Chaos & strange attractors

Imagine a population of insects that breed once a year. Call x the population as a fraction of the most the habitat can hold, so x runs from 0 to 1. Next year's population depends on this year's in two ways. More insects lay more eggs, so it grows with x. But a crowded habitat starves them, so it shrinks with 1 − x, the room left. Put the two together with a growth rate r and you have the logistic map.

In 1976 the ecologist Robert May published a review in Nature called "Simple mathematical models with very complicated dynamics", built around this map. His point was the title. Ecologists, economists and physicists all used equations like this one, and most assumed simple equations gave simple behaviour. May showed that as you turn up r, this one goes through every kind of behaviour there is, and he urged that it be taught early, so people would stop being surprised by it.

Here is what happens. Below r = 1 the population dies out: x shrinks toward 0 every year. Between 1 and 3 it settles to a steady value, x = 1 − 1/r, approached either smoothly or with a few dying wobbles. At r = 3 the steady state breaks and the population starts alternating between a high year and a low year, forever. At about 3.449 each of those values splits again and the cycle takes four years. Then eight, sixteen, thirty-two, faster and faster, until at about r = 3.5699 the cycle length becomes infinite: the population never repeats. That is chaos. Beyond it, out to r = 4, chaos is mixed with islands of order, the "windows", where a short cycle suddenly reappears.

You can iterate a map with a pencil and no arithmetic at all. Draw the curve y = f(x); for the logistic map it is an upside-down parabola peaking at x = ½. Draw the diagonal line y = x. Start at your x₀ on the horizontal axis and go straight up to the curve: the height there is x₁. Now go sideways to the diagonal, which turns that height back into a position on the x axis, then up or down to the curve again for x₂. Repeat. The zigzag path is called a cobweb.

Where the curve crosses the diagonal, f(x) = x, and the map stands still: a fixed point. Whether the cobweb spirals in toward it or away from it depends only on how steep the curve is there. If the slope is shallower than 45° either way, each step shrinks the distance and the point attracts. If it is steeper, each step grows the distance and the point repels. A negative slope makes the cobweb spiral, overshooting from side to side, and that is exactly what happens at r = 3.

Plot every value of r across the page. For each one, iterate the map a few hundred times to let it settle, then mark the next few hundred values of x above that r. Where the population is steady you get one dot. A 2-cycle gives two dots, a 4-cycle four. Chaos gives a smear. The result is the bifurcation diagram, one of the most reproduced pictures in science, and a fractal in its own right.

The single line forks at r = 3, each branch forks again at 3.449, and the forks come closer and closer together until they pile up at 3.5699. Past that point the smear has structure: bright curves where x lands most often, which are the images of the parabola's peak, and pale vertical gaps that are the windows. Zoom into one branch of the cascade, or into the middle of a window, and the whole diagram appears again, smaller and squashed but recognizably the same. That repeated self-similarity is what the instrument below lets you check for yourself.

In the mid-1970s Mitchell Feigenbaum, a physicist at Los Alamos, was computing the r values where the logistic map's cycle doubles. Call them r₁ = 3, r₂ ≈ 3.449490, r₃ ≈ 3.544090, r₄ ≈ 3.564407, and so on. The gaps between them shrink, and he noticed they shrink by almost the same factor each time. The ratio of one gap to the next settles toward a single number, about 4.669.

That alone would be a curiosity. The surprise came when Feigenbaum tried a different map, x → r·sin(πx), and got the same 4.669. It wasn't a property of the parabola at all. Any smooth map with a single hump whose top is quadratic, rounded like a parabola rather than pointed or flat, doubles its way into chaos with the same ratio. He published the result, with the theory of why, in 1978; Pierre Coullet and Charles Tresser found the same universality independently at about the same time.

There is a second constant for the other direction. At each doubling the forks in the diagram are narrower than the forks before, again by a fixed factor in the limit: α ≈ 2.502907875. (Some authors write it as negative, −2.5029, because each smaller copy is also flipped upside down.) δ says how the cascade shrinks along r; α says how it shrinks along x. Together they say that the region near the onset of chaos is self-similar: zoom in by δ one way and α the other, and the picture repeats.

Universality made the prediction testable in the real world, where nobody knows the exact equations. Around 1980 Albert Libchaber and Jean Maurer saw period doubling in convection rolls of liquid helium heated from below, with spacings close to Feigenbaum's ratio. Similar cascades turned up in dripping faucets, electronic circuits and lasers. Libchaber and Feigenbaum shared the 1986 Wolf Prize in Physics.

Scan the chaotic part of the diagram and you'll see pale vertical stripes. The widest one opens at r = 1 + √8 ≈ 3.8284. Inside it the smear collapses to just three values: the population runs in a steady 3-year cycle, right in the middle of chaos. Just to its left, the map behaves strangely in a different way. For long stretches it looks almost periodic, cycling close to three values, then bursts into chaos, then settles back. That pattern is called intermittency; it is the ghost of the 3-cycle that is about to be born.

The window then repeats the whole story in miniature. Each of the three branches doubles: period 6 near r ≈ 3.8415, period 12 near 3.8477, and so on, with Feigenbaum's ratio again in the limit, until chaos returns around r ≈ 3.8495. The window closes near r ≈ 3.857, where the three chaotic bands suddenly merge back into one. Zoom into the middle branch and you see a small, upside-down copy of the entire bifurcation diagram. There are infinitely many windows, one for every period, most of them far too narrow to see.

