Chapter 6.1, Problem 12E

Interpretation Introduction

Interpretation:

A certain system is known to have exactly two fixed points and they are saddle points. To sketch the phase portraits in which there is a single trajectory that connects the saddles and there is no trajectory that connects the saddles.

Concept Introduction:

Fixed points are nothing but the function equals to zero. In mathematical form is can be written as $\text{f}\left({\text{x}}^{*}\right)\text{ = 0}$.

The linear system $\dot{\text{x}}\text{ = f}\left(\text{x, y}\right)$ and $\dot{\text{y}}\text{ = g}\left(\text{x, y}\right)$ can be expressed in the form $\left(\begin{array}{l}\dot{\text{x}}\\ \dot{\text{y}}\end{array}\right)=\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial y}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)\left(\begin{array}{l}\text{x}\\ \text{y}\end{array}\right)$.

Here, $\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial y}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$ is Jacobian matrix at any fixed point ${\text{(x}}^{\text{*}}{\text{,y}}^{\text{*}}\text{)}$.