A flow that doubles like a parabola

A constant that does not care which map made a claim that sounds too strong to be true. Any family of maps of one variable with a smooth single hump passes from order to chaos through the same sequence of period doublings, and the gaps between doublings shrink by the same ratio, Feigenbaum’s 4.669 — the logistic map, the sine map, any of them. That essay told the story of Albert Libchaber’s liquid helium, which doubled its period with roughly the right ratio, and explained why: a physical system with many degrees of freedom, most of them damped, can behave like a map of one variable with a smooth maximum, and then it inherits the map’s constants.

That explanation was an argument. This essay makes it concrete in one of the smallest systems where it can be watched happening: a flow in three dimensions, defined by three differential equations, whose orbits are smooth curves in space and not points jumping on a line. The flow has nothing obviously to do with the logistic map, and it reproduces the logistic map’s entire picture. The way it does so — by quietly being a map of one variable — can be drawn.

Three equations and one product #

In 1976 Otto Rössler set out to write down the simplest flow he could find that behaves chaotically, simpler than the weather model Edward Lorenz had published in 1963. His equations are

$$x' = -y - z, \qquad y' = x + a\,y, \qquad z' = b + z\,(x - c).$$

Two of them are linear. The only nonlinear term is the product $z x$ in the third. With $a = b = 0.2$, everything depends on one number, $c$.

The mechanism is visible in the equations. The first two, ignoring $z$, make $x$ and $y$ rotate about the origin and spiral slowly outwards, because $a$ is positive. As long as $x$ is less than $c$, the third equation keeps $z$ small. When the spiral carries $x$ past $c$, the term $z(x - c)$ turns positive and $z$ shoots up; a large $z$ then pulls $x$ down through the first equation, throwing the orbit back towards the centre, where $z$ collapses and the spiral starts again. For these settings the fold always brings the orbit back, so in every computation it stays in a bounded region of space — the behaviour that a region the orbit cannot leave proved for Lorenz’s system and that, for Rössler’s, rests on the computations. Stretch outwards, lift, fold back in: the motion is the stretching and folding that makes a horseshoe, done continuously.

The hero figure records one number per loop: each time $x$ reaches a peak, its value. At $c = 2.5$ every peak is the same — one loop, repeating. At $c = 3$ the peaks alternate between two values: the orbit makes two loops of different sizes before repeating. By $c = 4$ there are four values, then eight, sixteen, and by $c \approx 4.2$ the peaks fill whole bands and no longer repeat at all. Beyond that, the bands widen, and narrow white stripes appear where the motion becomes regular again for a while. It is the picture of the road paved with doublings, drawn by a flow.

The orbit, loop by loop #

Projected onto the plane of $x$ and $y$, the orbits show the doubling in the flow’s own terms. At $c = 2.5$ the orbit is a single closed loop. At $c = 3.5$ it is a loop that goes round twice, once wide and once narrow, before closing; at $c = 4$, four times, each loop slightly different; at $c = 5.7$ it never closes, and the loops fill a band. A period doubling in a flow is this: a closed orbit that, as the knob turns, splits into one that goes round twice before closing, the two loops starting identical and drifting apart.

Nothing in the equations says “two” or “four”. The flow is smooth and continuous, defined at every instant, and its closed orbits can have any length. That its closed orbits arrive in a doubling sequence, with the same ratios as an iterated parabola, is what needs explaining.

Peaks against peaks #

The explanation is to look at the flow the way Lorenz looked at his own system in 1963 — the move the flow that is really a map made for his equations: record each peak, and plot it against the one before.

If the flow were genuinely three-dimensional in its long-run behaviour, a peak would not determine the next one: two orbits passing a peak at the same height but in different states of $y$ and $z$ would go on to different next peaks, and the plot would be a cloud. It is not a cloud. The 2,999 points lie on a thin curve, rising from the left, reaching a rounded top above a previous peak of about 7.69, and falling to the right — a single hump, fitted near its top by a downward parabola. Each peak determines the next, to the width of the line: the return map of the flow is a function of one variable, and it has exactly the shape of the logistic map’s parabola.

The reason is the flow’s contraction. The attractor of a flow has no volume, because the flow shrinks volumes at a steady rate; for Rössler’s system the shrinking is so strong in one direction that the attractor is very nearly a surface, a band that the flow stretches, lifts and folds back onto itself. A peak of $x$ is a crossing of a line drawn across that band, and the band is so thin that its position along the line says everything. The three dimensions have collapsed, as far as the long run is concerned, to one.

