= Solution
A <fixed point> $x_0$ is <Lyapunov stable> if for every $\varepsilon>0$ there is $\delta>0$ such that solutions starting within $\delta$ of $x_0$ stay within $\varepsilon$ for all forward times. It has <quasi-asymptotic stability> if some neighborhood of $x_0$ consists of initial points whose solutions converge to $x_0$. <Asymptotic stability> requires both properties.
Choose the <Lyapunov function>
$$
\boxed{V(x,y)=\frac{x^6}{3}+y^2.}
$$
It vanishes only at the origin, is positive elsewhere, and has compact sublevel sets. The vector field is polynomial, hence locally Lipschitz. Along a solution,
$$
\dot V=2x^5(-y-x^3)+2yx^5=-2x^8\leq0.
$$
A small sublevel set is forward invariant and can be fitted inside any prescribed neighborhood, proving <Lyapunov stability>; boundedness also prevents finite-time escape.
For attraction, a bounded forward solution has a nonempty compact limit set, and $V$ decreases to a limit. Every point in that limit set has that same $V$ value. Continuous dependence of solutions makes the limit set forward invariant, so $\dot V=0$ along every trajectory in it. Such a trajectory must have $x=0$ at all times. But then $\dot x=-y$, forcing $y=0$. Thus the only possible limit point is the origin, and every solution converges there. This verifies the invariant-set condition behind the <LaSalle invariance principle>, rather than incorrectly treating $\dot V$ as strictly negative at every nonzero point. The origin is therefore \b[asymptotically stable], indeed globally attracting.
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