One constant for each shape of top

A constant that does not care which map measured the gaps between successive period doublings in the logistic map and in the sine map and found them shrinking by the same ratio, 4.669, in both. A flow that doubles like a parabola found the same ratio emerging, more slowly, from three differential equations, because the flow’s peaks lie on a rounded hump. The claim is that any smooth map with a single hump does this, whatever its formula — and the earlier essay was careful to add the one condition that turns out to carry the weight. The hump must be rounded like a parabola: near its top the map must look like $1 - c\,x^2$. A map whose top looks like $1 - c\,x^4$ was said, in a single sentence, to have $\delta = 7.285$ instead.

That sentence is the subject here. If the shape of the top matters, then there is not one Feigenbaum constant but a family of them, one for each shape, and the family can be measured. The tool is the simplest map with a top of any shape: $f(x) = 1 - \mu |x|^z$, where $z$, the order of the maximum, says how the top is shaped. At $z = 2$ this is the logistic map in different coordinates. At $z = 4$ the top is flat to third order. At $z$ just above 1 it is nearly a corner. The order need not be a whole number, and every value from 1.1 to 16 tells a cascade of its own.

All three panels show the same sequence of events: one fixed point, a fork into two, four, eight, and a pile-up at a finite parameter beyond which the long-run behaviour is chaotic, with windows. The difference is in the spacing. Each panel is rescaled so that the period-2 marker and the accumulation point coincide, and in those common coordinates the flat top’s doublings bunch up against the end much more tightly than the sharp top’s. That is a difference in $\delta$, and it is large: 3.26, 4.67 and 9.30. The road paved with doublings first watched a cascade crowd towards its end; the panels here show that how hard it crowds is a property of the top of the map.

The tops, and nothing else #

The four maps below are drawn at the end of their own cascades, at the parameter where the doublings accumulate.

Away from $x = 0$ the four curves are the same kind of object, rising to a peak of 1 and falling towards $-1$ at the ends, and nothing about their overall shape suggests they should behave differently. The difference is concentrated at the top. The sharp map comes almost to a point; the flat map is very nearly level for half the interval. And the accumulation parameters themselves differ — 1.2095 for the sharp top, 1.4012 for the round one, 1.5949 for $z = 4$ and 1.7991 for $z = 12$ — because a flatter top has to be pulled down further before its orbits are disordered.

Why only the top should matter is the content of the renormalisation argument, which the earlier essay set out for the quadratic case. At a parameter of period $2^k$, the map applied twice, restricted to a small interval round the top, looks like a smaller, upside-down copy of the map itself. Repeat the doubling and rescale, and after many steps everything about the original map has been shrunk away except its behaviour in an ever smaller neighbourhood of its maximum. What survives is the shape of the top. Two maps with the same order of maximum are carried by repeated rescaling to the same limiting function, and their cascades end up governed by the same constants. Two maps with different orders are carried to different limiting functions, and the constants are those of their limits.

Six orders, six limits #

The constant $\delta$ is defined as a limit, and the right thing to check first is that each order has one.

Each estimate uses the superstable parameters, the values $\mu_k$ at which the top of the map, $x = 0$, lies on a cycle of period $2^k$. They are the easiest parameters in a cascade to locate exactly, because at them the condition is a single equation, $f^{2^k}(0) = 0$, and they interleave with the doublings themselves, so their gaps shrink by the same ratio. The slope of the cycle is zero there — the top of the map is on it — which is what how fast the staircase arrives called a superattracting point, where convergence doubles its correct digits every step. For $z = 2$, $\mu_1 = 1$ exactly, and each later $\mu_k$ was found as the first root above $\mu_{k-1}$, in a window predicted from the gaps so far.

The ratio of successive gaps is $\delta_k = (\mu_{k-1} - \mu_{k-2})/(\mu_k - \mu_{k-1})$. For the round top it runs 4.386, 4.601, 4.655, 4.666, 4.669 and then holds at 4.669 to the last place that double precision can resolve; this is Feigenbaum’s number, recovered from scratch. For $z = 4$ it settles at 7.2847 within four doublings — the value the earlier essay quoted, now measured. For $z = 1.5$ it settles at 3.8006, for $z = 3$ at 6.0847, for $z = 6$ at 9.2963 and for $z = 10$ at 12.341. The six curves do not approach one number at different speeds. They approach six numbers.

