Illustrated Physics
physics, drawn
Mechanics

The calm that is the ghost of a cycle

Just before a chaotic system settles into a stable cycle it does something stranger than either: it behaves perfectly periodically for long stretches, and then, at moments nothing in the record predicts, bursts into disorder and back. The calm is a cycle that does not exist yet, creeping through the narrow gap where it is about to be born, and how long each calm lasts is set by the square root of the distance to that birth.
17 min read 5 figures Order out of the randomThe same equation again

Assumes: The map a dripping tap turns out to be · The error that doubles on a schedule

The map a dripping tap turns out to be followed one route into chaos to its end: a strongly damped system, sampled once per cycle, doubles its period again and again until the doublings pile up and the motion becomes irregular. It ended by naming a second route, and the second route looks nothing like the first. There is no cascade. The system is periodic, and then — past a threshold, with nothing gradual about it — it is periodic most of the time, interrupted by bursts of disorder that arrive at no predictable moment and leave as suddenly as they came.

Yves Pomeau and Paul Manneville described this in 1980 and called it intermittency. The same year, measurements in a convection cell showed exactly the pattern they predicted: long stretches of regular oscillation separated by brief, irregular bursts, with the stretches lengthening as a control parameter approached a threshold. The explanation turns out to be one of the most portable arguments in dynamics, because the calm stretches are not a property of chaos at all. They are the ghost of a stable cycle that is about to exist.

Calm, then a burst, then calm

The logistic map, x↦rx(1−x)x \mapsto rx(1-x), is chaotic over most of the range of rr beyond its period-doubling cascade. At exactly r=1+8≈3.8284r = 1 + \sqrt 8 \approx 3.8284 a stable three-cycle appears out of that chaos: above that setting, every orbit settles onto three values visited in turn. Just below it, the orbit is doing something neither chaotic nor periodic.

Long stretches of order, broken without warning. 1800 successive values of the logistic map at r = 3.828427 − 0.00002, a distance of 2.0 × 10⁻⁵ below the setting at which its stable three-cycle is born. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them the orbit bursts through the whole interval with no discernible pattern, and then, at an unpredictable moment, is captured into another calm. In a run of 400,000 iterates at this setting the calms last 173 iterates on average, and the channel the orbit creeps through — measured on the map's own third iterate — has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates. Nothing random is added: the sequence is the same every time it is computed from the same start.
Fig. 1 1,800 successive values of the logistic map at r=1+8−2×10−5r = 1 + \sqrt{8} - 2 \times 10^{-5}. The shaded stretches are calm: the orbit repeats itself to within 0.004 every third step, cycling through three values as though the three-cycle already existed. Between them it bursts across the whole interval. In 400,000 iterates the calms average 173 iterates, and the channel the orbit creeps through has a gap of 4.1 × 10⁻⁵ and a longest passage of 253 iterates.

Two hundred-thousandths below the setting where the cycle is born, the orbit spends long stretches cycling through three values, repeating itself to three decimal places every third step. Then it leaves, wanders through every value between nought and one with no discernible pattern — stretched and folded exactly as the fold that has to be there requires, and as chaotic as anywhere in the map’s range — and is captured into another calm, which lasts about as long as the last one and ends as abruptly.

The map contains no randomness. The same starting value produces the same calms and the same bursts every time the sequence is computed. The irregular timing of the bursts is the deterministic chaos of the burst phase, an error that doubles on a schedule during each excursion, deciding where the orbit re-enters the region of calm and so how long the next calm will last.

A cycle that has not been born

The calms are explained by looking at the map three steps at a time. A three-cycle is a point that the third iterate f3f^3 returns to itself, so it lies where the graph of f3f^3 crosses the diagonal. A stable three-cycle is born at r=1+8r = 1 + \sqrt 8 in the simplest way a pair of fixed points can be born: the graph of f3f^3, rising as rr rises, touches the diagonal at three places and then pushes through, creating at each place a stable crossing and an unstable one. That is a saddle-node bifurcation, sometimes called a tangent bifurcation, and it is the commonest way anything in dynamics comes into existence.

