Chaos summary

Sensitive dependence on initial conditions is everywhere. If you’ve ever missed a bus, which made you miss a flight, which ruined an entire business trip, then you know: tiny errors can snowball into complete chaos.

The butterfly effect is one reason why Lorenz was fascinated by the weather. It means that even if we covered the earth in weather sensors a foot apart, we still couldn’t calculate the weather for a few weeks ahead. A second fascinating feature of the weather is that it’s aperiodic – it’s almost cyclical, but it never quite repeats itself.

Scientists like to have their expectations thwarted about as much as the rest of us. And they certainly weren’t expecting that some of the most fundamental physical systems in our world behave in completely chaotic, unpredictable ways. So naturally, most of them weren’t too thrilled about this new chaos theory embraced by younger, freethinking scientists from the 1970s onwards.

It sounded unscientific, unconventional, and most importantly, it seemed to complicate – even contradict – what they thought they knew about the universe. Up until Lorenz, most scientists had stuck to describing the world in linear ways.

Nonlinear dynamical systems aren’t just a pet project of mathematicians and physicists. As Lorenz showed when he studied the weather, nonlinear systems are fundamental to nature.

Animal populations, for example, change in a nonlinear, dynamical way. The subfield of biology that studies how they behave over time is called ecology – and it was one of the first fields to connect its findings to chaos theory.

Throughout his life, the mathematician and polymath Benoit Mandelbrot ended up in places where he wasn’t welcome. First, as a child, he fled with his family from Poland to France in the 1930s. After the war, he felt stifled by the intellectual climate at the École Polytechnique, where he studied mathematics. At the time, pure math was all the rage, and his colleagues didn’t appreciate his visual approach to mathematical problems. So Mandelbrot escaped to the US, where he found work at the IBM research center in New York.

In his research, too, Mandelbrot often chose unfriendly territory. One of his first interests at IBM was studying economic patterns, like income distribution and price changes. When he studied fluctuations in cotton prices in the nineteenth century, he got a first glimpse of the discovery that would make him famous: the intricately nested nature of our universe.

On his deathbed, quantum physicist Werner Heisenberg swore to ask God two questions about physics: Why relativity? And, why turbulence? “I really think He may have an answer to the first question,” Heisenberg joked.

Of all the longstanding problems in physics, turbulence is among the thorniest. It arises when a smooth flow of a gas or liquid suddenly turns messy, breaking up into whorls and eddies. Turbulence is everywhere, and poses a big problem for engineers. When the smoke ring of a cigarette rises steadily before breaking up into little curls, we might watch in fascination. But when the same thing happens beneath the wings of an airplane we’re traveling in, we’re more likely to panic.

At the Los Alamos National Laboratory in New Mexico in 1974, staff and police were growing a bit concerned. A disheveled-looking man was walking around the campus at night, chain-smoking and pacing erratically.

The eccentric turned out to be Mitchell Feigenbaum, a young mathematical physicist on the verge of a mind-boggling discovery.

By 1977, most scientists had heard about a strange new field of physics called chaos theory. The first big chaos conference was held in Como, Italy, that same year. But if you were a young math or physics student interested in chaos, there were still no mentors – let alone professors – to guide you.

At the new Santa Cruz campus of the University of California, a group of young mathematicians took matters into their own hands. It started with a quiet young graduate student named Robert Stetson Shaw. He’d heard about the Lorenz attractor, and he started playing around by plotting its equations on the big, analog computer on campus. Shaw could adjust the variables of the equations by turning knobs on the computer, and it helped him visualize the sensitive dependence on the initial conditions Lorenz had discovered.

Once the idea of chaos had made its way into the scientific mainstream, researchers began discovering nonlinear dynamical systems everywhere around them.

French physician Albert Libchaber teamed up with an engineer for an experiment meant to prove Feigenbaum’s theory of turbulent fluid motion. Libchaber constructed a tiny box that held liquid helium between two metal plates, which he could heat separately. Creating a few millidegrees of difference in temperature between the plates led the helium to start moving. First, as the warm liquid rose and the cool liquid sunk, the helium fluid organized itself into two rolling cylinders. But as Libchaber turned up the temperature, he observed the familiar pattern of period-doubling bifurcations.