for a.ii), I got that
F(theta,p) (approx) = p, and the approximation for G(theta,p) = -w^2*theta-kp (by setting the linear approximations for F and G equal to the actual equations for F and G). From this, I conclude that theta=sin(theta). However, in part iii), I'm not sure how to get solutions of this approximate system (theta=sin(theta)) by setting k=0. Perhaps the approximate system I have is incorrect.