Chapter 3.1, Problem 1E

Interpretation Introduction

Interpretation:

All the qualitatively different vector fields for ${\dot{\text{x}}\text{ = 1 + rx + x}}^{\text{2}}$ that occur as r varied, are to be sketched 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 points.

The stability of the dynamical systems can be studied using birurcation.

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