Chapter 6.8, Problem 8E

Interpretation Introduction

Interpretation:

Sketch the arrangement of the given cycles ${\text{C}}_{\text{1,}}{\text{C}}_{\text{2,}}{\text{C}}_{\text{3}}$, and show that there is at least one fixed point bounded by ${\text{C}}_{\text{1,}}{\text{C}}_{\text{2,}}{\text{C}}_{\text{3}}$

Concept Introduction:

Index of a closed curve C is an integer that measures the windings of the vector field on C, it also provides the information about any fixed point inside the curve.

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. As the point x moves in a counterclockwise direction around C, the angle $\varphi$ changes continuously because the vector field is smooth, and as x returns to its original place, the direction of the 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 the 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 $\varphi$ one cycle. ${\text{I}}_{\text{C}}$ is the net number of counterclockwise revolutions made by the vector field.

Properties of Indices

If contour C is continuously deformed into C^’ without passing through a fixed point, then ${\text{I}}_{\text{C}}{\text{= I}}_{{\text{C}}^{\text{'}}}$

If contour C doesn’t enclose any fixed points then ${\text{I}}_{\text{C}}\text{=0}$

If the directions of all arrows in the vector field are reversed by changing $t\to -t$, the index remains unchanged.

If the closed curved C is a trajectory for the system, then ${\text{I}}_{\text{C}}\text{= +1}$

Theorem: Any closed orbit in the phase plane mustenclose fixed points whose indices sum is $\text{+1}$.

${\displaystyle \sum _{\text{k=1}}^{\text{n}}{\text{I}}_{\text{k}}}\text{= +1}$

Theorem: If a close curve C surrounds n isolated fixed point $x_1^{*},x_2^{*},x_3^{*},\mathrm{.........}{x}_n^{*}$, then

$I_C=I_1+I_2+\mathrm{........}+I_n$ ;$k=1,2,\mathrm{........................},n.$