(i) Consider a system in $\mathbb{R}^{2}$ that is almost Hamiltonian:

$$\dot{x}=\frac{\partial H}{\partial y}+\epsilon g_{1}(x, y), \quad \dot{y}=-\frac{\partial H}{\partial x}+\epsilon g_{2}(x, y)$$

where $H=H(x, y)$ and $|\epsilon| \ll 1$. Show that if the system has a periodic orbit $\mathcal{C}$ then $\oint_{\mathcal{C}} g_{2} d x-g_{1} d y=0$, and explain how to evaluate this orbit approximately for small $\epsilon$. Illustrate your method by means of the system

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

(ii) Consider the system

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

(a) Show that when $\epsilon=0$ the system is Hamiltonian, and find the Hamiltonian. Sketch the trajectories in the case $\epsilon=0$. Identify the value $H_{c}$ of $H$ for which there is a homoclinic orbit.

(b) Suppose $\epsilon>0$. Show that the small change $\Delta H$ in $H$ around an orbit of the Hamiltonian system can be expressed to leading order as an integral of the form

$$\int_{x_{1}}^{x_{2}} \mathcal{F}(x, H) d x$$

where $x_{1}, x_{2}$ are the extrema of the $x$-coordinates of the orbits of the Hamiltonian system, distinguishing between the cases $H<H_{c}, H>H_{c}$.

(c) Find the value of $\alpha$, correct to leading order in $\epsilon$, at which the system has a homoclinic orbit.

(d) By examining the eigenvalues of the Jacobian at the origin, determine the stability of the homoclinic orbit, being careful to state clearly any standard results that you use.