Chapter 8.1, Problem 8E

Interpretation Introduction

Interpretation:

For $\text{ε > 0}$, all bifurcations of the system $\text{ε}\frac{{\text{d}}^{\text{2}}\varphi }{{\text{dτ}}^{\text{2}}}\text{ = -}\frac{\text{d}\varphi }{\text{dτ}}\text{ - sin}\varphi \text{ + γsin}\varphi \text{cos}\varphi$ are to be found and classified as $\text{ε and γ}$ vary. Also, the stability diagram of given system in the positive quadrant of the $\text{ε, γ}$ plane is to be plotted.

Concept Introduction:

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

In pitchfork bifurcation, the fixed points appear and disappear in symmetrical pairs.

There are two types of pitchfork bifurcation. One is supercritical, and another is subcritical.

In supercriticalbifurcation, a stable fixed point is present, and after changing parameters, it becomes unstable, and two new symmetric unstable points generate.

In subcriticalbifurcation, an unstable fixed point is present, and after changing parameters, it becomes stable, and two new symmetric stable points generate.