The system dx dt = −sx + sy dy dt = −xz + rx − y dz dt = xy − bz is the famous Lorenz system. It was developed as a model for we
ather prediction, but it has since become famous because its solutions display chaotic behavior. (a) (2 points) Let b = 8/3, s = 10, and r = 27. Use initial conditions x(0) = −4, y(0) = 5, and z(0) = 10. Use ode45 to solve this system on 0 ≤ t ≤ 30. Plot x(t). (b) (1 point) Change one of the initial conditions by 0.1, and solve it again. How do the two solutions for x(t) compare? This is an example of the chaotic behavior of this system of equations. (c) (2 points) Plot the 3D trajectory. Use the plot3 command with the x(t), y(t), and z(t) arrays as arguments. You will see the famous Lorenz strange attractor. Rotate it around a bit, and turn in a plot of an angle you like.
