How Chaos Theory Makes the Future Unpredictable

Each week Quanta Magazine explains one of the most important ideas driving modern research. This week, physics staff writer Charlie Wood traces the development of chaos theory and its metaphorical mascot, the butterfly.

Writers have long marveled at the possibility that minor events can have far-reaching consequences. A centuries-old proverb traces the fall of a kingdom to the “want of a nail” for securing the shoe of a fateful steed. In Ray Bradbury’s 1952 short story “A Sound of Thunder,” a time-traveling thrill seeker accidentally alters the course of an election by trampling on a prehistoric butterfly. And in 1961, Norton Juster wrote in The Phantom Tollbooth that “whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world.”

This phenomenon, popularly known as the butterfly effect, captures imaginations because it clashes with the way we intuitively understand the world to work. Small causes should have small effects, not big ones. As The Phantom Tollbooth headed to the printer, however, researchers at the Massachusetts Institute of Technology were discovering that the butterfly effect is more than just a romantic notion.

Edward Lorenz, the story goes, was harnessing the primitive computers of the early 1960s to run weather simulations when he noticed that simulated weather systems with identical starting points could end up producing dramatically different weather patterns. Further investigation revealed that the starting points only looked identical. The computer would print variables to three digits of accuracy on paper, but it held five digits in memory. So when two runs of a simulation appeared to start out with one variable set to 0.214, for instance, the computer might actually be computing weather patterns seeded with 0.21431 and 0.21423.  In Lorenz’s digital world, the values of those two extra digits might mean the difference between a clear day and a tornado. His research launched the field of chaos theory. It also classified weather as a “chaotic” system, meaning something so volatile that the metaphorical flap of a butterfly’s wings can put it on an entirely new course.

Chaos challenges the central objective of physics: making predictions. How can we ever trust our calculations if imperceptible initial uncertainties lead to completely different outcomes? But since the days of Lorenz, chaos theory has blossomed into a fully fledged discipline. Physicists have studied what makes a system chaotic and how to measure the degree of chaos. Along the way, they’ve come to appreciate that chaotic behavior is all around us, found in everything from irregular heartbeats to wild populations.

What’s New and Noteworthy

In 2017, the geophysicist Daniel Rothman noticed a strange acknowledgment in Lorenz’s seminal paper thanking Ellen Fetter for handling the numerical computations. Through some historical sleuthing, Rothman found that Fetter and a colleague, Margaret Hamilton, had been the researchers who actually ran the simulations that led to Lorenz’s chaos breakthrough. He shared his findings with Quanta, which covered this hidden history of chaos in 2019.

Last year’s hit TV show 3 Body Problem introduced many viewers to another famous example of chaos, originally discovered by the French mathematician Henri Poincaré in the late 1800s. When three stars are orbiting one another — even when they appear to have settled into a regularly repeating orbital pattern — you can never tell when they might suddenly start to behave erratically. (In the TV show and the earlier sci-fi novel by Liu Cixin, this poses an existential threat to the long-suffering Trisolaran civilization.) Poincaré’s work on the three-body problem brought chaos to the attention of mathematicians long before physicists came to appreciate its role in their simulations. In a 2022 column, the mathematician David Richeson explained the dynamical maps mathematicians have developed that get at the heart of chaos.

Chaotic systems stump humans because it’s not feasible to measure and track all their tiny but consequential details — all those fateful butterfly flaps. But that’s not necessarily the case for computers, physicists showed in 2018. Machine learning algorithms can’t fully lift the veil hiding the future of chaotic systems, but they can see eight times further than humans can.

Lately, physicists have been studying an even more dramatic type of unpredictability. An infamous result from computer science says it’s impossible to come up with a general method of predicting whether a computer program will halt or not. Physicists have shown that this undecidability also affects certain contrived materials and fluids flowing through strangely curved spaces. It’s fundamentally impossible to make certain predictions like whether a rubber duck floating through one of these fluids will ever reach a particular zone, for instance, even with perfect knowledge of the system’s initial state. One researcher described this phenomenon as “next-level” chaos.

Around the Web

• For a visual sense of how sensitive a chaotic system can be, check out this YouTube video from Eliza Diggins about the three-body problem with plenty of simulations of co-orbiting trios of worlds.
• Chaotic weather makes prediction hard, but not impossible. Weather forecasting has gotten a lot better over the last few decades, according to Our World in Data, due to much improved data collection and more powerful computers.
• Metaphors aside, could a butterfly really cause a hurricane? The answer is almost certainly no, the current Quanta Magazine physics editor Natalie Wolchover wrote for Live Science in 2011.