= Solution
At $\Omega=-2$, the origin has stable and unstable <eigenvalues> $1-\sqrt2$ and $1+\sqrt2$; the two nonzero <equilibria> are <unstable focus>[repelling foci] with <eigenvalues> $(1\pm i\sqrt7)/2$. 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> obeys
$$
\dot W=-gF-z^2+Fz+xg\leq-\tfrac12z^2-\frac5{18}x^6+\frac83x^4-x^2,
$$
using $Fz\leq(z^2+F^2)/2$. 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.
\Image[/past-exam-of-the-topics-in-applied-mathematics-course-of-the-university-of-oxford/2010-symmetric-oscillator.png]
{title=A compatible attracting cycle, <unstable focus>[repelling foci] and saddle separatrices at Omega minus two}
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