Lorenz attractor

What you are seeing: a simplified model of atmospheric convection, due to Edward Lorenz in 1963. The trajectory in $(x, y, z)$ never settles down and never repeats, but it stays trapped in a butterfly-shaped region of space. This was the first rigorously studied example of deterministic chaos: noise-free, completely deterministic, yet long-term prediction is impossible.

The equations are $\dot x = \sigma (y - x)$, $\dot y = x (\rho - z) - y$, $\dot z = x y - \beta z$, here projected onto the $(x, z)$ plane. The red dot is the current state. $\lambda_1$ in the corner is the running estimate of the largest Lyapunov exponent (analytic value about $0.906$); it tells you how fast two nearby starting points drift apart. Drop $\rho$ below $24.74$ for a stable spiral; push it past $100$ for periodic windows inside chaos.