Lorenz attractor

In 1963 the MIT meteorologist Edward Lorenz published three coupled equations, a drastically truncated model of convection in a fluid layer heated from below. Their solutions never settle down and never repeat, and two that start close together soon go separate ways. The paper, “Deterministic Nonperiodic Flow”, became one of the founding texts of chaos theory, and the two-lobed attractor it describes is the field’s best-known image.

What you see

Several trajectories, each in its own color, move on the attractor, projected onto a plane (x-z by default; x-y or y-z under projection). They start from points spread along the x axis and are run forward 1000 steps before the first frame, so they are already on the attractor when the display begins. Integration uses fourth-order Runge-Kutta by default.

Classic butterfly (σ=10, ρ=28): Lorenz’s original values. A trajectory circles one lobe a few times, then switches to the other at moments that are effectively unpredictable. Each trajectory switches on its own schedule.

Periodic orbit (ρ=99.96): in a window of ρ near 100 the chaos gives way to a stable periodic orbit, a single closed loop that every trajectory approaches. The motion is no longer chaotic: the largest Lyapunov exponent drops to zero, the value for any limit cycle, and the others are negative.

Transient chaos (ρ=22): below about ρ = 24.06 there is no strange attractor. Between about 13.93 and 24.06 a trajectory can still wander chaotically for a long time before it spirals into one of the two stable fixed points.

Governing equation

\frac{d x}{d t} = σ (y - x), \frac{d y}{d t} = x (ρ - z) - y, \frac{d z}{d t} = x y - β z .

At the classic values the attractor has a fractal dimension of about $2.06$ and a largest Lyapunov exponent $λ_{1} \approx 0.906$ . A small separation grows as $| δ 𝐳 (t) | \sim | δ 𝐳 (0) | e^{λ_{1} t}$ , doubling about every $0.76$ time units.

Parameters

σ (sigma): the Prandtl number, the ratio of the fluid’s viscosity to its thermal diffusivity, 1 to 20. It sets how quickly $x$ follows $y$ .

ρ (rho): the normalized Rayleigh number, how strongly the layer is heated, 0.5 to 60. It is the main control parameter. For σ = 10 and β = 8/3: below 1 the origin is the only fixed point and it is stable; above 1 two more fixed points appear, stable up to about 24.74; the strange attractor exists from about 24.06 upward, so the two coexist briefly.

β (beta): a geometric factor of the convection cell, 0.1 to 5. The classic value is 8/3.

trajectories: how many trajectories are drawn, 1 to 20. All share the same parameters; their starting points are spread over 6 units along x.