Chapter 12.5, Problem 4E

Interpretation Introduction

Interpretation:

To use numerical integration to compute $\dot{\theta }\left(\text{t}\right)$ for the damped driven pendulum $\ddot{\theta }\text{ + b }\dot{\theta }\text{ + sin θ = F cos t}$, with $\text{b = 0}\text{.22, F = 2}\text{.7}$. To show that the time series has an erratic appearance, and interpret it in terms of the pendulum’s motion. To plot the Poincaré section by strobing the system whenever $\text{t = 2πk}$, where $\text{k}$ is an integer. To zoom in on part of the strange attractor and enlarge a region that reveals the Cantor–like cross-section of the attractor.

Concept Introduction:

A damped driven pendulum given by $\ddot{\theta }\text{ + b }\dot{\theta }\text{ + sin θ = F cos t}$.

Here $\theta$ is the angle between the pendulum arm and the rest position, $\text{b}$ is the coefficient of friction, and $\text{F}$ is the strength of the forcing.