Illustrated Physics
physics, drawn
Concept

Lyapunov exponent — where it appears

The rate at which two neighbouring trajectories of a system separate, when the separation grows exponentially. A positive value guarantees that prediction degrades at a fixed rate — a double pendulum's is about 2 per second — and guarantees nothing at all about whether the motion is statistically random.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

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.

mechanics · Chaos Phase portraits of the standard map at 2 couplings. The standard map p → p + K sin θ, θ → θ + p, iterated 220 times from 12 starting points, at couplings of 0.6 and 1.3. Both coordinates run from 0 to 2π. An orbit that lies on a curve spanning the picture from left to right is an invariant circle, and nothing can cross it; an orbit that fills an area is chaotic; an orbit that circulates round a centre is trapped in a resonance island. At K = 0.6, orbits launched on p = 0 get no further than 1.55 in p, so a spanning curve is still there. At K = 1.3, orbits launched on p = 0 get no further than 10.78 in p, so a spanning curve is gone and transport is global. The point of the pair is that the change between them is not a change of character in any single orbit — chaotic orbits and regular ones coexist on both sides — but the loss of the barriers that kept the chaotic ones local.

The last curve to go

Chaos does not arrive all at once. Order is destroyed a resonance at a time, and there is a coupling — 0.971635, known to six figures — at which the final barrier separating one part of the phase space from another gives way. Below it a chaotic orbit is still trapped; above it nothing stops it.

mechanics · Chaos The set an orbit that never repeats settles onto. 24,000 successive positions of one orbit of the map x' = 1 − 1.4x² + y, y' = 0.3x, after five hundred steps of transient have been discarded. Nearby points separate at e^0.4188 per step, so the orbit is unpredictable in the way a doubling error measures; and every one of the 24,000 points lies inside a box 2.558 by 0.767, a diagonal of 2.670, so it is going nowhere. Those two statements are not compatible with a smooth stretching: something has to bring the separated points back, and the bringing back is the visible fold at the left-hand end. The curve is not a curve. Every strand of it is a bundle of strands at any magnification, which is what an area contraction of 0.3 per step leaves behind when the stretching along the other direction is e^0.419. The 24,000 points paint 7,352 distinct marks at the resolution this is drawn at, which is itself a measurement of how little of the plane the set occupies.

The fold that has to be there

Two trajectories that separate exponentially, in a region they can never leave, are being asked to do two incompatible things. The resolution is that the motion is folded back on itself over and over, and the object that survives infinitely many foldings is neither a curve nor a patch of surface — it has a dimension between the two, and the number can be measured two entirely different ways.

mechanics · Chaos 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.

mechanics · Chaos 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.

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.

mechanics · Chaos The noisy orbit a true orbit stays beside. A computation of the doubling map with an error of up to 0.001 added at every one of 400 steps, and two exact orbits compared with it. The exact orbit started from the computation's own first point parts from it at once, by more than 0.1 after 8 steps, because each error is doubled at every step after it. The shadow — the exact orbit built by running the map backwards from the computation's last point, where each error is halved instead — stays within 8.0 × 10⁻⁴ of the computation at every one of the 400 steps, closer than the noise added at any single step. Every point of the noisy computation is therefore a faithful picture of a real orbit of the map — just not of the one that started where the computation did.

The wrong orbit beside a right one

Every computer simulation of a chaotic system is wrong within a few dozen steps: rounding errors are stretched exponentially, and after fifty iterations of the simplest chaotic map two computations of the same equation disagree completely. Yet chaotic simulations are trusted everywhere, and for a reason that is a theorem rather than a hope. For a large class of chaotic systems the computed sequence, though it follows no true orbit from where it started, stays close to some other true orbit for as long as it runs.

mechanics · Chaos A bounce that turns into a swing and back. The share of a swinging spring's oscillation energy that is in its sideways swing rather than its up-and-down bounce, against time in units of the swing's period over 2π, for a spring started bouncing with a stretch of 0.08 of its length and almost no tilt (0.02 rad). When the bounce frequency is exactly twice the swing frequency (red), the energy drains out of the bounce into the swing until 100 per cent of it is swinging, then drains back, over and over. At a ratio of 2.4 (blue) the swing never takes more than 1.5 per cent. The bounce pumps the swing parametrically, twice per swing, which works only when the two keep step.

The spring that swings when it was only bounced

Hang a weight on a spring and pull it straight down: it bounces straight up and down. Unless its bounce happens to be twice as fast as its swing would be as a pendulum. Then the bounce dies away and the weight starts to swing, wider and wider, until the bounce has gone completely — and then the swing dies and the bounce comes back. Two motions that look independent are trading their energy through a resonance, regularly and for ever at small amplitude. Give the spring more energy and the trading becomes chaotic: the same simple rule, with nothing random in it, produces motion that never repeats and that a hundred-millionth of a radian can redirect.

mechanics · Chaos

Named alongside it

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

ChaosAttractorPhase spaceBifurcationDeterminismNonlinearityPredictabilityBox countingDiffusionDissipationErgodicityExponential sensitivity

All concepts