Chapter 9.2, Problem 1E

Interpretation Introduction

Interpretation:

To show that for the Lorenz equations, the characteristic equation for the eigenvalues of the Jacobian matrix at ${\text{C}}^{+}$, ${\text{C}}^{-}$ is ${\text{λ}}^3\text{ + }\left(\text{σ + b + 1}\right){\text{λ}}^{\text{2}}\text{ + }\left(\text{r + σ}\right)\text{ bλ + 2bσ}\left(\text{r - 1}\right)\text{ = 0}$. To show that there is a pair of pure imaginary eigenvalues when ${\text{r = r}}_{\text{H}}\text{ = σ }\frac{\left(\text{σ + b + 3}\right)}{\left(\text{σ - b - 1}\right)}$ for the solutions of the form $\text{λ = iω}$, where $\text{ω}$ is real. To explain why we need to assume $\text{σ > b + 1}$. To find the third eigenvalue.

Concept Introduction:

The Jacobian matrix is given by:

$\text{A = }\left(\begin{array}{ccc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}& \frac{\partial \dot{\text{x}}}{\partial \text{z}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}& \frac{\partial \dot{\text{y}}}{\partial \text{z}}\\ \frac{\partial \dot{\text{z}}}{\partial \text{x}}& \frac{\partial \dot{\text{z}}}{\partial \text{y}}& \frac{\partial \dot{\text{z}}}{\partial \text{z}}\end{array}\right)$

The Eigen value $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$