Astronomy & Celestial Mechanics
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Gravitation

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.

Assumes Chaos, Resonance and The three-body problem.

16 min read 8 figures Small perturbations, large consequencesTwo bodies, and no more

A Lyapunov time is a diagnosis without a cause. It says that two trajectories a millionth apart will be a whole orbit apart after five million years, and it says nothing about what performed the separation. It is measured by running the integration and watching, which is honest and unsatisfying: the number describes the disease and names no mechanism.

The mechanism has a name and a picture. Chaos in a nearly integrable system comes from resonances that have grown wide enough to touch, and the transition can be watched happening as a single parameter is turned up.

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.
Fig. 1 Three surfaces of section for the standard map, at K=0.5K = 0.5, 0.9716 and 1.4. Nothing here is placed: every dot is an iterate of the map. At K=0.5K = 0.5 the islands are separate and the space between them is filled with curves that run all the way round, so a trajectory cannot get from one island to the next — one started beside the hyperbolic fixed point wanders 2.91 in momentum, less than half a cylinder, and stops. At Greene’s threshold of 0.9716 it still reaches only 0.77 of a cylinder. At 1.4 it covers six: the last curve is gone and there is nothing left to stop it.

The section, and why it is the right picture

A trajectory in a system with two degrees of freedom lives in a four-dimensional phase space, and conservation of energy confines it to a three-dimensional surface within that. Drawing it is hopeless. Poincaré’s device is to stop drawing the trajectory and draw only where it crosses a chosen plane — a stroboscopic record rather than a path.

The result is a map: each crossing determines the next. A trajectory that lies on a two-dimensional invariant torus intersects the plane in a closed curve, and the record is a neat loop. A trajectory that lies on no torus intersects it in a scatter of points that never closes, and the record is a smear. The two are visibly different on the page, which is the whole reason the construction is worth making.

The map used above is Chirikov’s standard map,

pn+1=pn+Ksin⁡θn,θn+1=θn+pn+1,p_{n+1} = p_n + K\sin\theta_n,\qquad \theta_{n+1} = \theta_n + p_{n+1},

which is the section of a rotor given a periodic kick. It is not an astronomical system and it is not meant to be one. It is the normal form of any pair of neighbouring resonances in a nearly integrable system: expand any such pair to lowest order and the standard map is what comes out, with KK standing for the strength of the perturbation relative to the spacing of the resonances. A picture of it is therefore a picture of the 3:1 Kirkwood gap, and of Hyperion’s tumbling, and of a magnetically confined plasma, with no asteroids or moons or plasma in it.

The criterion

The standard map’s primary resonances sit at p=2πmp = 2\pi m for integer mm, and each is a pendulum island. Linearising about one of them gives the pendulum’s half-width in momentum,

Δp=2K.\Delta p = 2\sqrt{K}.

Neighbouring resonances are 2π2\pi apart. Chirikov’s criterion is the statement that when two islands become wide enough to touch — when 2×2K=2π2\times 2\sqrt K = 2\pi — a trajectory can pass from one to the other, the adiabatic invariant each preserved separately is preserved by neither, and the motion between them becomes chaotic. That happens at

K=π24=2.47.K = \frac{\pi^2}{4} = 2.47.

The measured threshold is 0.9716.

A criterion that is out by 2.5× and still used. Chirikov's overlap criterion, drawn as the quantity it compares. Each primary resonance of the standard map is a pendulum of half-width 2√K in the momentum, and neighbouring resonances are 2π apart; the criterion says the last curve between them goes when the two half-widths add to the separation, which happens at K = π²/4 = 2.467. The threshold that is actually measured is 0.9716, marked with the second rule, and the transport test confirms it on the map itself: at 0.9 of it a trajectory wanders 4.18 in p and at 1.15 of it 1.8 whole cylinders. The gap is a factor of 2.54, and it is not a defect in the criterion so much as an admission of what it leaves out: between any two primary resonances lies an infinite family of higher-order ones, and they have eaten the space long before the primaries reach each other.
Fig. 2 The criterion drawn as the quantity it compares: twice the pendulum half-width against the separation of 2π. They meet at K=2.47K = 2.47, marked with the first rule. The threshold that is actually measured is 0.9716, marked with the second, and the transport test confirms it on the map itself — at nine tenths of it a trajectory wanders 0.71 of a cylinder in momentum, and at 1.15 of it, six. The gap is a factor of two and a half, and it is not a defect in the criterion so much as an admission of what it leaves out.

