Define Lyapunov stability and quasi-asymptotic stability of a fixed point $\mathbf{x}_{0}$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$.

By considering a Lyapunov function of the form $V=g(x)+y^{2}$, show that the origin is an asymptotically stable fixed point of

$$\begin{aligned} &\dot{x}=-y-x^{3} \\ &\dot{y}=x^{5} \end{aligned}$$

[Lyapunov's Second Theorem may be used without proof, provided you show that its conditions apply.]