The figure also shows where the computation stops. Each doubling shrinks the gap between parameters by a factor $\delta$, and once the gap falls below about $10^{-15}$ of the parameter the arithmetic cannot separate the two roots. The round top gets fifteen doublings before that happens. The flat tops, with larger $\delta$, get fewer, and their estimates are correspondingly less precise: four or five significant figures at $z = 10$, against seven at $z = 2$.

The constant as a curve #

With the constant measured for each order, it can be drawn as a function of the order.

The curve rises steadily: 2.831 at $z = 1.1$, 3.801 at $z = 1.5$, 4.669 at $z = 2$, 6.085 at $z = 3$, 7.285 at $z = 4$, 9.296 at $z = 6$, 10.949 at $z = 8$ and 15.49 at $z = 16$. There is nothing special about whole-number orders. The map with $z = 2.5$ has $\delta = 5.413$, and the one with $z = 1.75$ has $\delta = 4.257$; the constant is defined as readily at a fractional order as at a whole one, and it varies smoothly between them.

So “universal” means something more precise than “the same everywhere”. Feigenbaum’s constant is universal within a class, and the class is fixed by one number, the order of the maximum. Every smooth map with a quadratic top has $\delta = 4.669$ — the logistic map, the sine map, the return map of the Rössler flow — and every map with a quartic top has $\delta = 7.285$, whatever else is true of it. The classes form a continuum, and the familiar constant is the one value on it that nature supplies most often.

The class is also inherited inside a single map. Every window is the whole diagram again found the period-three window of the logistic map cascading with 4.669, because the third iterate near its top is again a one-humped map with a quadratic top. The same argument works at any order: an iterate of $x \mapsto 1 - \mu|x|^z$ near the top of its hump has a top of the same order $z$, since composing with smooth monotone pieces cannot change how the function leaves its maximum. So every window of a quartic map doubles at 7.285, and every window of the map with $z = 1.5$ at 3.801. The order is a property of the map, carried into every window it has.

There is a reason the quadratic class is the most common. A smooth function’s maximum is quadratic unless its second derivative happens to vanish there, and a second derivative that vanishes is a coincidence that a single parameter cannot be expected to produce. To find a quartic top in a physical system, a second parameter has to be tuned until the curvature at the maximum disappears; at that special point the cascade switches to the quartic class, with its 7.285. This is why experiments on dripping taps, electronic circuits and convecting fluids all report numbers near 4.669. They are not being lucky. The quadratic class is where a one-parameter family of smooth systems almost always is.

How fast the cycle closes in #

The second Feigenbaum constant, $\alpha$, measures the cascade in space rather than in parameter. At the superstable parameter of period $2^k$, the top of the map is a point of the cycle, and the nearest other point of the cycle is a distance $d_k$ away. Each doubling shrinks that distance by a factor that settles at $\alpha$.

On a logarithmic scale each order gives a straight line, and the slopes differ. For the round top the distance shrinks by 2.503 per doubling, Feigenbaum’s second constant. For the sharp top with $z = 1.5$ it shrinks by 3.389, and for the flat top with $z = 6$ by only 1.468. The cycle crowds a sharp top hard and a flat top gently.

The direction makes sense once it is clear what the cycle is doing near the top. A superstable cycle passes through the maximum, and the points of the cycle near it are sent, one step later, to near the map’s highest value. A flat top sends a whole neighbourhood of points to almost exactly the same place, so a point of the cycle can sit relatively far from the top and still land where the self-similar structure needs it; a sharp top spreads its neighbourhood out, and only points very close to it land close enough. Doubling the period asks for the next level of that structure, and each level has to sit closer to the top by a factor set by how strongly the top spreads its neighbourhood — which is the order of the maximum.

Two constants, opposite ways #

The same twenty cascades give $\alpha$ as a function of the order.

Where $\delta$ rises with the order, $\alpha$ falls: 7.967 at $z = 1.1$, 3.389 at $z = 1.5$, 2.503 at $z = 2$, 1.690 at $z = 4$ and 1.189 at $z = 16$, levelling towards 1 as the top flattens. At the sharp end $\alpha$ grows quickly — the cycle has to close in very fast on a near-corner — while $\delta$ comes down towards small values. The two constants move in opposite directions, and the pair, not either alone, is what fingerprints a class: a measured cascade with $\delta = 7.28$ and $\alpha = 1.69$ is a quartic top, and one with $\delta = 4.67$ and $\alpha = 2.50$ is a quadratic one.

