One of the first discoveries of chaos in a mathematical model was by Lorenz (1963), who made a simple model of atmospheric convection. In this model, the temperature and velocity field are characterized by three variables, x, y, and z, which satisfy the coupled, nonlinear differential equations

(The precise definitions of x, y, and z need not concern us here.) Integrate these equations numerically to show that x, y, and z follow nonrepeating orbits in the 3-dimensional phase space that they span, and quickly asymptote to a 2-dimensional strange attractor. (It may be helpful to plot out the trajectories of pairs of the dependent variables.)

These Lorenz equations are often studied with the numbers 10, 28, 8/3 replaced by parameters σ, ρ, and β. As these parameters are varied, the behavior of the system changes.

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dx/dt = 10(y-x), dy/dt = -xz+ 28x - y, dz/dt = xy - 8z/3. (15.40)