Chapter 6.8, Problem 7E

Interpretation Introduction

Interpretation:

The system $\dot{\text{x}}\text{ = x}\left({\text{4 - y - x}}^{\text{2}}\right)$, $\dot{\text{y}}\text{ = y}\left(\text{x - 1}\right)$ has no orbits is to be shown by using index theory.

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

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

The Eigenvalue $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$

The solution for the quadratic equation can be calculated by $\text{λ = }\frac{\text{-b ± }\sqrt{{\text{b}}^{\text{2}}\text{- 4ac}}}{\text{2a}}$

Index theory provides global information about the phase portrait and fixed points.

The index of the closed curve $C$ is an integer which measures the winding of the vector field on closed curve C.

The index of the closed curve with respect to the vector field can be mathematically expressed as

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

Where, ${\text{I}}_{\text{c}}$ is the net number of counterclockwise revolutions that are made by the vector field, ${\left[\varphi \right]}_{\text{C}}$ is the net change in $\varphi$ around $\text{C}$.