Consider a modified van der Pol system defined by

$$\begin{aligned} &\dot{x}=y-\mu\left(\frac{1}{3} x^{3}-x\right) \\ &\dot{y}=-x+F \end{aligned}$$

where $\mu>0$ and $F$ are constants.

(a) A parallelogram PQRS of width $2 L$ is defined by

$$\begin{array}{ll} P=(L, \mu f(L)), & Q=(L, 2 L-\mu f(L)) \\ R=(-L,-\mu f(L)), & S=(-L, \mu f(L)-2 L) \end{array}$$

where $f(L)=\frac{1}{3} L^{3}-L$. Show that if $L$ is sufficiently large then trajectories never leave the region inside the parallelogram.

Hence show that if $F^{2}<1$ there must be a periodic orbit. Explain your reasoning carefully.

(b) Use the energy-balance method to analyse the behaviour of the system for $\mu \ll 1$, identifying the difference in behaviours between $F^{2}<1$ and $F^{2}>1$.

(c) Describe the behaviour of the system for $\mu \gg 1$, using sketches of the phase plane to illustrate your arguments for the cases $0<F<1$ and $F>1$.