Chapter 3.4, Problem 10E

Interpretation Introduction

Interpretation:

For $\dot{\text{x}}\text{ = rx + }\frac{{\text{x}}^{\text{3}}}{{\text{1 + x}}^{\text{2}}}$, the values of $\text{r}$ at which bifurcation occurs is to be determined. The bifurcation is to be classified as saddle-node, trans-critical, supercritical pitchfork, or subcritical pitchfork. The bifurcation graph of fixed points is to be ploted.

Concept Introduction:

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

In pitchfork bifurcation, the system transitions from one fixed point to three fixed points.

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.