Chaos Theory

Tiny, seemingly insignificant variations in the starting values of chaotic systems will produce drastically different results—a property known as sensitive dependence on initial conditions. Despite following classical physics, including Newton's laws of motion, the long-term unpredictability of these systems' behavior runs counter to Newton's orderly, "clockwork" universe.

In this simulation, users can set the release angles for the four pendulums in the red and blue pairs to six decimal places before release. Even when one of the angles across the pairs differs by just 0.000001 degrees, the double pendulums will begin swinging widely out of sync within a minute.

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

By accidentally finding that rounding one variable to its third decimal place and re-running a weather simulation produced a drastically different two-month forecast, the American mathematician established the field of chaos theory. His paper—"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"—introduced the well-known metaphor for the behavior of chaotic systems. (Some readers may experience a paywall.)

Margaret Hamilton and Ellen Fetter worked on MIT's 800-pound Royal McBee LGP-30 computer, which was fed binary code on paper tape to perform computations. Among the computations performed were those involving equations describing the motion of gas in a circulating fluid for Edward Lorenz, widely considered the father of chaos theory.

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

Although simple, linear equations reach an explicit value for a given input, such as f(x) = x + 1 equaling 4 when x equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

Earth's atmosphere is an example of a system whose evolution depends strongly on the initial conditions when numerically simulated for weather forecasting. Because it is impossible to know the properties of every single air molecule, meteorologists employ ensemble forecasting, in which multiple trials with slightly different initial conditions are run, and the probability of specific events is determined from all the results.

Oxford University complex systems professor Doyne Farmer discusses how predictions from many economic models are overly weighted toward simple relationships, such as supply and demand, and external shocks, such as the COVID-19 pandemic. However, real-world macroeconomic systems involve public and private actors interacting, producing feedback loops and cascading effects that resemble the oscillatory behavior seen in chaotic systems.