Chapter 3.4, Problem 11E

Interpretation Introduction

Interpretation:

For the system $\dot{\text{x}}\text{ = rx - sin x}$, when $\text{r = 0}$, all the fixed points are to be found and classified, and the vector field is to be sketched. When $\text{r > 1}$, it is to be shown that there is only one fixed point. As $\text{r}$ decreases from $\infty$ to $\text{0}$, all the bifurcations that occur are to be classified. For $\text{0 < r << 1}$, an approximate formula for values of $\text{r}$ at which bifurcations occur is to be found.

All the bifurcations that occur as $\text{r}$ decreases from $\text{0}$ to $\text{- }\infty$ are to be classified. The bifurcation diagram for $\text{- }\infty \text{ < r < }\infty$ is to be plotted and the stability of the various branches of fixed points is to be indicated.

Concept Introduction:

Fixed points are the points where $\dot{\text{x}}\text{ = 0}$.

The graph of the vector field for different values of $\text{r}$ can be plotted.

The saddle-node bifurcation is the basic mechanism by which fixed points are created and destroyed. As a parameter is varied, two fixed points move toward each other, collide, andmutually annihilate.

A subcritical pitchfork bifurcation occurs when there is a single unstable fixed point present, which after the change of parameters becomes unstable, and two new symmetric unstable fixed points appear.