The ghost of a cycle that has not been born. The logistic map's third iterate close to one of the three places where it nearly meets the diagonal, at r = 3.828427 − 0.0003, on a window 0.024 wide. The solid curve stays above the diagonal by a gap of 6.1 × 10⁻⁴, measured by searching the curve; the dashed curve is the same iterate at the window's edge, where the gap closes and a stable three-cycle appears at the point of contact. The staircase is an orbit passing through the channel between the curve and the diagonal, taking 17 steps of the third iterate to cross this window — tiny steps in the middle, where the gap is narrowest, and larger ones at the ends. Halving the distance to the window halves the gap, 3.0 × 10⁻⁴, and the curvature at the channel is 34.1, so the longest passage is π divided by the square root of curvature times gap, 22 steps.
Fig. 2 The map’s third iterate close to one of the three places where it nearly meets the diagonal, at r=1+8−3×10−4r = 1 + \sqrt{8} - 3 \times 10^{-4}. The solid curve clears the diagonal by a gap of 6.1 × 10⁻⁴; the dashed curve, at the window’s edge, touches it. The staircase is an orbit creeping through the channel, 17 steps of the third iterate to cross a window 0.024 wide — tiny steps where the gap is narrowest. Halving the distance halves the gap; the curvature there is 34.1, so the longest passage is 22 steps.

Just below the bifurcation the graph has not quite reached the diagonal. It passes above it at each of the three places, leaving a narrow channel between the curve and the line. An orbit iterated by f3f^3 moves, on each step, by the vertical distance between the curve and the diagonal — and in the channel that distance is tiny. So an orbit arriving at the channel creeps through it in a long series of small steps, each third iterate almost exactly equal to the one before. Seen one iterate at a time, that creep is a nearly perfect three-cycle.

The calm is the ghost of the cycle that will exist at the bifurcation. The fixed point of f3f^3 is not there yet; the place where it will be is there, and an orbit passing through that place behaves as though the fixed point were present, slowly, until it has crept out the other side of the channel and is thrown back into the chaotic part of the map. The figure measures both halves of the geometry directly on the map’s own iterate: the gap is proportional to the distance from the bifurcation, halving when the distance halves, and the curve near its closest approach is a parabola with a definite curvature.

Why the square root

Those two measured facts are the whole of the scaling law.

Near its closest approach the gap between the curve and the diagonal is, to leading order, g+a y2g + a\,y^2, where yy is the distance along the channel from the narrowest point, aa is the curvature, and gg is the gap at the narrowest point. An orbit’s step through the channel is that gap, so in a long channel the orbit’s position obeys, to a good approximation, dy/dn=g+ay2dy/dn = g + a y^2. Counting the steps from one end of the channel to the other is an integral,

n=∫dyg+ay2  →  πag,n = \int \frac{dy}{g + a y^2} \;\to\; \frac{\pi}{\sqrt{a g}},

the familiar arctangent, which reaches π/ag\pi/\sqrt{ag} for a channel long compared with its narrowest part. And since the gap grows in proportion to the distance ε\varepsilon below the bifurcation, the passage time grows as ε−1/2\varepsilon^{-1/2}.

A hundred times closer to the cycle’s birth, the calms last ten times longer. The exponent is not a property of the logistic map, of the three-cycle, or of anything chaotic. It comes from a parabola meeting a line and from the gap between them growing linearly, which is what happens at every saddle-node bifurcation in every smooth system. The chaos is needed only to bring the orbit back to the channel after each passage; the length of the calm is decided entirely by the channel.

The calms, measured

The derivation assumes a long channel, a small gap and a clean parabola. The map can be asked directly whether those assumptions hold.

