The Birth of Chaos Theory
In 1963, meteorologist Edward Lorenz was running a simplified computer model of atmospheric convection when he made a discovery that would reshape science. He re-entered initial conditions rounded to three decimal places instead of six — and the resulting weather prediction diverged completely from the original. This extreme sensitivity to initial conditions became known as the 'butterfly effect': the idea that a butterfly flapping its wings in Brazil could set off a tornado in Texas.
The Lorenz System
The Lorenz system is defined by three coupled ordinary differential equations: dx/dt = σ(y−x), dy/dt = x(ρ−z)−y, and dz/dt = xy−βz. The parameter σ (sigma) is the Prandtl number relating viscosity to thermal diffusivity, ρ (rho) is the Rayleigh number describing the temperature difference driving convection, and β (beta) is a geometric factor related to the aspect ratio of convection cells.
Understanding the Visualization
The simulation shows two trajectories: one in cyan (original) and one in red (perturbed by a tiny amount δ). At first, they appear identical — tracing the same butterfly-shaped path. But after some time, the perturbation grows exponentially and the two paths diverge completely, visiting different lobes of the attractor at different times. This is deterministic chaos: the equations are perfectly deterministic, yet long-term prediction is impossible.
The Lyapunov Exponent
The Lyapunov exponent λ quantifies this divergence rate. For the classic Lorenz parameters, λ ≈ 0.906, meaning nearby trajectories separate by a factor of e ≈ 2.718 roughly every 1.1 time units. This sets a fundamental horizon on predictability — no amount of computational power can overcome it. Increase the perturbation slider to see how even larger initial differences affect the divergence.