Chapter 3.1, Problem 2E

Interpretation Introduction

Interpretation:

For $\dot{\text{x}}\text{ = r - cosh x}$, 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 saddle-node bifurcation occurs is to be determined. The bifurcation diagram of fixed points is to be sketched.

Concept Introduction:

The basic mechanism through which fixed points are created and destroyed is the saddle-node bifurcation.

Two fixed points approach each other, collide, andmutually annihilate when a parameter is altered.

The graphs for all possible vector fields for $\text{r > 1}$, $\text{r < 1}$ and $\text{r = 1}$ can be plotted.

A saddle-node bifurcation arises at a critical value of $\text{r}$.