Chapter 8.5, Problem 5E

Interpretation Introduction

Interpretation:

• (a) To find and classify the fixed points in the $\text{(θ, v)}$ phase plane of a driven pendulum with quadratic damping.

• (b) To prove that the system has a stable and unique limit cycle for $\text{F > 1}$.

• (c) To deduce $\frac{\text{du}}{\text{dθ}}\text{= }\ddot{\text{θ}}$

• (d) An exact formula for the limit cycle when $\text{F > 1}$ is to be found. In the region $\text{v > 0}$, where the pendulum equation becomes $\frac{\text{du}}{\text{dθ}}\text{+ 2αu + sinθ = F}$

• (e) The limit cycle undergoes a homoclinic bifurcation at some critical value of F called F_c, as F decreases keeping $\text{α}$ fixed is to be shown. Also, give an exact formula for the bifurcation curve $\text{Fc(α)}\text{.}$

Concept Introduction:

• ➢ The fixed point of a differential equation is a point where, $\text{f(x*) = 0 ; while substitution f(x*) = }\dot{\text{x}}\text{ is used and x\&*\#x00A0;is a fixed point}$.

• ➢ Phase portraits represent the trajectories of the system with respect to the parameters and give qualitative idea about evolution of the system, its fixed points, whether they will attract or repel the flow etc.

• ➢ Limit cycle is the closed trajectory at which all a approach asymptotically.