The calms last as one over the square root of the distance to the window. The mean length of the calm stretches, in iterates, against the distance below the setting at which the logistic map's three-cycle is born, on logarithmic axes, measured from runs of the map long enough to contain at least forty calms at each setting. At 1 × 10⁻⁸ the calms average 8125 iterates over 2444 of them; at 3 × 10⁻⁸ the calms average 4707 iterates over 4195 of them; at 1 × 10⁻⁷ the calms average 2547 iterates over 7684 of them; at 3 × 10⁻⁷ the calms average 1484 iterates over 12976 of them; at 1 × 10⁻⁶ the calms average 804 iterates over 23098 of them; at 3 × 10⁻⁶ the calms average 456 iterates over 33630 of them; at 1 × 10⁻⁵ the calms average 245 iterates over 31072 of them. A straight line fitted to the measurements has slope −0.506. The upper line is the longest passage through the channel, π divided by the square root of curvature times gap, with both measured on the map's third iterate, whose slope is −0.500. Both are −½ because the gap grows in proportion to the distance from the window and a creep through a parabolic channel takes a time inversely proportional to the square root of its gap: a hundred times closer to the window, ten times longer calms.
Fig. 3 The mean length of the calms against the distance below the setting at which the three-cycle is born, on logarithmic axes, from runs containing at least two thousand calms at each setting: 245 iterates at 10⁻⁵, 804 at 10⁻⁶, 2,547 at 10⁻⁷ and 8,125 at 10⁻⁸. A line fitted to the measurements has slope −0.506. The upper line is the longest passage through the channel, from the measured gap and curvature, with slope −0.500.

Over three decades of distance the mean calm grows from 245 iterates to 8,125, and the fitted slope on logarithmic axes is −0.506. The longest possible passage through the channel, computed from the gap and curvature measured at each setting, has slope −0.500 exactly, and lies a little above the mean at every setting — because a calm begins wherever the burst happens to drop the orbit, which is usually somewhere inside the channel rather than at its entrance.

The law is a limit, and it is worth being plain about where. Measured further from the bifurcation, at a distance of 10−410^{-4} or 10−310^{-3}, the channel is no longer long compared with its narrowest part, the calms are only a few dozen iterates, and the local slope drifts away from −½. The figure’s settings were chosen where the measured slope has settled, which is below about 10−510^{-5} — and a sweep of real data that happened to sit in the drifting region would find an exponent that was not −½ without contradicting anything.

A waiting time with a ceiling

A sequence of bursts at irregular moments looks, from outside, like a random process, and the most familiar random waiting time is a nucleus with no clock: exponential, memoryless, with no longest wait. The calms are not that.

How long each calm lasts. The lengths of 19620 calm stretches in a run of six million iterates at a distance of 1 × 10⁻⁵ below the window, as a histogram. The dashed line is the longest passage through the channel, 358 iterates, computed from the gap and the curvature of the map's third iterate. The distribution is not centred on a typical value: it rises towards that longest passage and piles up just below it, with its tallest bin between 329 and 343 iterates. A burst that happens to throw the orbit back near the entrance of the channel gives a long calm, and one that drops it part-way through gives a short one, so calms longer than the channel allows are absent: the longest seen, 342, is also the length ninety-nine calms in a hundred fall short of.
Fig. 4 The lengths of 19,620 calms in six million iterates at 10⁻⁵ below the window, as a histogram, with the longest passage through the channel, 358 iterates, dashed. The distribution rises towards that passage and piles up just below it, with its tallest bin between 329 and 343 iterates; no calm is longer than 342.

The distribution of calm lengths is nearly the opposite of an exponential. It rises towards a maximum rather than falling from one, piles up just below the longest passage the channel allows, and stops — the opposite of an exponential that holds only in the middle and fails at its ends, this one has no middle to be exponential in. In six million iterates at a distance of 10−510^{-5} there are nearly twenty thousand calms, and not one of them is longer than 342 iterates against a longest passage of 358.

The shape comes from the reinjection. A burst drops the orbit back somewhere near the channel, and the calm that follows lasts as long as the remaining passage from that point. An orbit re-entering near the entrance creeps through the whole channel and gives a long calm; one re-entering near the middle gives a shorter one; the channel’s own shape — slow in the middle, fast at the ends — concentrates the lengths near the maximum. The waiting time between bursts carries a memory of the geometry that produced it, and a histogram of waiting times that stops sharply at a ceiling is a signature a record of bursts can be tested for.

