Chapter 3.1, Problem 3E

Interpretation Introduction

Interpretation:

To sketch all the qualitatively different vector fields for $\dot{\text{x}}\text{ = r + x - ln(1+x)}$ that occur as r varies, and a saddle-node bifurcation point which occurs ata critical value of r is to be determined. Also, the bifurcation diagram of fixed points ${\text{x}}^{*}\text{vs r}$ is to be sketched.

Concept Introduction:

The qualitative change in the dynamics of the flow with parameters is called bifurcation and the points at which this occurs is called bifurcation point.

Bifurcation is used to study the stability of the dynamical systems.

Saddle-node bifurcation is one of the bifurcation mechanism in which fixed points create, collide and destroy.