Existence follows from the argument above: the function $\Delta\hat{H}(y_0)$ has a unique zero $y^*$, and the symmetric trajectory starting at $(-\sqrt{3}, y^*)$ is a closed orbit. Uniqueness follows because any periodic orbit must cross the line $x = -\sqrt{3}$ (since it encircles the unstable origin and cannot remain in $\{|x| < \sqrt{3}\}$ where $\dot{H} > 0$). Each such crossing gives a value $y_0$ with $\Delta\hat{H}(y_0) = 0$, but there is only one such value. For stability: the origin is an unstable focus, so trajectories from inside the limit cycle spiral outward. For trajectories starting outside the limit cycle, $\Delta\hat{H}(y_0) < 0$ for $y_0 > y^*$, meaning $y_2 < y_0$ at each half-circuit, so trajectories spiral inward toward the limit cycle. Thus the limit cycle attracts all trajectories except the origin.