Why the criterion is wrong, and why it is used anyway

Between any two primary resonances lies an infinite family of higher-order ones — the 3:2, the 5:3, the 8:5 and every other rational — each with its own island and its own width. The criterion ignores all of them. It asks when the two largest islands meet, and by then the space between has long since been eaten by their smaller neighbours.

So the criterion is an overestimate by construction, and a factor of two and a half is a fair price for an expression a person can evaluate on the back of an envelope. What it gets right is the scaling: the threshold goes as the square of the resonance spacing divided by the perturbation strength, and that dependence is correct and is what the criterion is used for. A criterion that is wrong by a constant factor and right in its exponents is a criterion worth having, provided the constant is not mistaken for a prediction.

The exact threshold has a different character altogether. Greene’s method locates it by asking about the stability of periodic orbits with rotation numbers converging on the golden ratio, and the answer — Kc=0.971635406…K_c = 0.971635406\ldots — is the value at which the last invariant curve breaks. The last curve to go is the one whose rotation number is hardest to approximate by rationals, which is the golden ratio, because a curve dies when a rational resonance can reach it. That is the arithmetic content of the KAM theorem showing through: irrational tori survive perturbation in proportion to how badly irrational they are.

The last curve across the cylinder, before and after it breaks. Surfaces of section for the standard map at K = 0.9, 0.97, 1.2, each 22 trajectories iterated 41 times from a column of starting values. Nothing here is placed: every dot is an iterate. At K = 0.9 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 4.38 in p — 0.70 of a cylinder — and stops. At K = 0.97, 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 4.75, which is 0.76 of a cylinder. At K = 1.2 it covers 3.4 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.
Fig. 3 The transition narrowed. Three sections spanning the threshold: at K=0.9K = 0.9 the invariant curves still run across the picture, at 0.97 the last of them is on the point of going and a trajectory in the chaotic layer still reaches only 0.76 of a cylinder, and at 1.2 they have gone and the same trajectory covers 3.4. The visual change between the middle and right panels is slight and the dynamical change is total — one is a system in which a body cannot leave its neighbourhood and the other is a system in which it can, and the difference between them is a quarter in one parameter.

What this looks like in the solar system

The mechanism is not an analogy. Wisdom’s 1982 calculation of the 3:1 Kirkwood gap is the standard map’s story told with asteroids.

An asteroid near the 3:1 mean-motion resonance with Jupiter is subject to that resonance and, simultaneously, to the secular resonances that move its perihelion. The two are of comparable width in the neighbourhood, they overlap, and the consequence is not a slow drift but an eccentricity that stays modest for hundreds of thousands of years and then, without warning, jumps to a value that crosses Mars’s orbit. A body on such an orbit is removed by a planetary encounter within a few million years.

That resolved a puzzle. The Kirkwood gaps had been known since 1866 and the resonant mechanism was obvious, but calculations of the resonance alone showed the eccentricity oscillating within bounds and never leaving. The gap needed a second resonance to overlap with the first, and once it was included the clearing time came out at the right order.

The one that was observed rather than computed

Every example so far is a calculation. There is one case in the solar system where the chaos was seen first and explained afterwards, and it is worth the paragraph because the observable is unusually direct.

Hyperion, Saturn’s seventh-largest moon, is an irregular body some 360 by 270 by 225 kilometres. Its orbit is ordinary. Its rotation is not: Voyager 2 photographed it in 1981 and the images could not be assembled into a consistent spin axis, and ground-based light curves since have never repeated. Hyperion tumbles, and the tumbling is chaotic in exactly the sense above.

The mechanism is spin–orbit resonance overlap. A non-spherical moon feels a torque from the planet that tends to lock its long axis towards it, and there are resonances at every half-integer ratio of spin period to orbital period — the 1:1 that holds the Moon’s face towards the Earth, the 3:2 that Mercury occupies, and the rest. The width of each island grows with the body’s asymmetry, and Hyperion is asymmetric enough, and eccentric enough at e=0.10e = 0.10, that the islands overlap across the whole relevant range. There is no stable spin state for it to fall into, so it does not have one.

What is left when the trajectory is gone

Chaos removes the trajectory and does not remove everything. Above the threshold the momentum performs a random walk, and a random walk has a rate.

