(i) Describe the use of the stroboscopic method for obtaining approximate solutions to the second order equation

$$\ddot{x}+x=\epsilon f(x, \dot{x}, t)$$

when $|\epsilon| \ll 1$. In particular, by writing $x=R \cos (t+\phi), \dot{x}=-R \sin (t+\phi)$, obtain expressions in terms of $f$ for the rate of change of $R$ and $\phi$. Evaluate these expressions when $f=x^{2} \cos t$.

(ii) In planetary orbit theory a crude model of an orbit subject to perturbation from a distant body is given by the equation

$$\frac{d^{2} u}{d \theta^{2}}+u=\lambda-\delta^{2} u^{-2}-2 \delta^{3} u^{-3} \cos \theta$$

where $0<\delta \ll 1,\left(u^{-1}, \theta\right)$ are polar coordinates in the plane, and $\lambda$ is a positive constant.

(a) Show that when $\delta=0$ all bounded orbits are closed.

(b) Now suppose $\delta \neq 0$, and look for almost circular orbits with $u=\lambda+\delta w(\theta)+a \delta^{2}$, where $a$ is a constant. By writing $w=R(\theta) \cos (\theta+\phi(\theta))$, and by making a suitable choice of the constant $a$, use the stroboscopic method to find equations for $d w / d \theta$ and $d \phi / d \theta$. By writing $z=R \exp (i \phi)$ and considering $d z / d \theta$, or otherwise, determine $R(\theta)$ and $\phi(\theta)$ in the case $R(0)=R_{0}, \phi(0)=0$. Hence describe the orbits of the system.