Chapter 6.7, Problem 2E

Interpretation Introduction

Interpretation:

To find all the equilibrium points for the pendulum system equation $\ddot{\theta }\text{ + sin}\theta \text{ = }\gamma$ and classify them. Sketch the null clines and the vector field. Determine if the system is conservative or not? If so find conserved quantity. Find the system is reversible or not? Sketch the phase portrait on the plane as $\gamma$ varies. Find the approximate frequency of small oscillations about any centers in the phase portrait.

Concept Introduction:

Determine the fixed point of the system.

Use the linear system and determine the stability for every fixed point.

Represent the system equation in vector field on the phase plane.

Sketch the phase portrait.