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What you're looking at is a 2-dimensional simulation showing how the state of a dynamic system (the three variables x, y, and z of a three-dimensional system) evolves over time. In 3-D you would see that the lines don't actually cross or touch. If they did, at that point the system's state would be exactly the same as an earlier state, and chaotic systems never repeat themselves exactly. This is Lorenz's strange attractor. It's "strange" because its mathematical dimension is not an integer like we're used to, say 2 or 3. Depending on how it's measured, it's dimension is, strangely, about 2.06. Lorenz's equations are a simplification of the partial differential equations describing convective flow in a fluid heated from below -- for example, hot summer air building up to a thundercloud, or a pan of water on a stove. The three equations are just dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, and dz/dt = xy-βz. The constants have been given the values of (Prandtl number sigma) σ = 10.0, (beta) β = 8/3, and (Rayleigh number rho) ρ, which is varied. At ρ = 28.0 the system becomes chaotic. |
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