Chapter 4.3, Problem 3E

Interpretation Introduction

Interpretation:

The phase portrait of $\dot{\text{θ}}\text{ = μ sinθ - sin2θ}$ a function of the control parameter $\text{μ}$ is to be drawn, and bifurcation, which occurs as $\text{μ}$ varies is to be classified. Also, all the bifurcation values of $\text{μ}$ are to be obtained.

Concept Introduction:

The qualitative change in the dynamics of the flow with parameters is called bifurcation, and the points at which this occurs is called the bifurcation point.

To study the stability of the dynamical systems, bifurcation is used.

By changing the parameter, the fixed points move towards each other, collide, and mutually annihilate is known as Saddle Node Bifurcation.

When the single stable fixed point is present, and it turns to an unstable 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 occur is called Subcritical Pitchfork Bifurcation.