= Solution
An <equilibrium> is <asymptotically stable> when it is Lyapunov stable and attracts every initial state in some neighbourhood. <Lyapunov stability> means that sufficiently small initial displacement stays small for all future times; attraction additionally requires convergence to the <equilibrium>. A <Lyapunov function> is a differentiable positive-definite scalar function near the <equilibrium> whose <derivative> along solutions is nonpositive. Strict negativity, or an appropriate invariant-set argument for a non-strict <derivative>, gives <asymptotic stability>.
For $\nu=-1$, <equilibria> satisfy $y=0$ and $x+x^3=0$, so only $(0,0)$ is real. Its <Jacobian matrix> is $\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$, with <eigenvalues> $\pm i$ for every $\kappa$. It is a linear <centre equilibrium>, so <linearization> alone cannot decide nonlinear attraction.
Use the positive proper <energy>
$$
E=\tfrac12y^2+\tfrac12x^2+\tfrac14x^4,\qquad\boxed{\dot E=\kappa x^2y^2.}
$$
For $\kappa<0$, <energy> sublevel sets are <compact> and invariant. The zero-derivative set is the union of the axes, but its only invariant subset is the origin: on $x=0,y\ne0$ one immediately leaves that axis, and on $y=0,x\ne0$ one has $\dot y=-x-x^3\ne0$. <LaSalle's invariance principle> therefore proves global <asymptotic stability>. At $\kappa=0$ the conserved <energy> levels are closed <periodic orbits>, giving a nonlinear <centre equilibrium>. For $\kappa>0$, time reversal together with $y\mapsto-y$ converts the system to the attracting negative-$\kappa$ case, so the origin is a nonlinear repeller and is unstable. Nonzero forward trajectories cannot remain bounded; this does not assert existence for all time if finite-time escape occurs.
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