Chapter 3.2, Problem 3E

Interpretation Introduction

Interpretation:

For $\dot{\text{x}}\text{ = x - rx (1 - x)}$, all the qualitatively different vector fields that occur as $\text{r}$ is varied, are to be sketched. The critical value of $\text{r}$ at which the trans-critical bifurcation occurs is to be determined. The bifurcation diagram of fixed points is to be sketched.

Concept Introduction:

In trans-critical bifurcation, the two fixed points are not annihilated, they exchange their stability instead.

The trans-critical bifurcation occurs at the point at which the stability is exchanged.

The dotted line represents that the system is unstable and the solid line represents that the system is stable.