapunov timeResonanceResonance overlapStable chaosStandard mapSurface of section