Chapter 8.6, Problem 2E

Interpretation Introduction

Interpretation:

• (a) To show that the given system has no fixed points, given that ${\omega }_{1,}{\omega }_2>0{\text{ and K}}_1,{\text{K}}_{\text{2}}>0$.

• (b) To find the conserved quantity for the system.

• (c) To show that the system can be non-dimensionalized to $\frac{{\text{dθ}}_{\text{1}}}{\text{dτ}}={\text{1 + asin(θ}}_{\text{2}}{\text{-θ}}_{\text{1}}\text{) , }\frac{{\text{dθ}}_{\text{2}}}{\text{dτ}}={\text{ω + asin(θ}}_{\text{1}}{\text{-θ}}_{\text{2}}\text{)}$,

  if it is given that, ${\text{K}}_{\text{1}}{\text{= K}}_{\text{2}\text{.}}$

• (d) To find the winding number analytically. (winding number = $\underset{\text{τ}\to \infty }{\lim }\frac{{\text{θ}}_{\text{1}}\text{(τ)}}{{\text{θ}}_{\text{2}}\text{(τ)}}$).

Concept Introduction:

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

• ➢ The winding number determines the nature of trajectory. In the case of a rational winding number, the trajectory is a trefoil knot, while in the case of an irrational number, there is no closed solution.