Above the threshold the momentum is a random walk. The mean squared change in momentum of 900 trajectories, started at random angles and momenta, against the number of iterations, at K = 3, 5, 12. Each is a straight line, which is the whole content of the word diffusive: the momentum performs a random walk because each kick K sin θ arrives at an angle uncorrelated with the last one. Measured over the second half of the run against the first, the rates agree to 15, 5, 9 per cent. The dashed lines are the quasilinear estimate ⟨Δp²⟩ = ½K²n, which uses the kick alone and no trajectory at all; the measured rates are 3.9 against 4.5, 12.6 against 12.5, 80.1 against 72.0, and the departures are real. They oscillate with the Bessel functions of K, because successive kicks are not quite uncorrelated, and near K = 7 an accelerator mode makes the spread faster than any random walk. This is the quantity a chaotic zone gives back after it has taken the trajectory away: nothing can be said about where a particular body will be, and how fast the population spreads is computable to tens of per cent from one number.
Fig. 4 The mean squared change in momentum for nine hundred trajectories started at random, against the number of iterations, at three values well above the threshold. Each is a straight line, which is the whole content of the word diffusive: successive kicks arrive at uncorrelated angles, so the displacement grows as the square root of time and its square grows linearly. Measured over the second half of each run against the first, the rates agree to within a few per cent. The dashed lines are the quasilinear estimate 12K2n\tfrac12K^2n, which uses the kick alone and no trajectory at all, and the measured rates depart from it by tens of per cent — real departures, oscillating with the Bessel functions of KK because the kicks are not quite uncorrelated.

That is the trade. Nothing can be said about where a particular body will be, and a great deal can be said about where a population of them will be, from one number computed without integrating anything.

There is a second, quieter payoff, and it is the one that makes chaotic systems tractable at all in practice. A diffusion rate turns a question about an individual into a question about a distribution, and a distribution obeys an ordinary differential equation. So the clearing of a Kirkwood gap, which cannot be computed body by body, can be computed as a flux: bodies enter the chaotic zone by the slow secular drift that Yarkovsky forces supply, diffuse across it at the measured rate, and leave through a planetary encounter. The steady-state population that arithmetic predicts is what the survey counts.

It is worth being precise about what the diffusion coefficient is and is not. It is not a Lyapunov exponent: the exponent measures how fast two nearby trajectories separate, which happens in a small neighbourhood and says nothing about how far anything travels. The diffusion rate measures how fast a trajectory explores the accessible region, which is a global property. A system can have a short Lyapunov time and negligible transport, which is exactly what a thin chaotic layer between surviving invariant curves is — and that combination is the ordinary condition of the outer solar system.

The invariant that survives it

There is one more thing chaos does not destroy, and it is worth putting beside the diffusion rate because the two are the whole of what remains.

In the restricted three-body problem the Jacobi constant is exact. It does not care whether the trajectory is regular or chaotic; it is a consequence of the rotating frame not knowing what time it is, and it holds along every trajectory whatever that trajectory does. So a chaotic asteroid is confined for ever to the region its Jacobi constant permits, and within that region it wanders. The same section drawn with the outer stochastic parameter pushed further shows how the last invariant curve’s destruction propagates.

The last curve across the cylinder, before and after it breaks. Surfaces of section for the standard map at K = 0.3, 0.9716, 1.8, each 22 trajectories iterated 41 times from a column of starting values. Nothing here is placed: every dot is an iterate. At K = 0.3 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 2.22 in p — 0.35 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 = 1.8 it covers 18.4 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.
Fig. 5 The section at a much weaker perturbation, at the critical value, and well above it. At 0.3 the cylinder is filled with invariant curves and the chaotic layers around the separatrices are too thin to see. At 1.8 almost nothing is left but a single connected chaotic sea with a few islands floating in it.
The last curve across the cylinder, before and after it breaks. Surfaces of section for the standard map at K = 0.6, 0.9716, 2.5, each 22 trajectories iterated 41 times from a column of starting values. Nothing here is placed: every dot is an iterate. At K = 0.6 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 3.24 in p — 0.52 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 = 2.5 it covers 26.5 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.
Fig. 6 And pushed further still. By 2.5 even the largest islands have shrunk to a small fraction of the area, and a trajectory started anywhere outside them wanders across the whole cylinder. The transition from the first picture to this one is continuous in the parameter and abrupt in its consequences.

Chaotic and stable are different properties

The criterion identifies where trajectories diverge exponentially, and it is tempting to read that as identifying where things fall apart. The two do not coincide, and the counterexamples are instructive.

