Chapter 6.8, Problem 5E

Interpretation Introduction

Interpretation:

Locate the fixed point and calculate the index.

Concept Introduction:

An index of close curve is an integer that measures the windings of the vector field on C. It also gives the information about any fixed point is lying inside the curve. (Just like Gaussian surface in electrostatic)

To find the index of the curve draw a close contour C on the phase portrait. Then on each point “x” on the contour C, vector field $\dot{\text{x}}\text{ = (}\dot{\text{x}}\text{,}\dot{\text{y}}\text{)}$ makes an angle $\varphi ={\tan }^{-1}\left(\frac{\dot{\text{y}}}{\dot{\text{x}}}\right)$ with the positive x-axis. A the point x moves in counterclockwise direction around C, the angle $\varphi$ changes continuously because vector field is smooth, as x returns to its original place the direction of vector field comes to its original direction. Thus by completing one cycle $\varphi$ changes by an integer multiple of $2\pi$.

The index of closed curve C with respect to vector field is given as

${\text{I}}_{\text{C}}\text{=}\frac{\text{1}}{\text{2π}}{\left[\varphi \right]}_{\text{C}}$

Here, ${\left[\varphi \right]}_C$ is the net change in $\varphi$ in one cycle. ${\text{I}}_{\text{C}}$ is the net number of counterclockwise revolutions made by the vector field.