Where a doubling happens on the hump #

The return map turns the question of when the flow doubles into a question about one curve. A closed orbit of one loop has every peak the same, so on the return map it is a point where the hump crosses the diagonal: a peak that returns itself. Whether that orbit is stable depends on the slope of the hump at the crossing. If the slope lies between −1 and 1, a peak slightly off the crossing returns slightly closer to it, and the single loop attracts its neighbours. When the slope passes through −1, a peak slightly above the crossing returns slightly below it by the same amount, and slightly above again after that: the orbit begins to alternate, and the single loop gives way to a pair of loops of different sizes.

That is the period-doubling at $c = 2.83$, read off a curve. The crossing sits on the falling side of the hump, to the right of its top, and as $c$ rises the hump steepens there until its slope reaches −1. The same reasoning applied to two steps of the return map instead of one gives the next doubling, and so on down the cascade: each doubling is the moment some iterate of the hump becomes too steep where it meets the diagonal. No part of that argument mentions three dimensions, a spiral or a product $zx$. It is the logistic map’s argument, made about a different hump.

The fitted parabola near the top has coefficient −0.556 in units of the peak heights: a definite, nonzero curvature, which is what “rounded” means. If the coefficient were zero the top would be flatter than a parabola, and if the hump came to a point there would be no parabola to fit at all. Either would put the flow in a different class.

The return map is also why the flow is sensitive to its starting point at $c = 5.7$ and not at $c = 3.5$. On the steep sides of the hump, two nearby peaks lead to next peaks further apart; iterated, the gap grows until it is as large as the hump. At $c = 3.5$ the orbit lives at two points where the combined slope over two steps is shallower than one, and gaps shrink instead. Chaos and order are the same curve, visited in different places.

A hump that rises with the knob #

Since the return map has the logistic map’s shape, turning $c$ should act on it as turning $r$ acts on the logistic map.

It does. At $c = 3.5$ the orbit visits only two points of the return map, swapping between them; at $c = 4$, four. At $c = 3.5$ the peaks run from 4.97 to 7.07. At $c = 4.6$ the chaotic orbit traces out a hump whose peaks reach 9.3, and at $c = 5.7$ the hump has grown taller and steeper, with peaks up to 11.4. Raising $c$ raises and steepens the hump, exactly as raising $r$ raises the logistic parabola $rx(1-x)$; the periodic orbits are the places where iterating the hump comes back on itself; and the sequence in which they appear is the sequence the hump dictates. Everything the dark lines in the logistic diagram said about the orbit of the top of the parabola applies, with the top of this hump in its place.

The match is not exact — the return map is not literally a parabola, it is a smooth hump that is approximately one near its top and differs from one in its tails — and that is why the universality theorem is needed rather than an identity. The theorem says that the details of the hump do not matter, only that its top is smooth and rounded, and the fitted parabola in the return-map figure is the evidence that this one qualifies.

The ratios, measured #

The period doubles at $c = 2.8286$, $3.8347$, $4.1230$, $4.1865$ and $4.2004$ — from one peak to two, two to four, and so on up to thirty-two. The gaps between them shrink: 1.006, then 0.288, then 0.0635, then 0.0138. The ratios of consecutive gaps are 3.49, 4.54 and 4.59. For the logistic map the same ratios are 4.75, 4.66 and 4.67, already close to the limit 4.6692. The flow approaches the same constant, more slowly and from below.

The slower approach is the price of the flow being only approximately a map with a single hump at the first doublings. Feigenbaum’s constant describes the limit of infinitely many doublings, where only a vanishingly small neighbourhood of the hump’s top matters; the first gaps of any particular system carry corrections that die away as the doublings shrink into that neighbourhood. The logistic map, being itself a parabola, starts almost at the limit; the flow starts further off and closes in. The measurement stops at thirty-two because the next doubling, a gap of about 0.003 in $c$, sits inside the precision with which the flow’s long-run period can be decided by integrating it.

A window that holds the whole diagram #

The white stripes in the hero figure are windows of periodic behaviour inside the chaos, and the widest of them begins near $c = 5.19$ with an orbit of three peaks. As $c$ rises the three-peak orbit doubles to six near $c = 5.38$, to twelve near 5.47, and dissolves back into chaos by about 5.5. That is a copy of the whole cascade, with periods multiplied by three, compressed into an interval a few tenths wide — exactly the structure every window is the whole diagram again found in the logistic map. Elsewhere in the chaotic band there are narrower windows — one of period six near $c = 4.385$, one of period five near 4.695 — each with its own cascade.