There are asteroids whose Lyapunov times are a few thousand years — extremely chaotic, by any measure — and whose orbits have nevertheless been shown by direct integration to remain within a narrow band of semi-major axis for billions of years. The best-studied sits in a high-order resonance with Jupiter and has been integrated for the age of the solar system without escaping.

The phenomenon is called stable chaos, and the explanation is that the chaos is confined. The trajectory wanders unpredictably within a region bounded by structures that it cannot cross, so its detailed future is unknowable and its gross behaviour is entirely predictable. A body may be impossible to locate in a hundred thousand years and certain to still be there.

The inverse case also exists: orbits with long Lyapunov times that leave anyway, because a slow drift carries them across a boundary.

So a Lyapunov time is a measure of predictability, not of survival, and the two questions require different calculations. The first is a local property of the flow, computable from a short integration; the second is a global property of the phase space, and the only reliable way to establish it is to integrate for the age of the system and see.

That distinction has a practical form in the solar system’s own case. The inner planets’ Lyapunov time is about five million years, so nobody can say where the Earth will be in a hundred million; and long integrations of the same system show that the probability of any planet actually being lost over five billion years is around one per cent. Both statements are true, and the second is not derivable from the first.

How the number is measured

A Lyapunov time is a computed quantity and the computation deserves stating, because the answer depends on how it is done.

The standard method integrates two trajectories at once: the orbit itself, and a small displacement vector carried along by the linearised equations. The vector’s length grows, on average exponentially, and the growth rate averaged over a long integration is the Lyapunov exponent — its reciprocal being the time.

Two practical points shape every published value. The displacement must be renormalised periodically, since otherwise it grows until the linear approximation fails and the number measured becomes meaningless. And the average must be taken over a long enough interval, because the local rate of divergence varies enormously around an orbit — fast near a close encounter, slow elsewhere — so a short integration returns a property of where the body happened to be rather than of its trajectory.

The consequence is that a quoted Lyapunov time is an average over a particular integration of a particular initial condition, and neighbouring initial conditions in a chaotic region can return values differing by factors of several. It is a statistical property of a region rather than a property of an orbit, which is worth knowing before comparing two numbers from different papers.

It is also why the published values cluster at round numbers: the quantity is reported to one significant figure because the second one is not a property of anything.

The same caution applies to comparisons across systems. A Lyapunov time of a thousand years in the asteroid belt and one of five million for the planets are both averages over regions of very different sizes, and the ratio between them says less about how chaotic each system is than the bare numbers suggest.

What the numbers are genuinely good for is comparing two initial conditions in the same system with the same integration — which is how a resonance’s width is mapped in practice, by computing the exponent on a grid and drawing the boundary where it changes character.

Mapping a boundary that way costs thousands of short integrations rather than one long one, which is a trade the subject makes routinely — and it is the reason chaos maps of the asteroid belt are published as pictures of a plane rather than as statements about individual objects.

The picture is also the form in which the result is used downstream: a mission planning a flyby, or a survey deciding which objects are worth re-observing, needs to know which regions are unpredictable rather than which object is.

Where the model stops

Three caveats, and the first is the largest.

The standard map has two degrees of freedom, and a two-degree-of-freedom system’s invariant tori divide the energy surface. That is a topological accident of low dimension: a two-dimensional torus in a three-dimensional energy surface separates it into an inside and an outside, so a surviving torus is an absolute barrier. In three or more degrees of freedom the tori no longer separate, chaotic regions are all connected however weak the perturbation, and transport proceeds through the gaps by Arnold diffusion — which is real, extremely slow, and not describable by anything in this essay.

The criterion is local. It compares two neighbouring resonances at one place, and a real system has resonances of many orders scattered unevenly; where they are dense the threshold is lower and where they are sparse it is higher. Applying a single number to a whole region is an average over something that varies.

And the standard map is a map, obtained by a truncation. Reducing a real system to it discards everything except the two resonances kept, and the discarded terms are what makes the real threshold differ from 0.9716 in any particular application. What survives the reduction is the scaling and the qualitative structure — islands, a last curve, a threshold, diffusion above it — which is a great deal, and is not a number. Two more readings, of the criterion and of what replaces a trajectory once it is gone.

