x˙ = σ(y − x) (1) y˙ = rx − y − xz (2) z˙ = xy − bz (3)
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. 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, namely that several different initial chaotic conditions evolve in phase space in a way that never repeats, so all chaos is unpredictable. This underscores that chaotic systems can be completely deterministic and yet still be inherently unpredictable over long periods of time. Because chaos continually increases in systems, we cannot predict the future of systems well. E.g., even the small flap of a butterfly’s wings could set the world on a vastly different trajectory, such as by causing a hurricane. The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly. In 1963, Edward Lorenz, with the help of Ellen Fetter who was responsible for the numerical simulations and figures,[1] and Margaret Hamilton who helped in the initial, numerical computations leading up to the findings of the Lorenz model,[2] developed a simplified mathematical model for atmospheric convection.[1] The model is a system of three ordinary differential equations now known as the Lorenz equations: x˙ = σ(y − x) (1) y˙ = rx − y − xz (2) z˙ = xy − bz (3) MATH IS BEAUTIFUL
