In an attempt to predict weather patterns, Edward N. Lorenz developed a model in 1969 with...

In an attempt to predict weather patterns, Edward N. Lorenz developed a model in 1969 with the following three coupled equations (Lorenz model) in x(t), y(t), and z(t):

where σ, r, and b are positive constants and x, y, and z are real. Lorenz chose, for physical reasons, σ = 10 and , and the parameter r is increased from 0. Let x(0) = 2, y(0) = 5, and z(0) = 5. Investigate the behavior for

(a) r = 0, 10, and 20, 0 ≤ t < 20

---

(b) r = 28, 0 ≤ t < 20, where chaotic behavior sets in for t ≈ 7.

In both cases, investigate the trajectories by using either three-dimensional plots of the coordinates x(t), y(t), z(t) for different time steps or, if your numeric programs do not generate such plots, plot x(t) versus t.