Chapter 6.8, Problem 4E

Interpretation Introduction

Interpretation:

For the systems ${\dot{\text{x}}\text{ = y}}^{\text{3}}$, $\dot{\text{y}}\text{ = x}$, locate the fixed points and calculate the index.

Concept Introduction:

Index theory provides the global information about the phase portrait.

The index of the closed curve can be defined as the net number of counter-clockwise revolutions made by vector field. In mathematical form it can be written as,

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

Where ${\text{I}}_{\text{c}}$ is the index of the closed curve and ${\left[\varphi \right]}_{\text{c}}$ is the net change in $\varphi$ around C.