= Solution
The planar <divergence> is $\nabla\cdot F=-(\Omega+x^2)<0$ for $\Omega>0$. Since the plane is simply connected, the <Bendixson-Dulac theorem> excludes any nonconstant <periodic orbit>: the flux across a closed trajectory would be zero because the <vector field> is tangent, whereas <Green's theorem> makes it the strictly negative integral of the <divergence> over the enclosed region.
Alternatively define $E=y^2/2-x^2/2+x^4/4$. Direct differentiation gives
$$
\boxed{\dot E=-(\Omega+x^2)y^2.}
$$
A nonconstant periodic solution would require the integral of this strictly dissipative expression over a period to vanish. That forces $y\equiv0$ and hence an <equilibrium>, a contradiction. This energy argument identifies the damping mechanism behind the geometric criterion.
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