Lorenz System

3 min read#chaos

Three coupled equations, distilled from a weather model, whose solutions trace the iconic butterfly-shaped strange attractor.

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Lorenz System

The Lorenz system is three simple-looking differential equations that launched modern chaos theory. In 1963 the meteorologist Edward Lorenz stripped a model of atmospheric convection down to three variables and found something nobody expected: a deterministic flow that never repeats, never settles, and never escapes — instead winding forever around a Strange Attractor shaped like a butterfly's wings.

The equations

\dot x = \sigma(y - x), \qquad \dot y = x(\rho - z) - y, \qquad \dot z = xy - \beta z.

Here x, y, and z are abstract measures of convection intensity and temperature variation. The three constants are the knobs:

  • \sigma — the Prandtl number (fluid viscosity vs. thermal diffusivity), classically \sigma = 10.
  • \rho — the Rayleigh number (the strength of the driving temperature difference), classically \rho = 28.
  • \beta — a geometric aspect ratio, classically \beta = 8/3 \approx 2.667.

At the classic values the system is chaotic. Lorenz discovered this almost by accident: he restarted a run from numbers rounded to three decimals and the new trajectory diverged completely from the old — the first clear sighting of sensitive dependence.

Flying the attractor

Drive the simulation below. Notice that the trajectory orbits one wing for a while, then flips to the other — but when it flips is effectively unpredictable. The set of all these orbits is the Lorenz attractor, a Strange Attractor of fractal dimension \approx 2.06: thinner than a volume, thicker than a surface.

The Lorenz attractor at the classic parameters σ=10, ρ=28, β=8/3. Two wings, infinitely many orbits, never a repeat.

Structure inside the chaos

The Lorenz flow has three fixed points: the origin, plus a symmetric pair C^\pm at the centers of the two wings. At \rho = 28 all three are unstable, so trajectories are perpetually repelled — yet bounded, so they can never leave. The resolution is to circle the unstable points forever, switching wings, tracing the butterfly.

The volume of any blob of initial conditions shrinks at a constant rate, since

\nabla\!\cdot\mathbf{f} = \frac{\partial \dot x}{\partial x}+\frac{\partial \dot y}{\partial y}+\frac{\partial \dot z}{\partial z} = -(\sigma + 1 + \beta) < 0.

A shrinking volume that nonetheless stretches in one direction is the engine of chaos: contract overall, stretch along one axis, fold the result back on itself. That stretch-and-fold is what gives the Attractor its layered, fractal cross-section.

The Lorenz flow contracts phase-space volume everywhere, yet two nearby trajectories still diverge. How is that possible?

See also