 Solution

At $\Omega =-2$, the origin has stable and unstable eigenvalues $1-\sqrt{2}$ and $1+\sqrt{2}$; the two nonzero equilibria are repelling foci with eigenvalues $(1\pm i\sqrt{7})/2$. A compatible portrait has one attracting closed orbit surrounding all three equilibria. Its total enclosed index is $+1+1-1=+1$, as a periodic orbit requires. Its half-turn image is also attracting, so uniqueness of the attracting cycle makes that cycle half-turn symmetric. Orbits spiral away from either focus equilibrium and approach the surrounding cycle; the stable separatrices approach the saddle instead. The figure integrates representative trajectories and shows the local saddle directions.

These trajectories cannot escape to infinity. A Liénard boundedness function supplies one explicit boundedness argument using the Liénard coordinate $z=y+F(x)$ with $F(x)=x^3/3-2x$ and $g(x)=x^3-x$. Then $\dot{x}=z-F(x)$ and $\dot{z}=-g(x)$. Put $V=x^4/4-x^2/2$ and $W=V+z^2/2-xz$. This function is coercive after addition of a constant. Its derivative obeysusing $Fz\le (z^2+F^2)/2$. The right side is negative outside a compact region, yielding bounded forward motion.

In the displayed portrait, generic non-equilibrium initial states tend to the cycle; points on the saddle's stable manifold tend to the origin; and initial data exactly at a repelling focus equilibrium remain there. The Poincaré-Bendixson theorem excludes a chaotic compact limit set for this smooth planar flow. Strictly, the assumption gives uniqueness only of an attracting periodic orbit: it does not by itself exclude exceptional nonattracting periodic or separatrix limit sets. Those would need additional global information if one wanted a classification stronger than the compatible portrait requested.