Chapter 7.6, Problem 1P

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 eigenvalues of the linear system $\left(5\right)$, valid near the origin, are given by Eq. $\left(7\right)$

(b) Determine the corresponding eigenvectors.

(c) Determine the eigenvalues and eigenvectors of the system $\left(5\right)$ in the case where $r=28$.

${\left(\begin{array}{l}x\\ y\\ z\end{array}\right)}^{\text{'}}=\left(\begin{array}{ccc}-10& 10& 0\\ r& -1& 0\\ 0& 0& -8/3\end{array}\right)\left(\begin{array}{l}x\\ y\\ z\end{array}\right)$. $\left(5\right)$

${\lambda }_1=-\frac{8}{3}$, ${\lambda }_2=\frac{-11-\sqrt{81+40r}}{2}$, ${\lambda }_3=\frac{-11+\sqrt{81+40r}}{2}$. $\left(7\right)$