Lorenz Attractor & Butterfly Effect Simulator

The Lorenz system is the original model of deterministic chaos: three simple nonlinear ordinary differential equations, derived in 1963 from a truncated model of atmospheric convection, that produce trajectories tracing an endlessly complex "butterfly" shape in 3D phase space without ever repeating. This simulator integrates the equations in real time and draws the trajectory as a glowing trail, while a second, nearly identical point started just 0.0001 away traces its own path alongside it. Watch the two trajectories track each other closely at first, then rapidly diverge — a direct demonstration of sensitive dependence on initial conditions, the mechanism behind the "butterfly effect" and the reason long-range weather forecasting has a hard predictability limit. Adjust σ, ρ and β to reshape the attractor, or toggle the twin trajectory off to study a single path in isolation.