Chaos Theory

The origin of chaos theory, colloquially known as the 'butterfly effect'

The theory explains that systems following well-understood physical laws can still produce unpredictable outcomes due to small variations in their initial conditions. Originally identified in weather data, elements of chaos have been observed in stock market fluctuations, cardiac arrhythmias, and the evolution of online social movements.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

How a competition honoring a king's birthday uncovered history's first chaotic system

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.

The life of Edward Lorenz, who originally used a seagull in his 'butterfly effect' metaphor

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.)

The story of two female programmers running the simulations behind chaos theory's discovery

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.

View the Lorenz attractor, a graphical solution to chaotic equations that resembles a butterfly

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.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

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.

A mathematical and visual analysis distinguishing chaos from randomness

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.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

How chaos helps explain why small market corrections can produce major economic instability

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.