Lorenz Attractor: Butterfly Effect Simulator

The Lorenz system is the classic entry point into chaos theory: three simple, fully deterministic differential equations that nonetheless produce trajectories no one can predict far into the future. This simulator integrates two nearly identical starting points through the same equations side by side — one violet, one cyan, separated by just 0.0001 in each coordinate at t=0. Watch how closely they track each other at first, then how rapidly they diverge as both settle onto the two-lobed, butterfly-shaped strange attractor, never repeating, never crossing themselves, and never converging back together. Adjust σ, ρ and β to explore the parameter space, including the boundary near ρ≈24.74 where the system transitions from stable equilibrium to full chaos.