Chaos that fades rather than stops

At the bifurcation the three-cycle becomes stable and the chaos ends. Approaching from below, it does not end abruptly; it thins out.

Chaos that fades as the square root. The Lyapunov exponent of the logistic map — the average rate at which neighbouring orbits separate — against its setting, across the edge of the three-cycle window at 3.828427, one plus the square root of eight, from 60,000 iterates at each of 161 settings. Below the edge it is positive and the orbit is chaotic; across the edge it drops below zero, and inside the window the stable three-cycle pulls neighbouring orbits together. Close to the edge the exponent is measured separately at 7 settings from 1 × 10⁻⁸ to 1 × 10⁻⁵ below it: 1 × 10⁻⁸ gives 0.0033, 3 × 10⁻⁸ gives 0.0058, 1 × 10⁻⁷ gives 0.0101, 3 × 10⁻⁷ gives 0.0174, 1 × 10⁻⁶ gives 0.0334, 3 × 10⁻⁶ gives 0.0537, 1 × 10⁻⁵ gives 0.0914, and a line fitted on logarithmic axes has slope 0.48. The chaos does not switch off; it fades as the square root of the distance, because separation happens only in the bursts and bursts become rarer as the calms lengthen. The law is a limit: between 3 × 10⁻⁵ and 10⁻⁴ below the edge the same exponent grows only as the 0.32 power, because there the calms are too short for the channel's parabola to dominate them.
Fig. 5 The Lyapunov exponent of the logistic map across the edge of the three-cycle window at 1+81 + \sqrt{8}, from 60,000 iterates at each of 161 settings: positive below the edge, negative inside the window. Measured at seven settings from 10⁻⁸ to 10⁻⁵ below the edge, it is 0.0033 at 10⁻⁸ and 0.0914 at 10⁻⁵, and a line fitted on logarithmic axes has slope 0.48. Between 3 × 10⁻⁵ and 10⁻⁴ below the edge the same exponent grows only as the 0.32 power.

The Lyapunov exponent measures the average rate at which neighbouring orbits separate, and in an intermittent orbit the separation happens almost entirely during the bursts: through the channel, neighbouring orbits creep along together. As the calms lengthen, bursts become rarer — in proportion to the square root of the distance, since that is how often a calm ends — and the average separation rate falls with them. So the exponent should vanish as the square root of the distance, and measured at settings from 10−810^{-8} to 10−510^{-5} below the bifurcation it falls from 0.0914 to 0.0033 with a fitted slope of 0.48.

Further from the edge the same measurement grows only as the 0.32 power, for the reason the calm lengths drift: the calms are too short for the channel to dominate them. The chaos does not switch off at the bifurcation as a light does. It fades, and the fading has an exponent that belongs to the channel rather than to the chaos.

The same bottleneck, elsewhere

The argument used nothing about maps except a parabola near a line, and the same parabola appears wherever something is pushed just past the point at which a stable state disappears.

A pendulum with strong friction — the overdamped case of the three ways of coming to rest — pushed by a steady torque, settles at an angle where the torque balances gravity. Increase the torque past the largest gravity can balance and the pendulum goes over the top — but just past that threshold it lingers, for a long time, near the angle where the balance used to be, before swinging round and returning to linger again. The time per revolution grows as the inverse square root of the excess torque, by the same integral, because the stable and unstable resting angles have merged and left a narrow gap in the pendulum’s equation of motion.

That equation is exactly the equation of a Josephson junction. The phase difference across a superconducting junction obeys the driven, overdamped pendulum equation, with the bias current playing the torque and the critical current playing the largest torque gravity can resist. Above the critical current the phase slips, and the voltage that is a frequency is the rate of those slips. Just above the critical current, the slips are separated by long pauses in which the phase lingers near its vanished resting value, and the average voltage rises as the square root of the excess current, I2−Ic2\sqrt{I^2 - I_c^2}. A superconducting junction just above its critical current and the logistic map just below its three-cycle window are the same bottleneck, with a voltage measuring in one what the calm lengths measure in the other.

