= Solution
The <Bendixson-Dulac theorem> states that if $(P,Q)$ is a continuously differentiable planar <vector field> on a <simply connected> open region $D$, and a continuously differentiable <Dulac function> $B$ makes
$$
\frac{\partial(BP)}{\partial x}+\frac{\partial(BQ)}{\partial y}
$$
have one weak sign throughout $D$, without vanishing identically on any nonempty open subset, then there is no nonconstant <periodic orbit> contained in $D$. To see why, a hypothetical simple <periodic orbit> bounds a region in $D$. The <divergence theorem> gives the <integral> of this expression as the outward flux of $B(P,Q)$ across the orbit. The flux is zero because the <vector> field is tangent to its orbit, whereas the one-sign area <integral> is nonzero, a contradiction. The usual stricter version assumes a strictly positive or strictly negative expression everywhere.
For the printed <polynomial> system, the origin is an <equilibrium> and
$$
M(0,0)=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad\lambda=\pm i.
$$
Thus \b[its <linearization> is a <centre equilibrium>]. For the full system take $B=1$. The <divergence> is
$$
P_x+Q_y=x^4-3x^2+x^2y^2=x^2(x^2+y^2-3).
$$
On the open disk $x^2+y^2<3$ this is nonpositive, and is negative wherever $x\ne0$. Its zero set inside the disk is a line, so it cannot vanish on any open subset. The <Bendixson-Dulac theorem> therefore proves
$$
\boxed{\text{There are no nonconstant periodic orbits contained in }x^2+y^2<3.}
$$
This also exposes a terminology problem in the source: a nonlinear <centre equilibrium> has surrounding closed orbits, so the printed claim of a centre cannot hold in that sense. The imaginary <eigenvalues> establish only a centre of the linearized system. In fact the full <equilibrium> is an \b[asymptotically stable <weak focus>], as the following direct check shows.
A <quartic Lyapunov correction for a weak focus> makes the nonlinear damping visible. Put $r^2=x^2+y^2$ and consider the <Lyapunov function>
$$
V(x,y)=\frac{x^2+y^2}{2}-\frac{5x^3y+3xy^3}{8}=\frac{r^2}{2}+O(r^4).
$$
It is positive definite sufficiently close to zero. Differentiating along the <polynomial> <vector> field gives
$$
\dot V=-\frac38(x^2+y^2)^2+O(r^6)=-\frac38r^4+O(r^6).
$$
For clarity, the quartic terms come from $-x^4$ in the <derivative> of $r^2/2$, and from applying $y\partial_x-x\partial_y$ to the quartic correction. Their sum is $-3x^4/8-3x^2y^2/4-3y^4/8$; all other terms have degree at least six. Therefore $\dot V<0$ in a sufficiently small punctured neighbourhood, proving <asymptotic stability> by <Second Lyapunov theorem>. Finally
$$
\dot\theta=\frac{xQ-yP}{r^2}=-1+O(r^2),
$$
so nonconstant nearby orbits turn clockwise as they converge. The <equilibrium> is a weak focus despite the centre in its <linearization>. This interpretation resolves both the local dynamics and the absence of closed orbits without changing the printed <coefficients>.
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