Chapter 6.7, Problem 3E

Interpretation Introduction

Interpretation:

To analyze the system for nonlinear damping $\ddot{\text{θ}}\text{ + }\left(\text{1 + a cosθ}\right)\dot{\text{θ}}\text{ + sin θ = 0}$, for all $\text{a}\ge \text{0}$.

Concept Introduction:

Fixed points occur where $\dot{\text{x}}\text{ = 0}$ and $\dot{\text{y}}\text{ = 0}$.

The Jacobian matrix at a general point $\left(\text{x, y}\right)$ is given by

$\text{A = }\left(\begin{array}{cc}\frac{\partial \dot{\text{x}}}{\partial \text{x}}& \frac{\partial \dot{\text{x}}}{\partial \text{y}}\\ \frac{\partial \dot{\text{y}}}{\partial \text{x}}& \frac{\partial \dot{\text{y}}}{\partial \text{y}}\end{array}\right)$

The Eigen value $\text{λ}$ can be calculated using the characteristic equation

$\left|\left(\text{A - λI}\right)\right|\text{ = 0}$

The solution for the quadratic equation can be calculated by $\text{λ = }\frac{\text{-b ± }\sqrt{{\text{b}}^{\text{2}}\text{- 4ac}}}{\text{2a}}$