Neurons of one common type begin to fire repetitively the same way. Below a threshold current the cell rests; just above it, the resting state has vanished in a saddle-node, the membrane voltage lingers in the ghost of that state between spikes, and the firing rate starts from zero and rises as the square root of the excess current. The long quiet intervals of a slowly firing cell are ghosts of a resting state, as the calms are ghosts of a cycle.

The lingering near a vanished fixed point is related to the slowing down near a critical point that the point at which the two become one describes, and both are the dynamical signature that a system is close to a change in what it is able to do — which is why ecologists and climate scientists look for lengthening recovery times as a warning that a threshold is near.

Other kinds of intermittency

The saddle-node route is the intermittency Pomeau and Manneville classified as type I, to distinguish it from two others. All three belong to damped systems; a frictionless one reaches chaos by the destruction of invariant curves instead, and has no calms of this kind. In type II, a stable fixed point loses stability to a spiral, and the orbit spends its calms spiralling slowly away from the ghost of a steady state; in type III, the instability is of the period-doubling kind, and the calms are nearly periodic with a slowly growing alternation. Both have calms whose mean length grows as the inverse of the distance to threshold rather than its inverse square root, because the orbit’s escape is governed by a linear instability rather than by a parabolic channel.

The distinction matters practically. An experiment that measures how calm lengths scale as a control parameter approaches threshold can identify which kind of bifurcation it is near, without knowing the system’s equations — the same move the map a dripping tap turns out to be made with the Feigenbaum ratio. The exponent is a fingerprint of the local geometry of the bifurcation, and it is visible in a record of when bursts happened.

Where the map stops describing an experiment

Noise closes the channel. The passage time grows without limit only in a noiseless system. Any real noise kicks the orbit across the narrow part of the channel, and once the distance to the bifurcation is small enough that the gap is comparable with the noise, the calms stop lengthening. A measured mean calm length that grows as the inverse square root and then levels off is reading the noise level, not a failure of the theory.

The reinjection is global. The scaling exponent belongs to the channel; the distribution of calm lengths, and so the mean’s prefactor, depends on where the bursts drop the orbit, which depends on the whole map. Two systems near the same kind of bifurcation share an exponent and not a histogram.

A calm has to be defined. Detecting calms requires a threshold — here, repeating to within 0.004 every third step — and the measured mean depends on it: with a threshold of 0.001 the mean at 10−810^{-8} is about ten per cent longer. The exponent does not depend on the threshold once the calms are long; the numbers do.

Real systems are not one-dimensional maps. The logistic map stands for any system whose sampled dynamics is effectively one-dimensional, which is the condition the dripping tap essay set out. A system with several unstable directions can pass near a saddle-node in more complicated ways, and a convection cell’s intermittency is a map only after the same contraction of phase space that made its period doubling one.

And the law is a limit. The measured slopes settle at −½ and ½ only close to the bifurcation, and drift measurably away from it. That is a statement about the approach to the asymptote, not a failure of it, and it means an exponent measured over a range that includes the drift will be wrong in a definite direction.

What a time series cannot show

The series figure shows calms and bursts and cannot show why the calms are calm. The channel lives in the graph of the third iterate, which the series does not contain: seen one value at a time, the creep through the channel is a near-repetition, and nothing about it announces that the repetition is running out. The geometry that decides the calm’s length is invisible from inside the calm.

Nor can any finite record show the ghost’s defining property, which is that it becomes a real cycle at a setting just beyond the record’s range. A system that has lingered near a three-cycle for a hundred thousand iterates is not distinguishable, by that record alone, from one that will linger there forever. The bifurcation is inferred from how the calms change as the setting changes, never from one setting.

Still open: telling a ghost from noise

Bursts separated by quiet intervals are found in records of many systems whose equations nobody has written down — lasers, turbulent boundary layers, the firing of neurons, climate proxies with abrupt transitions — and the question each time is whether the quiet intervals are ghosts of a nearby bifurcation or simply a noisy system switching between two states. The two produce superficially similar records and imply different things: a ghost means the system is close to acquiring a new stable state, and its calms will lengthen predictably as a parameter changes; a noise-driven switch means no such change is coming.

