Consider the system

$$\dot{x}=y, \quad \dot{y}=x-x^{3}+\epsilon\left(1-\alpha x^{2}\right) y,$$

where $\alpha$ and $\epsilon$ are real constants, and $0 \leqslant \epsilon \ll 1$. Find and classify the fixed points.

Show that when $\epsilon=0$ the system is Hamiltonian and find $H$. Sketch the phase plane for this case.

Suppose now that $0<\epsilon \ll 1$. Show that the small change in $H$ following a trajectory of the perturbed system around an orbit $H=H_{0}$ of the unperturbed system is given to leading order by an equation of the form

$$\Delta H=\epsilon \int_{x_{1}}^{x_{2}} F\left(x ; \alpha, H_{0}\right) d x$$

where $F$ should be found explicitly, and where $x_{1}$ and $x_{2}$ are the minimum and maximum values of $x$ on the unperturbed orbit.

Use the energy-balance method to find the value of $\alpha$, correct to leading order in $\epsilon$, for which the system has a homoclinic orbit. [Hint: The substitution $u=1-\frac{1}{2} x^{2}$ may prove useful.]

Over what range of $\alpha$ would you expect there to be periodic solutions that enclose only one of the fixed points?