Dynamical systems
The Lorenz attractor
April 12, 2026
The Lorenz system is three coupled ordinary differential equations, introduced by Edward Lorenz as a toy model of convection. The same parameters can produce trajectories that never repeat and never settle: they trace a thin structure in space (the famous "butterfly") called a strange attractor.
The system is deterministic (no random terms), yet chaotic: two states that start almost on top of each other can diverge quickly. Below, two trajectories use the same equations and parameters; the only difference is a tiny nudge in the starting value of x. Watch how the orange and blue paths peel apart while still living on the same attractor.
t = 25.00 / 96.00 (simulation units)
Parameters (inputs)
The equations use σ, ρ, β. Changing them changes the whole motion. ε is the gap in starting x between the orange and blue paths (sensitivity).
Equations (standard form)
dx/dt = σ (y − x) dy/dt = x (ρ − z) − y dz/dt = xy − β z
Classic chaotic parameters: σ = 10, ρ = 28, β = 8/3.
For a readable overview, see Wikiwand → Lorenz system.