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# Lorenz Attractor, Warm Framed & Mounted Print

## Lorenz Attractor, Warm Framed & Mounted Print

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Simple equations have simple solutions. The assertion sounds reasonable, but over the 20th century mathematicians discovered how profoundly complicated dynamical systems can be merely by involving quadratic terms. Let $$(x, y, z)$$ denote spatial position, and let $$t$$ represent time. In the 1960s, meteorologist Edward Lorenz and computer scientists Ellen Fetter and Margaret Hamilton studied a particularly simple-looking system of differential equations as a model of atmospheric convection: \begin{align*} \frac{dx}{dt} &= -10x + 10y, \\ \frac{dy}{dt} &= 28x - y - xz, \\ \frac{dz}{dt} &= xy - \tfrac{8}{3}z. \end{align*} An individual solution, rather than settling down to a point or loop as one might expect, instead moved unpredictably. Further, the qualitative behavior of solutions changed chaotically when the initial conditions were slightly varied. The image depicts randomly-chosen numerical solutions to the system. The two-lobed "strange attractor" that solutions "tend toward" can be glimpsed in the two eye-like loops near the center.