The calm-length distribution is the obvious discriminator — a ghost’s calms pile up below a ceiling, a noise-driven switch gives an exponential tail — and in clean simulations it works. In real records it is weakened by exactly the noise that closes the channel, by short records with few calms, and by the threshold that defines a calm in the first place. How reliably intermittency can be identified, and its kind determined, from a single noisy record of finite length is an open problem in the analysis of data from systems near tipping points.

The habit worth carrying away is to read long quiet intervals as a clue about what is nearby. A system that lingers has usually lost a stable state recently, or is about to gain one, and the rate at which the lingering grows as a parameter changes is a measurement of the geometry of that loss.

Part 5 of 5

This essay is one argument about Chaos. The others:

Every value the orbit of x → r x(1 − x) settles on. The values a long orbit of x → r x(1 − x) visits, one column of the picture for each of 320 settings of r between 2.8 and 4. A single point means the orbit settles to one value, two means it alternates, and each branching doubles the count with the gaps shrinking by a constant factor. The superstable settings marked run 3.23607, 3.49856, 3.55464, located by bisection on the map itself. They accumulate at r = 3.569946, and past it the orbit visits a band of values rather than a list of them. The bands are not noise: the map has no random number in it, and the same initial value gives the same orbit every time. The map a dripping tap turns out to be Part 4 — A universal result about one-dimensional maps is worth nothing to a physicist unless a real system is one, and a tap, a convection cell and a driven circuit are continuous systems with no map in sight. What makes them maps is dissipation — and the systems that have none take a different route entirely.

Where this goes next

Every value the orbit of x → r x(1 − x) settles on. The values a long orbit of x → r x(1 − x) visits, one column of the picture for each of 320 settings of r between 2.8 and 4. A single point means the orbit settles to one value, two means it alternates, and each branching doubles the count with the gaps shrinking by a constant factor. The superstable settings marked run 3.23607, 3.49856, 3.55464, located by bisection on the map itself. They accumulate at r = 3.569946, and past it the orbit visits a band of values rather than a list of them. The bands are not noise: the map has no random number in it, and the same initial value gives the same orbit every time. The map a dripping tap turns out to be A universal result about one-dimensional maps is worth nothing to a physicist unless a real system is one, and a tap, a convection cell and a driven circuit are continuous systems with no map in sight. What makes them maps is dissipation — and the systems that have none take a different route entirely. A current with no voltage, and then a voltage that is a frequency. Current against mean voltage for a Josephson junction shunted by a resistance, in units of its critical current and of the critical current times the resistance. Up to the critical current the junction carries a supercurrent at exactly zero voltage, set by the difference of the two superconductors' phases. Above it a mean voltage appears, growing as R√(I² − Ic²) and approaching the ohmic line at large current; the curve is integrated from the junction equation and checked against that result at three currents. The voltage is not steady. It is a train of pulses, each the phase slipping by one turn, at a frequency of 483.6 GHz per millivolt: for a junction with Ic = 1 mA and R = 1 Ω, one unit of the voltage axis is 1 mV and 484 GHz. The voltage that is a frequency Two superconductors separated by a barrier a nanometre thick carry a current with no voltage at all, set by the difference of their quantum phases. Push harder and a voltage appears — and a voltage makes that phase difference run, so the current oscillates at 483.6 gigahertz for every millivolt. Shine microwaves on the junction and the voltage locks to exact multiples of the frequency divided by a ratio of fundamental constants, with nothing about the junction in it. That is why a volt is now counted in cycles. Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve. The error that doubles on a schedule Two double pendulums released a hundred-millionth of a degree apart follow one curve for ten seconds and then have nothing to do with each other. The separation grows exponentially the whole time, including while the picture shows a single trace — which turns unpredictability into a rate, and makes the length of a forecast the logarithm of the precision rather than anything proportional to it.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

AttractorBifurcationChaosIntermittencyJosephson effectLogistic mapLyapunov exponentSaddle node bifurcation