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 k is varied from 0 to ∞ , 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 θ ˙ ( θ ) and θ ˙ ( θ + 2π ) must be same. The dimensionless equation of the pendulum can be found by setting the coefficient of dθ dt as unity. The behavior of the pendulum can be determined by varying the values of ξ .

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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 k is varied from 0 to ∞, 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 θ˙(θ) and θ˙(θ + 2π) must be same.

The dimensionless equation of the pendulum can be found by setting the coefficient of dθdt as unity.

The behavior of the pendulum can be determined by varying the values of ξ.

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