= Solution
On each side of the stated parallelogram, substitution into the outward normal component of
$$
(\dot x,\dot y)=\left(y-\mu(x^3/3-x),-x+F\right)
$$
shows that the leading $L^3$ terms point inward; the remaining terms are $O(L)$, so the region is positively invariant for sufficiently large $L$.
There is one equilibrium,
$$
(x_*,y_*)=(F,\mu(F^3/3-F)).
$$
Its Jacobian has determinant one and trace $\mu(1-F^2)$. If $F^2<1$ the equilibrium is unstable. A trajectory starting nearby remains in the compact trapping region, and its omega-limit set contains no stable equilibrium. The <Poincaré-Bendixson theorem> therefore supplies a periodic orbit.
Back to article page
