The Angle That Kept Time: Two Attractors, and Whether Complexity Can Be Told Where It Lives

Two pictures and one question

Both of the objects in this chapter are strange attractors: shapes traced out over time by three coupled equations that have no random term in them anywhere, and that nonetheless never repeat. Deterministic, and aperiodic. Both are drawn from a family the volume has already used for training — a small number of rules producing motion no shorter description can summarise.

Put them side by side and one question sharpens. When a trajectory looks tangled, where is the tangle? Is complexity something a system has, distributed through all of it at once, or is it something that can be found in particular variables and, once found, subtracted from the rest? The two attractors answer that question differently, and the difference is the chapter.

The clock hand that does not wobble

The Aizawa system is a constructed example — no physical derivation stands behind it — and its horizontal equations have a particular shape. The rate of change of x carries a term in x and a term in minus y; the rate of change of y carries the same coefficients with the signs swapped. That is the signature of a rotation. Rewrite the horizontal motion in polar coordinates, keeping the radius and the angle rather than the two Cartesian positions, and the two equations come apart: the radius obeys one equation, and the angle obeys the equation that its rate of change equals the constant d.

That is worth stating plainly. The angle advances at a fixed rate. It does not speed up, slow down, hesitate or reverse. At the usual setting the hand goes round at three and a half radians per unit time and it will do so forever, no matter what the rest of the system is doing. Nothing feeds back into it.

So the visible swirl is a deception about which part is complicated. The eye reads the winding as disorder. The winding is the metronome. Everything genuinely unpredictable in that system is happening in two other quantities — how far out the orbit reaches, and how high above or below the plane it climbs. The orbit breathes and rises chaotically while the turn keeps perfect time. Complexity in this system can be told where it lives: in the radius and in the third coordinate, and not in the angle.

Convection cut down to three numbers

The Lorenz system has a different provenance. Edward Lorenz published it in 1963 in the Journal of the Atmospheric Sciences, under the title “Deterministic Nonperiodic Flow,” as a severe truncation of a fluid layer heated from below — Rayleigh–Bénard convection, cut down until only three numbers were left. The variables are not positions in a room. One tracks the intensity of the rolling motion; the other two track temperature differences across the layer. The parameters are a fluid property, a geometric ratio, and a measure of how hard the layer is being driven; at the values Lorenz used the motion never settles and never repeats.

Everything about the object is settled work. The two-lobed shape is standard. The correlation dimension is estimated at roughly two and a bit — Grassberger and Procaccia in 1983 — which says the attractor is more than a surface and less than a volume. And in 2002 Warwick Tucker supplied a computer-assisted proof, using interval arithmetic together with analytic estimates, that the Lorenz equations really do support the strange attractor the numerical pictures had been showing for forty years. That is the kind of settlement this volume asks for and rarely gets: the picture was not merely persuasive, it was eventually made to stand.

Why no angle survives

Now try on Lorenz what worked on Aizawa. It fails, and the way it fails is instructive.

The Lorenz attractor has two lobes. A trajectory winds around one of them for a while, then crosses over and winds around the other, then comes back — and the sequence of crossings is the chaos. There is no single centre to measure an angle around. Worse, the quantity that would have to play the part of the angle is not independent of anything: the rolling intensity and the temperature differences are multiplied together in the equations, so the turning is driven by the very quantities whose variation is unpredictable. Turn and tangle are the same motion.

This is not a failure of ingenuity. The standard way to read Lorenz is to stop looking for a smooth clock and record instead which lobe the orbit is on — to write the motion as a sequence of discrete choices, one per circuit. The unpredictability is in the sequence, and the sequence is not a coordinate. One takes the successive high points of one variable and asks what the next high point will be given the last; that map, not any angle, is where the system's future is decided.

So the two systems do not differ in how chaotic they are. They differ in whether the chaos can be given an address. In Aizawa the steady part detaches cleanly and can be set aside. In Lorenz there is nothing to set aside.

Separability is a property of what was written down

Here is the chapter's own claim, and it is modest. When a steady part comes away from a chaotic part, that is a fact about the particular equations someone wrote, not a fact about chaos. Aizawa's horizontal equations were built with a rotation sitting inside them, so a rotation could be lifted out. Lorenz's were derived from a fluid problem that had no such term to give, so nothing lifts out.

The honest qualification runs the other way too. Partial separation of a steady turn from everything else is a real and general technique for oscillators that are only weakly disturbed — phase reduction, in the literature, with a considerable body of work behind it. What is not general is the exact separation: an angle that advances at a constant rate and never receives a correction, across the whole attractor, as in Aizawa. That is the special case, and the chapter's claim is about that.

What the pairing changes is a verb. The volume has been asking, chapter after chapter, where complexity lives — in the phase question, in the staircase inside the ringing, in the tuned skin. This chapter says the question has a prior question underneath it: before asking where, ask whether the system admits a where at all. Some do. Some have no such coordinate, and insisting on one is the error.

The circle that keeps time, and what rides on it

One image survives out of the pair, and it is worth keeping because it costs nothing. In the separable case the circle is the part that keeps time — the clean, cheap, perpetual-looking motion — and the life of the thing is in the radial breathing and the vertical climb riding on top of it. The steady turn is not the interesting part; it is the condition under which the interesting part can be read.

That is as far as the image goes, and the next paragraph is the fence. Neither system is a model of anything in this volume. Aizawa has no physical derivation. Lorenz is a truncation of convection so severe that it is not used to predict weather and never was — its standing is as an existence proof that simple deterministic rules can defeat prediction, which is exactly why it changed the century's understanding of forecasting. Nothing here licenses a claim about fluids, membranes, phonons, or any substrate. What the two attractors supply is a discipline for reading: a single distinction, checkable in any system anyone writes down, about whether a complication has an address or only a history.