Chapter 9.2, Problem 6E

Interpretation Introduction

Interpretation:

To show that the system $\dot{\text{x}}\text{ = -vx + zy}$, $\dot{\text{y}}\text{ = -vy + }\left(\text{z - a}\right)\text{x}$, $\dot{\text{z}}\text{ = 1 - xy}$ is dissipative. To show that the fixed points may be written in parametric form as ${\text{x}}^{*}\text{ = ±k}$, ${\text{y}}^{*}{\text{ = ± k}}^{\text{- 1}}$, ${\text{z}}^{*}{\text{ = vk}}^2$, where $\text{v}\left({\text{k}}^{\text{2}}{\text{- k}}^{\text{- 2}}\right)\text{ = a}$. To classify the fixed points.

Concept Introduction:

• ➢ The divergence of a vector field is

  $\nabla \text{. f = }\frac{\partial }{\partial \text{x}}\dot{\text{x}}\text{ + }\frac{\partial }{\partial \text{y}}\dot{\text{y}}\text{ + }\frac{\partial }{\partial \text{z}}\dot{\text{z}}$

• ➢ The fixed points are calculated as $\dot{\text{x}}\text{ = 0}$, $\dot{\text{y}}\text{ = 0}$, $\dot{\text{z}}\text{ = 0}$

• ➢ 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

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