Chapter 3.1, Problem 4E

Interpretation Introduction

Interpretation:

For $\dot{\text{x}}\text{ = r + }\frac{\text{1}}{\text{2}}\text{ x - }\frac{\text{x}}{\text{1+x}}$ all the qualitative different vector field is to be sketched that occurs as r is varied. Also shown the all saddle-node bifurcation occurs at a critical value of r.

Concept Introduction:

The saddle-node bifurcation is the basic mechanism by whichfixed points are created and destroyed.

As a parameter is varied, two fixed points move towards each other, collide, and mutually annihilate. Where r is a parameter which may be positive, negative, or zero.

Solve the given equation $\dot{\text{x}}\text{ = r + }\frac{\text{1}}{\text{2}}\text{ x - }\frac{\text{x}}{\text{1+x}}$ for x and then we got parameter as $\text{r = }\frac{\text{3}}{\text{2}}\pm \frac{\text{2}}{\sqrt{\text{2}}}$.