Chapter 4.4, Problem 4E

Interpretation Introduction

Interpretation:

To find if the given equation of motion for the overdamped pendulum connected to torsional spring gives a well-defined vector field on the circle. To nondimensionalize the equation. To find what the pendulum does in the long run. To show that many bifurcations occur as $\text{k}$ is varied from $\text{0}$ to $\infty$, and to find the kind of these bifurcations.

Concept Introduction:

For the equation of motion to give a well-defined vector field on the circle, the values of $\dot{\text{θ}}\left(\text{θ}\right)$ and $\dot{\text{θ}}\left(\text{θ + 2π}\right)$ must be same.

The dimensionless equation of the pendulum can be found by setting the coefficient of $\frac{\text{dθ}}{\text{dt}}$ as unity.

The behavior of the pendulum can be determined by varying the values of $\text{ξ}$.