(i) A system in $\mathbb{R}^{2}$ obeys the equations:

$$\begin{aligned} &\dot{x}=x-x^{5}-2 x y^{4}-2 y^{3}\left(a-x^{2}\right) \\ &\dot{y}=y-x^{4} y-2 y^{5}+x^{3}\left(a-x^{2}\right) \end{aligned}$$

where $a$ is a positive constant.

By considering the quantity $V=\alpha x^{4}+\beta y^{4}$, where $\alpha$ and $\beta$ are appropriately chosen, show that if $a>1$ then there is a unique fixed point and a unique limit cycle. How many fixed points are there when $a<1$ ?

(ii) Consider the second order system

$$\ddot{x}-\left(a-b x^{2}\right) \dot{x}+x-x^{3}=0,$$

where $a, b$ are constants.

(a) Find the fixed points and determine their stability.

(b) Show that if the fixed point at the origin is unstable and $3 a>b$ then there are no limit cycles.

[You may find it helpful to use the Liénard coordinate $z=\dot{x}-a x+\frac{1}{3} b x^{3}$.]