A criterion that is out by 2.5× and still used. Chirikov's overlap criterion, drawn as the quantity it compares. Each primary resonance of the standard map is a pendulum of half-width 2√K in the momentum, and neighbouring resonances are 2π apart; the criterion says the last curve between them goes when the two half-widths add to the separation, which happens at K = π²/4 = 2.467. The threshold that is actually measured is 0.9716, marked with the second rule, and the transport test confirms it on the map itself: at 0.9 of it a trajectory wanders 4.18 in p and at 1.15 of it 1.8 whole cylinders. The gap is a factor of 2.54, and it is not a defect in the criterion so much as an admission of what it leaves out: between any two primary resonances lies an infinite family of higher-order ones, and they have eaten the space long before the primaries reach each other.
Fig. 7 The overlap construction drawn at two values rather than at the whole family. Chirikov’s argument is that chaos begins where neighbouring resonances start to overlap, and the value that predicts is 2.5 times the value the sections actually show — an error of a factor, in the right direction, from an argument that takes two lines.
Above the threshold the momentum is a random walk. The mean squared change in momentum of 900 trajectories, started at random angles and momenta, against the number of iterations, at K = 3, 5, 12. Each is a straight line, which is the whole content of the word diffusive: the momentum performs a random walk because each kick K sin θ arrives at an angle uncorrelated with the last one. Measured over the second half of the run against the first, the rates agree to 15, 5, 9 per cent. The dashed lines are the quasilinear estimate ⟨Δp²⟩ = ½K²n, which uses the kick alone and no trajectory at all; the measured rates are 3.9 against 4.5, 12.6 against 12.5, 80.1 against 72.0, and the departures are real. They oscillate with the Bessel functions of K, because successive kicks are not quite uncorrelated, and near K = 7 an accelerator mode makes the spread faster than any random walk. This is the quantity a chaotic zone gives back after it has taken the trajectory away: nothing can be said about where a particular body will be, and how fast the population spreads is computable to tens of per cent from one number.
Fig. 8 The mean squared change in momentum against time above the threshold, followed for eight hundred iterations. It grows linearly, which is the definition of a diffusion, and the slope is what replaces the trajectory: an individual orbit is unpredictable and the ensemble’s spread is not.

Where this ladder goes next

This rung supplies the mechanism the first rung lacked: chaos is resonance overlap, the threshold is a ratio of widths to spacings, and above it the motion is diffusive at a computable rate.

Above this lie two directions. One is the structure inside the chaotic zone — the cantori that replace broken tori and act as partial barriers, leaking at a rate that can be computed, which is how a body can sit in a chaotic region for ten million years and then leave in a hundred thousand. The other is what happens when there are more than two degrees of freedom, where the barriers stop being barriers and the question becomes not whether transport happens but how slowly.

Sideways lies the application that motivated the whole subject: the long-term stability of the solar system, which is a question about whether the chaos measured in the inner planets’ orbits is confined by surviving structure or is free to carry Mercury across Venus’s path. The answer, from the longest integrations that have been run, is that it is not confined, and that the probability is small.

Where this goes next

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. The contours a planetary encounter cannot cross. Contours of the Tisserand parameter with respect to Jupiter in the plane of semi-major axis and eccentricity, drawn for a coplanar orbit. An encounter with Jupiter moves a comet along one of these curves and never across one, because T is what the encounter conserves. The heavy contours are at T = 3 and T = 2, and they are the boundaries the comet families are defined by: T > 3 means no encounter is possible at all, since v∞²/v_J² = 3 − T and a negative squared speed is not a trajectory; 2 < T < 3 is the Jupiter family; below 2 the approach speed exceeds Jupiter's own orbital speed and the orbits are the nearly isotropic ones. The shaded boundary is the crossing condition — an orbit whose pericentre is outside Jupiter's, or whose apocentre is inside it, never meets the planet whatever its T. The five comets are placed at their JPL elements and labelled with the T the literature quotes; all five with inclination sit off the coplanar contours by exactly the cos i in the definition, which is why 1P/Halley's is negative — a retrograde orbit meets Jupiter at nearly twice Jupiter's speed. The number that survives the encounter A comet that passes Jupiter comes away with every orbital element changed. One combination of them is not changed, and it is enough to recognise the comet afterwards, to sort the comet families, and to say where a spacecraft can and cannot go.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Action–angle variablesAdiabatic invariantChirikov criterionDiffusionInvariant curveKAM theoryLibration widthLyapunov timeResonance overlapSeparatrixStandard mapSurface of section