We have been studying a Hamiltonian system that possesses a one-parameter family of periodic orbits, depending on the energy level $h$. We "know" via various non-rigorous means that these periodic orbits are stable for $h<\frac{1}{8}$ and unstable for $h>\frac{1}{8}$ but haven't been able to prove it.

We know that if the linearization possesses a periodic orbit at the critical value $h=\frac{1}{8}$, then this value of $h$ lies on the boundary between stability and instability. We have numerically calculated this periodic orbit using ultra-high-precision arithmetic and a 30th order ODE solver. We have also used Hill's method of harmonic balance to high order in computer algebra to show that $h_{\rm critical}$ agrees with $1/8$ to double precision, using no ODE solves.

If we could simply show that the following ODE has a $2\pi$-periodic orbit, we would have a proof. Does anybody know how find such a solution?

$$\frac{d}{d\theta} \vec{x} = A(\theta) \vec{x},$$

where $$ A(\theta) = \frac{1}{4\sqrt{17+8 \cos{2 \theta} }} \begin{pmatrix} - \sin{2 \theta} & \frac{7+12 \cos{2 \theta} -4 \cos{4 \theta}-3\sqrt{17+8 \cos{2 \theta} } }{2-2\cos{2 \theta}} \\ \frac{3-4 \cos{2 \theta} -4\cos{4 \theta}-\sqrt{17+8 \cos2 \theta }} {2+2\cos 2\theta}& \sin{2 \theta} \\ \end{pmatrix}. $$

This would answer the one big question we left unanswered in this paper also available on my website.