Chapter 6.8, Problem 10E

Interpretation Introduction

Interpretation:

Check the validity of theorem for various types of closed orbits on a torus, cylinder, and sphere.

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 closed 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. 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.

Properties of Indices

If contour C 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 remain 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 to $\text{+1}$.

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

Theorem: If a close curve C surrounds n isolated fixed point ${\text{x}}_{\text{1}}^{\text{*}}{\text{,x}}_{\text{2}}^{\text{*}}{\text{,x}}_{\text{3}}^{\text{*}}\text{,}\mathrm{.........}{\text{x}}_{\text{n}}^{\text{*}}$, then

${\text{I}}_{\text{C}}{\text{=I}}_{\text{1}}{\text{+I}}_{\text{2}}\text{+}\mathrm{........}{\text{+I}}_{\text{n}}$ ;$\text{k=1,2,}\mathrm{........................}\text{,n}\text{.}$