Chapter 8.6, Problem 3E

Interpretation Introduction

Interpretation:

To show that each trajectory is dense by considering flow on torus given by, ${\dot{\text{θ}}}_{\text{1}}{\text{=ω}}_{\text{1}}{\text{, }\dot{\text{θ}}}_{\text{2}}{\text{=ω}}_{\text{2}}$

Concept Introduction:

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

• ➢ Dense Trajectory on the torus means trajectory comes arbitrarily close to any given point on the torus but it does not pass through that point and orbit is never a closed one.