The Lorenz Model is a simple model for atmospheric convection developed by Edward Lorenz in 1963. The system is given by three equations:

\[\begin{aligned} \frac{d x}{d t} & =\sigma(y-x) \\ \frac{d y}{d t} & =x(ho-z)-y \\ \frac{d z}{d t} & =x y-\beta z \end{aligned}\]

a. Find the equilibrium points of the system.

b. Find the Jacobian matrix for the system and evaluate it at the equilibrium points.

c. Determine any bifurcation points and describe what happens near bifurcation point(s). Consider \(\sigma=10, \beta=8 / 3\), and vary \(ho\).

d. This system is known to exhibit chaotic behavior. Lorenz found a so-called strange attractor for parameter values \(\sigma=10, \beta=8 / 3\), and \(ho=28\). Using a computer, locate this strange attractor.