(a) State and prove Dulac's criterion. State clearly the Poincaré-Bendixson theorem.

(b) For $(x, y) \in \mathbb{R}^{2}$ and $k>0$, consider the dynamical system

$$\begin{aligned} &\dot{x}=k x-5 y-(3 x+y)\left(5 x^{2}-6 x y+5 y^{2}\right) \\ &\dot{y}=5 x+(k-6) y-(x+3 y)\left(5 x^{2}-6 x y+5 y^{2}\right) \end{aligned}$$

(i) Use Dulac's criterion to find a range of $k$ for which this system does not have any periodic orbit.

(ii) Find a suitable $f(k)>0$ such that trajectories enter the disc $x^{2}+y^{2} \leqslant f(k)$ and do not leave it.

(iii) Given that the system has no fixed points apart from the origin for $k<10$, give a range of $k$ for which there will exist at least one periodic orbit.