Chapter 3.5, Problem 2E

Interpretation Introduction

Interpretation:

To do the linear stability analysis for all the fixed points of the equation $\frac{\text{d}\varphi }{\text{dt}}\text{ = f(}\varphi \text{)}$, where $\text{f(}\varphi \text{) = - sin }\varphi \text{ + γ}\sin \varphi \text{ cos }\varphi \text{ = sin }\varphi \text{ (γ cos }\varphi \text{ - 1)}$.

Concept Introduction:

Fixed points come off when $\text{f(}\varphi \text{) = 0}$.

If ${\text{f}}^{\text{'}}\text{(}{\varphi }^{*}\text{) > 0}$, then ${\varphi }^{\text{*}}$ is an unstable fixed point

If ${\text{f}}^{\text{'}}\text{(}{\varphi }^{*}\text{) < 0}$, then ${\varphi }^{\text{*}}$ is a stable fixed point.