Chapter 3.2, Problem 1E

Interpretation Introduction

Interpretation:

All the qualitatively different vector fields for ${\dot{\text{x}}\text{ = rx + x}}^{\text{2}}$ that occur as r varied are to be sketched and a point at which transcritical bifurcation occurs 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 bifurcation.

Transcritical bifurcation is one of the bifurcation mechanism in which two fixed points exchange their stability instead of destroying.

In the transcritical bifurcation, fixed points exist for all values of the parameter.