Lorenz System

σ (sigma) 10.0

ρ (rho) 28.0

β (beta) 2.667

Δx₀ (initial separation) 0.001

Trajectory 1

x —

y —

z —

Trajectory 2

x —

y —

z —

Divergence

Distance —

Time —

Steps —

Trajectories

Trajectory 1

Trajectory 2

Current position

Drag sliders to adjust σ, ρ, β parameters · Δx₀ sets the initial separation between the two trajectories · Switch views to see different phase space projections · Divergence shows how the distance between trajectories grows over time

What’s happening

The Lorenz equations

In 1963, Edward Lorenz was running a simplified model of atmospheric convection on an early computer. He rounded an intermediate value from 0.506127 to 0.506 and restarted the simulation. The result diverged wildly from the original run. He had stumbled onto sensitive dependence on initial conditions — the hallmark of chaos.

The equations

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

Three coupled ordinary differential equations. Three variables (x, y, z). Three parameters (σ, ρ, β). The standard values σ = 10, ρ = 28, β = 8/3 produce the classic butterfly-shaped attractor. The system is deterministic — the same initial conditions always produce the same trajectory — yet nearby trajectories diverge exponentially.

The butterfly effect

Two trajectories starting just 0.001 apart will track each other closely for a while, then suddenly diverge. This is not randomness — it is deterministic chaos. The divergence rate is measured by the Lyapunov exponent. For the classic parameters, the largest Lyapunov exponent is about 0.9, meaning nearby trajectories separate by a factor of e roughly every 1.1 time units.

The strange attractor

Despite the chaos, trajectories don’t fly off to infinity. They are confined to a strange attractor — a fractal set with Hausdorff dimension approximately 2.06. The trajectory never crosses itself (in the full 3D space), never repeats, yet stays within a bounded butterfly-shaped region forever. This is the geometric signature of deterministic chaos.

Parameter regimes

The behavior depends critically on ρ. For ρ < 1, the origin is the only attractor and all trajectories decay to rest. At ρ = 1, the origin loses stability through a pitchfork bifurcation. Between 1 < ρ < 24.74, two stable fixed points appear. At ρ ≈ 24.74, a subcritical Hopf bifurcation creates the chaotic attractor. The classic value ρ = 28 is solidly in the chaotic regime.