Consider the nonlinear oscillator

$$\begin{aligned} \dot{x} &=y-\mu x\left(\frac{1}{2}|x|-1\right), \\ \dot{y} &=-x \end{aligned}$$

(a) Use the Hamiltonian for $\mu=0$ to find a constraint on the size of the domain of stability of the origin when $\mu<0$.

(b) Assume that given $\mu>0$ there exists an $R$ such that all trajectories eventually remain within the region $|\mathbf{x}| \leqslant R$. Show that there must be a limit cycle, stating carefully any result that you use. [You need not show that there is only one periodic orbit.]

(c) Use the energy-balance method to find the approximate amplitude of the limit cycle for $0<\mu \ll 1$.

(d) Find the approximate shape of the limit cycle for $\mu \gg 1$, and calculate the leading-order approximation to its period.