The constant $\alpha$ also decides the geometry of what is left at the accumulation point. The attractor there is a Cantor set, built by repeatedly keeping two scaled copies of the previous level, and a dimension that is not a whole number measured such sets by their scaling factors. For the quadratic class the dimension is 0.538. The two copies are not scaled alike — one sits near the top, where the map compresses by roughly $\alpha^z$, and one near the highest value, where it compresses by $\alpha$ — so the dimension depends on the order through both, and each class has its own.

A relation the numbers suggest #

The figure’s last line records something the measurements show and no argument here proves. For flat tops, $\delta$ is very nearly a power of $\alpha$, and the power is very nearly the order: $\log\delta/\log\alpha$ is 7.82 at $z = 8$, 11.82 at $z = 12$ and 15.81 at $z = 16$, the order less about 0.18 each time. At the round top the same ratio is 1.68, nowhere near 2, and at $z = 1.1$ it is 0.50.

A rough reason can be given for the flat end. The point of a superstable cycle nearest the top sits a distance $d$ from it, and one step later it lands a distance $\mu d^z$ below the map’s highest value, 1. So near the highest value the cycle shrinks by $\alpha^z$ per doubling, not by $\alpha$. The parameter $\mu$ multiplies exactly that quantity, $|x|^z$, and when the top is flat almost the whole map is plateau, so the cycle’s sensitivity to $\mu$ is concentrated where the plateau’s edge meets the highest value. If the parameter gaps simply tracked the shrinking there, $\delta$ would equal $\alpha^z$. They come close for large orders and not for small ones, where the rest of the cycle still matters. That is a heuristic, not a derivation, and the gap of 0.18 in the exponent is a number the heuristic does not explain.

What happens at the ends #

The two ends of the range are where the cascade is under most strain.

As the order falls towards 1, the map approaches $1 - \mu|x|$, which is the tent map in different coordinates: a corner instead of a top. The tent family has no cascade at all. For slopes below 1 its fixed point attracts everything, and for slopes above 1 every orbit is pulled apart at a constant rate and no stable cycle exists. The maps with orders just above 1 still double — the cascade survives as long as there is any rounding at the top — but the whole cascade is squeezed into a shrinking interval of parameters just above $\mu = 1$: at $z = 1.25$ the accumulation point is 1.2095, at $z = 1.1$ it is 1.1255. And the constants show the strain. The estimates near $z = 1.1$ settle more slowly than those at $z = 2$, and $\alpha$ climbs steeply, which is the cycle having to close in faster and faster on a top that is becoming a corner.

As the order grows, the top becomes a long flat plateau with steep sides, and the map approaches a step: everything near the middle goes to 1, everything near the ends to nearly $-1$. The cascade still happens, at parameters approaching 2, but $\delta$ grows and the number of doublings arithmetic can resolve shrinks; at $z = 16$ the estimate is good to about three figures, 15.49, and still rising. Numerical studies using the renormalisation equation directly, rather than counting doublings, suggest that $\delta$ approaches a finite limit, around thirty, as the order grows without bound, and that $\alpha$ approaches 1.

Still open: what has been proved for which orders #

For the quadratic order, universality is a theorem. Oscar Lanford proved in 1982, with a computer-assisted estimate, that the renormalisation operator has a fixed point and that its one expanding eigenvalue is 4.669…; Dennis Sullivan, Curtis McMullen and Mikhail Lyubich, over the following two decades, replaced the computation by a conceptual proof that every real-analytic one-humped map with a quadratic top is drawn to that fixed point. Much of the argument carries over to maxima of higher even whole-number order, where the top is still a power of $x$ that is smooth through zero.

For orders that are not even whole numbers the picture is less complete. Fixed points of renormalisation exist for every order greater than one — Marco Martens proved this in 1998 — and the numbers in this essay are what the cascades give. But the full statement that makes $\delta(z)$ a universal constant, that the renormalisation is hyperbolic at its fixed point with exactly one expanding direction so that every map of order $z$ is drawn to it at the rate $\delta(z)$, has not been established for general real orders. The smoothness of the curve in the figures, the limits of $\delta$ and $\alpha$ as the order goes to 1 or to infinity, and whether those limits are themselves given by some simpler equation, are known numerically and are not theorems.

The top is the class #

The classical constant is one point on a curve. A smooth map with a single maximum doubles its way into chaos at a rate fixed entirely by how its top is shaped, and the twenty cascades here measure that dependence across the whole range: a sharp top crowds its doublings slowly and its cycles fast, a flat top the reverse, and the round top of every generic smooth system sits in between at 4.669 and 2.503. Universality is real, and it is exactly as wide as the order of the maximum allows. Change that one number and every constant changes with it; change anything else and nothing does.