xₙ₊₁ = rxₙ(1−xₙ) — period-doubling cascade to chaos
The logistic map shows period-doubling bifurcations at r₁≈3.0, r₂≈3.449, r₃≈3.544... converging to chaos at r∞≈3.5699. The ratio of consecutive bifurcation intervals converges to the Feigenbaum constant δ≈4.66920. Lyapunov exponent λ>0 (red) signals chaos; λ<0 (blue) signals stable orbits. Click bifurcation diagram to zoom.