The Lorenz Attractor

The Lorenz system is a model of thermal convection that emerged from the study of equations used in the prediction of weather. It was one of the first systems in which chaotic behaviour was observed and studied.

The Lorenz attractor is generated using the following system of equations:

x' = -S x + S y
y' = R x - y - x z
z' = -B z + x y

The numbers S, B, and R are the system's physical parameters, fixed by Lorenz at

s = 10, B = 8/3, R = 28.

The Lorenz attractor features two outward spiraling trajectories. As the solution spins further and further from the center of one spiral, it becomes attracted by the other spiral, where the process is repeated. The chaotic behavior comes from the fact that the number of turns the trajectory spends in one spiral before it is attracted by the other is undetermined. It is hypothsized that for any sequence of reasonably small positive numbers, there exists a trajectory on the Lorenz attractor with precisely that these numbers as turns around the spirals.

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