Chapter 3.4, Problem 8E

Interpretation Introduction

Interpretation:

For the given function $\dot{\text{x}}\text{= rx - }\frac{\text{x}}{{\text{1+x}}^2}$ the value of $\text{r}$ at which bifurcation occur is to be determined. The bifurcation is to be classified as saddle-node, trans-critical, supercritical pitchfork, or subcritical pitchfork. The bifurcation diagram of fixed points ${\text{x}}^{\text{*}}\text{vs}\text{. r}$ is to be sketched.

Concept Introduction:

Bifurcation theory is used to study the stability of dynamical systems.

The phenomenon in which fixed points are created and destroyed by varying the control parameter is termed as saddle-node bifurcation.

The transcritical bifurcation occurs when the fixed points exchange their stabilities as the parameter is changed.

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

A pitchfork bifurcation occurs where 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 are stable.

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