Logistic Map

The logistic map, xn+1 = r xn (1 − xn), is a canonical example from Strogatz’s Nonlinear Dynamics and Chaos showing period-doubling and the onset of chaos. Tweak the parameter r and initial condition x₀ to see the cobweb iterates and how long-term behavior changes.

Cobweb — author notes for this tab.
Cobweb — definitions, hints, references.

Cobweb Diagram

Unit square with y = rx(1 − x) and y = x. Cobweb shows successive iterates.

Bifurcation Diagram

For each r, iterate x ← r x (1 − x), discard transients, then plot the long‑term x values. Period‑doubling leads to chaos as r increases.

Bifurcation diagram: Each vertical slice corresponds to a single r. After skipping transient iterations, the remaining iterates reveal the attractor: a single point (fixed point), two points (period‑2), four points, and so on, accumulating into a fractal set of bands — the onset of chaos.

Lyapunov Exponent λ(r)

λ(r) ≈ (1/N) ∑ log |f’(xₙ)| after skipping transients; f’(x)=r(1−2x). Positive λ indicates chaos.
Cobweb — connect intersections with y=x to orbits.
Cobweb — suggested sweeps and reading.
Bifurcation — author notes/headline for this tab.
Bifurcation — definitions, period‑doubling route, hints.
Bifurcation — windows, Feigenbaum constants.
Bifurcation — how to zoom and interpret bands.
Lyapunov — author notes/headline for this tab.
Lyapunov — definition, computation recipe.
Lyapunov — λ(r) sign, relation to chaos.
Lyapunov — skip transients, averaging details.
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