The Lorenz equation is widely used in the theory of non-linear equations and was encountered in modeling of natural convection. While working with these equations Lorenz discovered the phenomena of chaos. The system is described by the following set of differential equations:

\[\begin{aligned}& \frac{d x}{d t}=\alpha(y-z) \\& \frac{d y}{d t}=-x z+r x-y \\& \frac{d z}{d t}=x y-b z\end{aligned}\]

where \(\alpha, r\), and \(b\) are the parameters.

Fix two parameters \(\alpha=10\) and \(b=-8 / 3\) and examine the stability of the system with \(r\) as a parameter. Write MATLAB code to track the trajectories and prepare phase-plane plots.