The Road Into Chaos

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the Lyapunov exponent λ(r) · the witness for chaos — λ>0 is sensitive dependence

Turn one knob, r, and a single equation — x → r·x(1−x) — walks the whole road from order into chaos. A stable point splits into a 2-cycle, then a 4-cycle, an 8-cycle … the period-doubling cascade. The windows between doublings shrink by a universal ratio, the same for any single-hump map: Feigenbaum's δ ≈ 4.6692. This bench doesn't draw that number — it measures it from the cascade and proves it. Drag across the diagram to move the cursor.

The map

Jump to a landmark

The parameter r

r 3.5700

mark the cascade ladder on the diagram draw the Lyapunov curve λ(r)

Measured from the cascade

The logistic map. The superstable parameters R_n of each 2ⁿ-cycle are root-found to machine precision (the critical point x=½ sits on the cycle), and their ratios converge to δ. The Lyapunov exponent λ = ⟨log|f′|⟩ is the honest test for chaos — λ<0 is order, λ>0 is sensitive dependence. A self-test proves δ from two different maps, λ=ln2 at r=4, and that a linear map fakes no cascade. · part of the Workbench.