Perfectly deterministic systems that produce exponential divergence and the butterfly effect.
Three coupled equations, no randomness, the same starting values producing the same trajectory every time — and yet the behavior looks random. This is deterministic chaos.
The trajectory never repeats, never settles, never escapes to infinity. It traces a shape of infinite complexity inside a finite region of space. That shape is the strange attractor.
Edward Lorenz produced this butterfly in 1963 from a three-variable weather model on a Royal McBee computer. The trajectory orbits one lobe, crosses to the other, loops back, and never repeats the same number of turns before switching.
$\sigma$ governs rotation rate. $\rho$ sets the height and whether chaos occurs at all (below about 24.7 the system settles to a fixed point). $\beta$ sets the lobe aspect ratio. The classic butterfly values are $\sigma = 10$, $\rho = 28$, $\beta = 8/3$.
Drag to orbit the attractor. Adjust parameters to reshape it.
Figure 1. The Lorenz attractor, rendered as a continuous trajectory with depth shading. The butterfly shape emerges from the three coupled equations shown in the top-right overlay.
Launch two trajectories from points 0.001 apart in one coordinate. For a while they track each other through the same loops. Then they split, one going left while the other goes right, and within a few orbits they have nothing in common.
The chart below plots the distance between them on a log scale. A straight line means exponential growth, and its slope is the Lyapunov exponent — the mathematical definition of chaos.
Increase the initial separation: the trajectories diverge sooner, but the slope stays the same. That slope is a property of the attractor, not the starting conditions.
Figure 2. Two trajectories start slightly apart. They track together, then diverge exponentially. The chart below shows their distance over time on a log scale. A straight line on this log plot means exponential growth; the slope is the largest Lyapunov exponent.
Many systems of three or more coupled nonlinear equations produce strange attractors. Six examples follow, each fully deterministic and each tracing a non-repeating shape.
Figure 3. Six strange attractors. Each follows deterministic equations, but each traces a unique geometric signature. Click to expand.
Lorenz discovered this in 1963 when he re-entered weather-model initial conditions rounded to three decimal places instead of six and watched the simulation diverge entirely from the original. That run is what ended long-range deterministic weather forecasting as a goal.