Chapter 3.4, Problem 4E

Interpretation Introduction

Interpretation:

All the qualitatively different vector fields for $\dot{\text{x}}\text{ = x + }\frac{\text{r x}}{{\text{1+ x}}^{\text{2}}}$ that occur as r varied are to be sketched, the value of $r$ at which pitchfork bifurcation occurs is to be determined and the bifurcation is to be classified as supercritical or subcritical. Also, the bifurcation diagram of fixed points ${\text{x}}^{*}\text{vs r}$ is to be sketched.

Concept Introduction:

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

In pitchfork bifurcation, the fixed points appear and disappear in symmetrical pairs.

There are two types of pitchfork bifurcation, one is supercritical and another is subcritical.

In supercritical bifurcation, a stable fixed point is present and after changing parameters it becomes unstable and two new symmetric unstable points generate.

In subcritical bifurcation, an unstable fixed point is present and after changing parameters, it becomes stable and two new symmetric stable points generate.