Chapter 7.6, Problem 2P

Problems $1$ through $3$ ask you to fill in some of the details of the analysis of the Lorenz equations in this section:

(a) Show that the linear approximation valid near the critical point $P_2$ is given by Eq. $\left(8\right)$.

(b) Show that the eigenvalues of the system $\left(8\right)$ satisfy Eq. $\left(9\right)$.

(c) For $r=28$, solve Eq. $\left(9\right)$ and thereby determine the eigenvalues of the system $\left(8\right)$.

${\left(\begin{array}{l}u\\ v\\ w\end{array}\right)}^{\text{'}}=\left(\begin{array}{ccc}-10& 10& 0\\ 1& -1& -\sqrt{8(r-1)/3}\\ \sqrt{8(r-1)/3}& \sqrt{8(r-1)/3}& -8/3\end{array}\right)\left(\begin{array}{l}u\\ v\\ w\end{array}\right)$. $\left(8\right)$

$3{\lambda }^3+41{\lambda }^2+8\left(r+10\right)\lambda +160\left(r-1\right)=0$, $\left(9\right)$