The celebrated Lorentz attractor is described by the differential equations

\[\frac{d x}{d t}=-\sigma x+\sigma y \quad, \quad \frac{d y}{d t}=-x z+\alpha x-y \quad, \quad \frac{d z}{d t}=x y-\beta z,\]

and is used to described chaotic fluid dynamics involving heat flow. It is parameterized by \(\alpha, \beta\), and \(\sigma\).

(a) Solve this system of equations numerically and plot, for example, \(x\) vs \(y\) and \(z\) vs \(y\). Determine the onset of chaos by testing super-sensitivity to initial conditions.