Lorenz attractor simulation

Edward Lorenz found these three equations while modelling convection in 1963, and discovered that a tiny rounding error changed his forecast completely. The butterfly-shaped attractor became the symbol of chaos theory.

Interactive, free and runs in your browser. Part of the Dynamical Systems simulations.

What you will see

1,800 particles, each started at a slightly different point, race around the attractor with glowing trails, while a faint shape behind them shows the long-run density of a single trajectory. The two equilibria at the centres of the wings are marked; the motion circles one wing, then flips unpredictably to the other.

Try this

  1. Drop ρ below about 24.7 and watch the particles settle onto the two equilibria.
  2. Raise ρ back to 28 and watch chaos return.
  3. Follow two neighbouring particles and see how soon they end up on different wings.

Common misconception

The butterfly effect does not mean a butterfly can cause a hurricane on demand. It means small uncertainties grow so fast that long-range prediction is impossible, even with perfect equations.

Exam-style question

For σ = 10, β = 8/3 and ρ = 28, where are the two non-trivial equilibria of the Lorenz system?

C± = (±√(β(ρ − 1)), ±√(β(ρ − 1)), ρ − 1) = (±√72, ±√72, 27) ≈ (±8.49, ±8.49, 27). For ρ above about 24.74 they become unstable and the motion is chaotic.

Key equations

  • dx/dt = σ(y − x)
  • dy/dt = x(ρ − z) − y
  • dz/dt = xy − βz

Curriculum

Enrichment: nonlinear dynamics and chaos.

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