Lyapunov Exponent Explorer

Chaos theory isn't about randomness — it's about deterministic systems whose nearby trajectories separate exponentially fast, making long-term prediction impossible even though every step follows a fixed rule. This simulator makes that idea measurable. A 3D terrain plots the bifurcation diagram of the logistic (or sine) map across its parameter r, colored by the Lyapunov exponent λ computed for every column: blue where trajectories converge, orange where they diverge. A live pair of points starting a hair's-width apart (ε) runs the same map in real time above the terrain, visibly splitting apart and resetting, while a Benettin-algorithm running average turns that splitting into the same exact number the analytic formula predicts — the butterfly effect, quantified.