= Solution
The <Poincaré-Bendixson theorem> states that the omega-limit set of a forward orbit confined to a compact region of the plane, containing no <equilibrium points>, is a <periodic orbit>.
Put $x=R\cos\theta$, $y=R\sin\theta$. Direct calculation gives
$$
\dot R=R[1-H(\theta)R^2],\quad H=1+\tfrac12\sin^2\theta-\sin^4\theta,\qquad\dot\theta=1+R^2\cos\theta\sin^3\theta.
$$
Here $1/2\leq H\leq17/16$ and $|\cos\theta\sin^3\theta|\leq3\sqrt3/16$. Thus on the inner circle $R=\sqrt3/2$ the radial velocity is positive, and on the outer circle $R=3/2$ it is negative. In their closed annulus,
$$
\dot\theta\geq1-\frac{27\sqrt3}{64}>0,
$$
so there is no equilibrium. The theorem gives a <periodic orbit> in this trapping annulus.
It is asymptotically stable: the vector-field <divergence> is
$$
2-4x^2-3y^2\leq2-3R^2\leq-1/4
$$
throughout the annulus. Its nontrivial <Floquet multiplier> is $\exp\int_0^T\operatorname{div}F\,dt<1$. This proves \b[an attracting <periodic orbit> exists]. There can be only one there: two such nested orbits would enclose an annular region with zero outward flux, contradicting the strictly negative <divergence> via the <divergence theorem>.
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