Chapter 6.8, Problem 14E

Interpretation Introduction

Interpretation:

To classify the fixed point at the origin as a function of $\text{α}$ for the system given by $\dot{\text{x}}\text{ = x cos α - y sin α,}$ $\dot{\text{y}}\text{ = x sin α + y cos α}$, where $\text{α}$ is a parameter that runs over the range $\text{0 }\le \text{ α }\le \text{ π}$. If $\text{C}$ is a simple closed curve that does not pass through the origin, to show that ${\text{I}}_{\text{c}}$ is independent of $\text{α}$. If $\text{C}$ is a circle centered at the origin, to compute ${\text{I}}_{\text{c}}$ explicitly by evaluating the integral for any convenient choice of $\text{α}$.

Concept Introduction:

Fixed points occur where $\dot{\text{x}}\text{ = 0}$ and $\dot{\text{y}}\text{ = 0}$.

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

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