Chaos Theory

Chaos theory studies systems that behave unpredictably despite being deterministic.

The Butterfly Effect

A butterfly flaps its wings in Brazil and sets off a tornado in Texas. This is not literally true, but it captures the core idea: tiny changes in initial conditions produce wildly different outcomes.

Lorenz System

$$\begin{aligned} \frac{dx}{dt} &= \sigma (y - x) \ \frac{dy}{dt} &= x (\rho - z) - y \ \frac{dz}{dt} &= xy - \beta z \end{aligned}$$

Three simple equations. Infinitely complex behavior. The weather, the stock market, and turbulence all live in this space.