For a dynamical system of the form $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$, give the definition of the alpha-limit set $\alpha(\mathbf{x})$ and the omega-limit set $\omega(\mathbf{x})$ of a point $\mathbf{x}$.

Consider the dynamical system

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

where $\mathbf{x}=(x, y) \in \mathbb{R}^{2}$ and $k$ is a real constant. Answer the following for all values of $k$, taking care over boundary cases (both in $k$ and in $\mathbf{x}$ ).

(i) What symmetries does this system have?

(ii) Find and classify the fixed points of this system.

(iii) Does this system have any periodic orbits?

(iv) Give $\alpha(\mathbf{x})$ and $\omega(\mathbf{x})$ (considering all $\mathbf{x} \in \mathbb{R}^{2}$ ).

(v) For $\mathbf{x}_{0}=\left(0, y_{0}\right)$, give the orbit of $\mathbf{x}_{0}$ (considering all $y_{0} \in \mathbb{R}$ ). You should give your answer in the form $y=y\left(x, y_{0}, k\right)$, and specify the range of $x$.