Lorenz System & Bifurcation

The Lorenz system models convection in a fluid layer. As the Rayleigh number ρ increases, the system transitions from stable fixed points to periodic orbits to the famous chaotic attractor. Drag the ρ slider to explore the full bifurcation sequence.

Dynamical regime
Chaos (ρ=28)
28.0
10.0
2.7
2000
8
Bifurcation: x-extrema vs ρ
Lorenz equations:
ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz

Phase transitions: ρ < 1: origin stable. 1 < ρ < 13.9: two stable fixed points C±. ρ ≈ 24.7: subcritical Hopf. ρ = 28: canonical chaos (Lyapunov exponent λ₁ ≈ 0.906). ρ > 313: periodic windows re-emerge. The attractor has Hausdorff dimension ≈ 2.06.