Chapter 6.8, Problem 12E

Interpretation Introduction

Interpretation:

To classify all fixed points of a given system for a parameter a from $-\infty \text{ to +}\infty$,

To show that sum all of indices are conserved as “a” varies, and to generalize this result for a system $\dot{\text{x}}\text{ = f(x, a) where x}\in {\text{ R}}^2\text{ and a is parameter}\text{.}$

Concept Introduction:

The fixed point of a differential equation is a point where, $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$

The index of a closed curve Cis an integer that measures the winding of the vector field on C.

The of a fixed point is defined as I_c, where Cis anyclosed curve that encloses one and only one fixed point.