Lorenz Attractor - 3D Acrylic Desk Sculpture with Diverging Twins and a Poincare Plane

Setting up the desk...

Presets

Live readouts

Graphs

x(t) of orbit A and of its twin B, started delta0 apart

Distance between A and B (log scale) against the prediction delta0 e^(lambda t)

Lorenz map: each z maximum against the next one (the chaos is a 1-D tent-like map)

Experiment: bifurcation diagram, z maxima against rho (green line in the sim = current rho is the dashed vertical)

Experiment: time for two orbits to separate by 1, against the initial separation

Experiments

Both run real integrations for the current sigma and beta (under a second).

Numerical checks

running...

How to use

The sculpture shows the Lorenz state space on a desk: the acrylic frame is the box x in [-24, 24], y in [-30, 30], z in [0, 52]. The orange head is orbit A, the cyan head its twin B; the glowing violet cloud is an ensemble of initial states that is stretched and folded over the attractor. The two brass pedestals mark the fixed points C+ and C-, the translucent disc is the Poincare plane z = rho - 1, and each upward crossing leaves a spark on it. Drag to orbit, wheel or pinch to zoom, Space pauses, R restarts. Everything else works as in the 2D lab.

The desk, frame and pedestals were built in Blender (baked PBR textures); all dynamics run in JavaScript.

The science

The model. Edward Lorenz (1963) truncated the equations of thermal convection in a fluid layer heated from below to three modes: dx/dt = sigma (y - x) dy/dt = x (rho - z) - y dz/dt = x y - beta z x is the amplitude of the convective roll, y the temperature difference between rising and sinking fluid, z the distortion of the vertical temperature profile. sigma is the Prandtl number, rho the scaled Rayleigh number (heating), beta a geometric factor (8/3 for the classic roll).

Fixed points and Hopf threshold. The origin is the only equilibrium for rho <= 1. For rho > 1 two more appear, C+- = (+-sqrt(beta (rho-1)), +-sqrt(beta (rho-1)), rho-1), the two steady convection rolls. Their Jacobian has the characteristic polynomial l^3 + (sigma + beta + 1) l^2 + beta (sigma + rho) l + 2 sigma beta (rho - 1) = 0 and they lose stability at rho_H = sigma (sigma + beta + 3) / (sigma - beta - 1) = 24.737 for the classic values. Above it every orbit is repelled from both rolls but cannot escape to infinity, so it wanders between them forever.

Dissipation. The flow contracts phase-space volume at the constant rate div F = -(sigma + 1 + beta), so a volume shrinks as V(t) = V0 e^(-(sigma+1+beta) t) (13.67 per time unit for the classic values). The attractor therefore has zero volume; its Kaplan-Yorke dimension is 2 + lambda1 / |lambda3| = 2.06.

Sensitive dependence. Two nearby orbits separate like d(t) = d0 e^(lambda1 t) with the largest Lyapunov exponent lambda1 = 0.9056 for the classic values (the others are 0 and -14.57). The page measures lambda1 live with the Benettin method (a shadow orbit renormalised every 10 steps) and predicts the time at which the twins split, ln(1/d0)/lambda1. Each factor of 10 in precision buys only 2.5 more time units of forecast.

Lorenz map. Plotting each local maximum of z against the next one gives a nearly one-dimensional curve with slope above 1 everywhere (a tent-like map), which is why the long-term behaviour is chaotic but the next maximum is predictable from the last.

Honest simplifications. (1) The three-mode truncation is a toy for real convection; it is accurate only near the onset of convection and at Prandtl 10 it is not quantitatively right for water. (2) The integration is RK4 with a fixed step; for chaotic orbits any numerical orbit follows a true orbit only for a limited time (shadowing), but statistics such as lambda1 are robust. (3) The Benettin estimate has a statistical error of about one percent after T=400. (4) The ensemble is a Gaussian blob and its spreading shows mixing, not a rigorous measure. (5) The Poincare section collects upward crossings of z = rho - 1 only.

Checks run in the page. The RK4 error ratio when the step is halved (16 for a fourth-order method); the contraction of a tiny tetrahedron against e^(-(sigma+1+beta) t); the Benettin exponent against the literature 0.9056; relaxation to the analytic fixed point at rho = 15; and the Hopf threshold from the closed form against a bisection on the real part of the roots of the cubic.