Period 3 has a special place in the theory. In 1975 Tien-Yien Li and James Yorke proved that if a continuous map of an interval has a point of period 3, it has points of every period, and an uncountable set of starting points that never settle into any cycle. Their paper was called "Period three implies chaos", and it is where the word "chaos" got its mathematical meaning. It later emerged that Oleksandr Sharkovsky had proved a stronger ordering of periods in 1964, in a paper little known in the West: 3 comes first, so period 3 forces every other period.

The defining property of chaos is that two starts which are almost the same end up completely different. Take r = 4 and two starting values that differ in the tenth decimal place. After ten steps they differ in about the seventh; after thirty-three steps the difference is the size of the whole interval, and knowing one tells you nothing about the other.

The rate of that separation is the Lyapunov exponent, written λ. If nearby orbits separate like e^λn after n steps, λ is the average of the logarithm of how much each step stretches. For a one-dimensional map that stretch is just the slope of the curve at each point the orbit visits.

At r = 4 the exponent is exactly ln 2 ≈ 0.693. Each step doubles the error on average, so each step loses one binary digit of what you knew about the start. A double-precision number holds about 53 bits, so a computer simulation of the logistic map at r = 4 has forgotten its starting value entirely after about 53 steps, and from then on it is following some other true orbit, not the one you asked for. (A result called shadowing says that, for many chaotic systems, there is always a true orbit that stays close to the computed one, which is why the statistics of such simulations can still be trusted even though the individual path cannot.)

Edward Lorenz, a meteorologist at MIT, was running a small weather model on a Royal McBee computer in the early 1960s. As he told the story later, he wanted to look at one run again, so he typed in numbers from a printout and restarted it in the middle. The printout showed three decimals, 0.506, where the machine had kept six, 0.506127. The new run tracked the old one for a while, then drifted, and before long had nothing to do with it. A rounding error of one part in ten thousand had grown to the size of the weather.

In 1963 he published "Deterministic Nonperiodic Flow", built on a stripped-down model of convection, fluid heated from below, derived from a model by Barry Saltzman. It has three variables: x measures how fast the convection rolls turn, y the temperature difference between rising and falling fluid, z how far the vertical temperature profile is from a straight line. Lorenz chose σ = 10, ρ = 28 and β = 8/3.

Plot the solution in three dimensions and it traces two lopsided loops, like the wings of a butterfly. It circles one wing a few times, swaps to the other, circles that a few times, and swaps back, with no pattern to the number of turns. It never crosses its own path, because in a three-dimensional flow, a point's position fixes its whole future. And it never leaves this shape: start anywhere and the trajectory is pulled onto it. The shape is called the Lorenz attractor.

The two wings circle two unstable steady states, at x = y = ±√72 ≈ ±8.49, z = 27, where convection would turn steadily one way or the other. For ρ below about 24.74 those steady states are stable and the solution spirals into one of them. Above it they repel, and the solution is left wandering between them forever. The famous image of a butterfly flapping its wings in Brazil comes from the title of a talk Lorenz gave in 1972, "Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"

An attractor is where a system ends up: a point, for a pendulum with friction; a loop, for a heartbeat or a cycle of the logistic map. In 1971 David Ruelle and Floris Takens, writing about how turbulence starts, called the other kind strange attractors: sets that pull everything in, but on which the motion is chaotic. The name stuck. Nearly all of them turn out to be fractals.

The reason is a tension. Chaos needs stretching, so nearby points separate. But the attractor is bounded, so whatever is stretched must be folded back. Stretch, fold, stretch, fold: it is the motion of a baker kneading dough, and Stephen Smale's "horseshoe" map of the 1960s is the cleanest mathematical version. After many folds, a cross-section through the dough is a Cantor-like stack of layers, layers within layers. That is the fractal.

Michel Hénon made the process visible in 1976 with a map of the plane simpler than Lorenz's flow: x → 1 − a·x² + y, y → b·x, with a = 1.4 and b = 0.3. Its attractor looks like a curved boomerang drawn with a single line, until you zoom in and the line turns out to be several lines, and each of those several more. Its dimension is about 1.26.

Later, people went looking for pretty ones. The Clifford attractors, named after Clifford Pickover, and the Peter de Jong attractors are two-dimensional maps built from sines and cosines with four free parameters. Most parameter choices give a boring point or loop; a lucky few give lace. They have been a staple of generative art ever since, and the instrument below draws all three families.

In 1989 Mario Markus and Benno Hess asked what happens when the growth rate itself changes from year to year in a fixed pattern. Pick two rates, a and b, and a sequence of letters such as AB or AABAB. Iterate the logistic map using r = a on the A years and r = b on the B years, repeating the pattern forever. Then compute the Lyapunov exponent of the result. Do that for every (a, b) in a square and colour each pixel by λ: one family of colours where λ is negative and the orbit settles into order, another where λ is positive and it is chaotic.

The pictures, now called Markus–Lyapunov fractals, look like nothing else in this book: smooth, glossy shapes like bent metal or folded silk, edged with fine chaotic filigree. They reached a wide audience in A. K. Dewdney's "Computer Recreations" column in Scientific American in 1991. Each one is a map of how forced population cycles respond to their two rates, and the bright, sharp-edged spines are where the exponent dives toward minus infinity: superstable cycles, where the orbit passes exactly through the top of the parabola.