Lyapunov Spectrum & Bifurcation

Logistic map xₙ₊₁ = r·xₙ·(1−xₙ) — where chaos lives

λ > 0 → chaos (red)
λ < 0 → stable (blue)
λ = 0 → period-doubling

Feigenbaum constant δ ≈ 4.6692
Onset of chaos: r ≈ 3.5699
Period-2: r ≈ 3.0
Period-4: r ≈ 3.449
Chaotic (λ>0)
Stable (λ<0)
λ=0 line