chosen coprime integers $n$ and $m$. A representative quasi-periodic orbit found this way using the PaGMO optimiser [@biscani_global_2010] is displayed in Figure \[fig:quasi\_periodic\].

Periodic orbits can also be found in a similar way by imposing the additional condition $p \phi(T) = 2\pi$, where $p \in \mathbb Z$. For any triplet of $\left( n,m,p \right)$ integers, one has then to solve numerically an optimisation problem that yields periodic orbits such as the one displayed in Figure \[fig:periodic\] for a case $n=1$, $m = 2$, and $p = 7$.

![image](quasi_periodic){width="100.00000%"}

![image](periodic){width="100.00000%"}

Bound and unbound orbits
------------------------

The solution of the Stark problem in terms of the Weierstrassian functions allows to determine the conditions under which the motion is bound. As we have seen in the previous sections, the parabolic coordinate $\eta$ is always bound, whereas $\xi$ can be either bound or unbound. From the general solution , it is easily deduced that the formula for $\xi\left(\tau\right)$ has a pole (and thus $\xi$ is unbound) when the denominator is zero, i.e., under the condition $$\wp_{\xi}\left(\tau\right)-\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{0}^{2}}{2}\right) = 0.$$ Recalling now that $\wp_\xi\left(\tau\right)$ is analytical everywhere except at the poles (where it behaves like $1/\tau^2$ around $\tau = 0$), it can be deduced from the 