Lorenz system

A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β =  ⁠8/3⁠ The Lorenz system is a set of three ordinary differential equations, first developed by the meteorologist Edward Lorenz while studying atmospheric convection. It is a classic example of a system that can exhibit chaotic behavior, meaning its output can be highly sensitive to small changes in its starting conditions.

For certain values of its parameters, the system's solutions form a complex, looping pattern known as the Lorenz attractor. The shape of this attractor, when graphed, is famously said to resemble a butterfly. The system's extreme sensitivity to initial conditions gave rise to the popular concept of the butterfly effect—the idea that a small event, like the flap of a butterfly's wings, could ultimately alter large-scale weather patterns. While the system is deterministic—its future behavior is fully determined by its initial conditions—its chaotic nature makes long-term prediction practically impossible.