Strange Attractor Zoo

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.

Figure 1. strange attractor in the (x, z) projection. Method: classical fourth-order Runge-Kutta from shared/js/engine/ode-rk.js, fixed dt = 0.005. Live max-Lyapunov estimator via tangent-vector renormalization .

attractor

sigma 10.0

rho 28.0

beta 2.67

speed 0.2

WHAT TO TRY

Watch the orbit wind around one wing, then jump to the other with no warning: the bottom x(t) strip flips sign at every jump. That irregular switching, from a fully deterministic system, is the Lorenz signature.
Lower rho below about 24.7: the wings collapse and the orbit spirals into a fixed point. Push rho up and the butterfly reappears with a positive max-Lyapunov exponent.
Switch attractor from the menu (Roessler, Aizawa, Thomas, Halvorsen, Chen-Ueta): each is a different dissipative chaotic flow with its own x(t) fingerprint.