(i) What is a Liapunov function?

Consider the second order ODE

$$\dot{x}=y, \quad \dot{y}=-y-\sin ^{3} x$$

By finding a suitable Liapunov function of the form $V(x, y)=f(x)+g(y)$, where $f$ and $g$ are to be determined, show that the origin is asymptotically stable. Using your form of $V$, find the greatest value of $y_{0}$ such that a trajectory through $\left(0, y_{0}\right)$ is guaranteed to tend to the origin as $t \rightarrow \infty$.

[Any theorems you use need not be proved but should be clearly stated.]

(ii) Explain the use of the stroboscopic method for investigating the dynamics of equations of the form $\ddot{x}+x=\epsilon f(x, \dot{x}, t)$, when $|\epsilon| \ll 1$. In particular, for $x=R \cos (t+\theta)$, $\dot{x}=-R \sin (t+\theta)$ derive the equations, correct to order $\epsilon$,

$$\dot{R}=-\epsilon\langle f \sin (t+\theta)\rangle, \quad R \dot{\theta}=-\epsilon\langle f \cos (t+\theta)\rangle$$

where the brackets denote an average over the period of the unperturbed oscillator.

Find the form of the right hand sides of these equations explicitly when $f=$ $\Gamma x^{2} \cos t-3 q x$, where $\Gamma>0, q \neq 0$. Show that apart from the origin there is another fixed point of $(*)$, and determine its stability. Sketch the trajectories in $(R, \theta)$ space in the case $q>0$. What do you deduce about the dynamics of the full equation?

[You may assume that $\left\langle\cos ^{2} t\right\rangle=\frac{1}{2},\left\langle\cos ^{4} t\right\rangle=\frac{3}{8},\left\langle\cos ^{2} t \sin ^{2} t\right\rangle=\frac{1}{8}$.]