Lorenz Attractor Ensemble

Edward Lorenz found in 1963 that three simple equations for convection rolls never settle into a repeating cycle. This shows the result as a swarm. A few thousand trajectories (4096) all start inside a ball one part in a thousand wide, around $(1, 1, 1)$, and are stepped forward with the same RK4 integrator at the classic parameters $\sigma=10$, $\rho=28$, $\beta=8/3$. Within a few seconds the cloud is stretched across the whole Lorenz attractor, the two-lobed butterfly surface. Nearby starts pull apart exponentially fast (the readout estimates the rate, $\lambda_{\max}$ is about 0.9), yet every trajectory stays on that same shape forever: that is what deterministic chaos means, exquisitely sensitive to the start but bounded and structured. The GPU draws the swarm as a glowing density field (brighter where trajectories crowd) and re-projects a short 3D trail so the motion reads while the view slowly turns. The substeps slider sets how many integration steps run per frame, so how fast time flows; trail decay sets how long the comet tails linger. Drag to orbit, scroll to zoom.

Figure 1. Ensemble density of

4096

Lorenz trajectories (

\sigma=10

\rho=28

\beta=8/3

): brighter regions are where trajectories crowd on the attractor. The cloud starts in a

10^{-3}

ball and is stretched across the whole butterfly within seconds, the signature of deterministic chaos. Method: CPU RK4 per trajectory, GPU additive density splatting with ACES tonemapping.

WHAT TO TRY

Vary each control and watch the rail readouts respond.
Compare the diagnostic plot against the live scene.