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The closed annulus $1/\sqrt{3}\le r\le 1$ is positively invariant by the radial signs, and $\dot{\theta }=2$ excludes equilibria there. Poincaré-Bendixson therefore yields a periodic orbit. Its stability can be established explicitly rather than inferred merely from trapping.

For the exact anisotropic radial limit cycle, set $w=r^{-2}$. Then $\dot{w}=-2w+4-2\cos 2\theta$, or $dw/d\theta +w=2-\cos 2\theta$. Its unique periodic solution isIt is positive and stays within the trapping annulus. Any other solution at the same phase has $w-w_p=C{e}^{-2t}$, proving radial attraction and uniqueness of the positive periodic orbit. The full planar period is $\pi$, since a full angular revolution takes that time; the half-turn symmetry of the radius does not halve the period of the position vector. The nontrivial return periodic-orbit multiplier is $e^{-2\pi }<1$. Thus the periodic orbit is asymptotically stable, and all nonzero forward trajectories approach it up to phase.