Which windows appear, and in which order along $c$, is constrained for every one-humped map by a universal ordering of the patterns in which the top of the hump can return to itself — the theory windows counted like necklaces used to count them. That a flow obeys it is a consequence of the same collapse onto a hump. The period-3 window has a special place in that ordering: once a one-humped map has an orbit of period three, it has orbits of every period, a theorem of Sharkovskii published in 1964 and rediscovered by Li and Yorke in 1975 as “period three implies chaos”. The Rössler flow’s period-3 window is that theorem made visible in a system of differential equations.

What the reduction does and does not prove #

The figures establish an empirical fact: Rössler’s flow behaves, peak to peak, like a one-humped map, and therefore shows the universal features of such maps to the precision measured. They do not prove that it must. Proving that a particular flow has a return map that is exactly a smooth one-humped function is hard, because the attractor is not exactly a surface — it has a thin layered structure, a fractal thickness — and the return map is only approximately a curve. For the Lorenz system, the corresponding statements waited until 1999, when Warwick Tucker proved by a computer-assisted argument that the Lorenz attractor really exists with the structure the pictures suggest.

The comparison with Lorenz’s own system is instructive. Lorenz plotted consecutive peaks of his variable $z$ in 1963 and found not a rounded hump but a sharp cusp — a tent with a corner at the top. A map with a corner at its maximum is in a different class from one with a rounded top: it does not period-double towards chaos in the same way, and Feigenbaum’s constant does not apply to it. So two flows with three variables and one or two products can belong to different universality classes, decided by the shape of the top of a curve that no equation mentions. The Rössler flow happens to have a rounded top, and that is why it inherited the logistic map’s numbers.

Why a smooth top is the whole condition #

The rounded top matters because of what iterating a hump near its top does. Near a smooth maximum, any one-humped function looks like a parabola; two steps of it, rescaled, look like a parabola again, with a definite rescaling factor. Repeating the rescaling converges to a fixed shape, and the doubling ratios are properties of that fixed shape, not of the original hump. That is the renormalisation argument that the essay on the constant described as a fixed point in a space of functions. A corner instead of a rounded top changes the local shape, the rescaling converges somewhere else, or nowhere, and the constants change.

So the content of “universality” for a physical system is a single geometric question: after the fast directions have collapsed, does the slow dynamics reduce to a one-humped map with a quadratic top? For Rössler’s flow the return-map figure answers yes, and the doubling ratios follow. For Libchaber’s helium the answer was inferred from the ratios themselves. For any new system the return map is the thing to compute first.

What the pictures cannot show #

The bifurcation diagrams show peaks after a fixed settling time at finitely many values of $c$. Near a doubling the flow settles slowly, and a short settling time blurs the split; near the edge of chaos the long-run period can exceed anything the run sees. The figures therefore show the cascade only down to the scale the integration resolves, and the claim that it continues infinitely, accumulating at a definite $c_\infty$ near 4.2, is the universality theory’s prediction, consistent with the five measured doublings but not established by them.

The return map, likewise, looks like a curve to the resolution of a picture, and is known to have a fine layered structure beneath that resolution. The fitted parabola describes the visible top; whether the exact return map is smooth there, in the sense the universality theorem needs, is not something a picture can decide.

Still open: proofs for flows #

For the Rössler flow at its standard parameters, it has not been proved that the attractor is chaotic in the strict sense, or that its period-doubling cascade has Feigenbaum’s ratios in the limit. Computer-assisted proofs have established chaotic behaviour for some parameter values, using rigorous interval arithmetic to bound every step of the integration, and rigorous enclosures of individual periodic orbits exist; the full cascade, with its universal ratios, is supported by computations like these and by the theory for one-dimensional maps, not by a theorem about the flow.

More generally, there is no theorem saying which flows in three dimensions reduce to one-humped maps, or for which parameters. The reduction depends on strong contraction transverse to the attractor, and the conditions under which a flow’s long-run dynamics can be proved to be governed by a one-dimensional map — so that universality theorems for maps apply to it — are understood for special classes of systems and not in general. The flows that people write down to describe chemical oscillators, lasers and electrical circuits show the logistic cascade again and again, and in almost every case the evidence is the kind this essay presents: the return map, drawn.