Lyapunov Exponent Explorer

Parameter r 3.700

Initial separation ε 1e-4

Iteration speed 8/s

Lyapunov λ (Benettin)

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Regime

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Trajectory separation |δₙ|

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How it works

Every column of the orange-to-blue terrain is one value of the parameter r. For each column the map is iterated past its transient and the surviving values are plotted — the classic bifurcation diagram. Each column is also colored by its Lyapunov exponent λ, the number that actually defines chaos: it measures the average exponential rate at which two infinitesimally close starting points pull apart.

λ = lim(N→∞) (1/N) Σ ln|f'(xₙ)|
logistic: f(x)=r·x(1−x), f'(x)=r(1−2x)
sine map: f(x)=r·sin(πx), f'(x)=r·π·cos(πx)

The white marker on the terrain tracks the slider's r. Above it, two spheres run the same map from x and x+ε — the visible pair resets whenever they drift apart, so you can watch the split happen again and again. The number in the panel is not that visual guess: it's the Benettin algorithm, the standard numerical method — re-normalize the separation to ε after every step and average the log-growth, which converges to the exact same λ as the analytic formula above.

δₙ₊₁ = f(r,x+δₙ) − f(r,x)
λ ≈ (1/N) Σ ln(|δₙ₊₁|/ε),  then reset δ→ε

Parameter r — slides through periodic windows (λ<0, terrain reads blue-green, trajectories converge) into chaos (λ>0, terrain reads red-orange, trajectories explode apart).
Initial separation ε — how close the two starting points begin; smaller ε only delays the visible split, it never changes λ, which is exactly the point of sensitive dependence.
Iteration speed — how fast the live pair steps through the map and the Benettin average accumulates.
Map — switch between the logistic map and the sine map; both are unimodal but their bifurcation structure and λ(r) curve differ in detail.

λ < 0 — periodic λ > 0 — chaotic Trajectory pair x, x+ε

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