Collection: Lorenz System

• dx / dt = a (y – x)
• dy / dt = x (b – z) – y
• dz / dt = xy – c z

It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system.

In popular media the ‘butterfly effect’ stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects.

The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.