Consider the dynamical system

$$\begin{aligned} &\dot{x}=x(y-a), \\ &\dot{y}=1-x-y^{2}, \end{aligned}$$

where $-1<a<1$. Find and classify the fixed points of the system.

Use Dulac's criterion with a weighting function of the form $\phi=x^{p}$ and a suitable choice of $p$ to show that there are no periodic orbits for $a \neq 0$. For the case $a=0$ use the same weighting function to find a function $V(x, y)$ which is constant on trajectories. [Hint: $\phi \dot{\mathbf{x}}$ is Hamiltonian.]

Calculate the stable manifold at $(0,-1)$ correct to quadratic order in $x$.

Sketch the phase plane for the cases (i) $a=0$ and (ii) $a=\frac{1}{2}$.