Consider the dynamical system

$$\begin{aligned} &\dot{x}=-x+x^{3}+\beta x y^{2} \\ &\dot{y}=-y+\beta x^{2} y+y^{3} \end{aligned}$$

where $\beta>-1$ is a constant.

(a) Find the fixed points of the system, and classify them for $\beta \neq 1$.

Sketch the phase plane for each of the cases (i) $\beta=\frac{1}{2}$ (ii) $\beta=2$ and (iii) $\beta=1$.

(b) Given $\beta>2$, show that the domain of stability of the origin includes the union over $k \in \mathbb{R}$ of the regions

$$x^{2}+k^{2} y^{2}<\frac{4 k^{2}\left(1+k^{2}\right)(\beta-1)}{\beta^{2}\left(1+k^{2}\right)^{2}-4 k^{2}} .$$

By considering $k \gg 1$, or otherwise, show that more information is obtained from the union over $k$ than considering only the case $k=1$.

$\left[\right.$ Hint: If $B>A, C$ then $\left.\max _{u \in[0,1]}\left\{A u^{2}+2 B u(1-u)+C(1-u)^{2}\right\}=\frac{B^{2}-A C}{2 B-A-C} \cdot\right]$