Background
The butterfly effect reveals that deterministic equations can produce essentially unpredictable states due to extreme sensitivity to initial conditions. Chaos is structured randomness deep within topology.
Core Theory
1. Lorenz System & Strange Attractor
$$\dot{x} = \sigma(y - x), \quad \dot{y} = x(\rho - z) - y, \quad \dot{z} = xy - \beta z$$2. Lyapunov Exponents
Measure the exponential divergence of nearby trajectories: $\lambda_1 > 0$ for chaos.
Figure
Figure 1: Lorenz attractor trajectory showing fractal self-similarity.