Chapter 3.4, Problem 5E

Interpretation Introduction

Interpretation:

The value of $r$ at which bifurcation occurs is to be determined and the bifurcation is to be classified as saddle-node, transcritical, subcritical pitchfork or supercritical pitchfork bifurcation. Also, bifurcation diagram of fixed points ${\mathrm{x}}^{\mathrm{*}}$ vs. r is to be sketched.

Concept Introduction:

By changing the parameter, the fixed points move towards each other, collide and mutually annihilate is known as saddle-node bifurcation.

The stabilities of the fixed points interchanged by changing the parameter is known as transcritical bifurcation.

When the single stable fixed point is present and it turns to an unstable fixed point due to change in parameter and two new symmetric stable fixed points occurs is called supercritical pitchfork bifurcation.

When the single unstable fixed point is present and it turns to a stable point due to change in parameter and two new symmetric unstable fixed points occurs is called subcritical pitchfork bifurcation.