= Solution
The <Poincaré-Bendixson theorem> states that a nonempty compact $\omega$-limit set of a planar $C^1$ flow which contains no equilibrium is a periodic orbit.
The origin is the only equilibrium: at a nonzero equilibrium, putting $q=x^2+y^2-a$ would require
$$
\begin{pmatrix}q&3\\-1&q\end{pmatrix}
\binom xy=0,
$$
but its determinant $q^2+3$ is positive. By local asymptotic stability, choose a small simple closed Lyapunov level curve on which the original vector field points inward. On a sufficiently large circle,
$$
\frac12\frac d{dt}(x^2+y^2)
=(x^2+y^2)^2-a(x^2+y^2)+2xy>0,
$$
so the original vector field points outward.
Reverse time. The annulus between these two curves is then a compact positively invariant <trapping region>: the reversed field points into it at both boundaries. It contains no equilibrium. The Poincaré--Bendixson theorem applied to any reversed trajectory in the annulus produces a periodic orbit. Reversing time does not change its image, so the original system also has a periodic orbit.
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