Chapter 3.4, Problem 3E

Interpretation Introduction

Interpretation:

The vector field for the function ${\dot{\text{x}}\text{ = rx + 4x}}^3$ as r varies has been sketched qualitatively. Pitchfork bifurcation occurs at a critical value of r, and classification of bifurcation as supercritical or subcritical to be determined. Bifurcation diagram of ${\text{x}}^{\text{*}}\text{ vs}\text{. r}$ has been sketched.

Concept Introduction:

Supercritical pitchfork bifurcation occurs when the single fixed points are present it is stable, and after parameters change, this fixed point becomes unstable, and two new symmetric fixed points which appear are stable.

Subcritical pitchfork bifurcation occurs when there is a single fixed point present which is unstable, and after parameters change, this fixed point becomes stable and two new symmetric fixed points which appear become unstable.

Pitchfork bifurcation is the bifurcation in which, when parameters are varied then the fixed points tend to appear and disappear in symmetrical pairs.