properties of parity and periodicity that $\wp_\xi\left(\tau\right)$ must have a global minimum within the real period $2\omega_R$. Moreover, since $\wp_\xi$ satisfies the differential equation , the condition for the existence of a stationary point is $$\wp_\xi\left(\tau\right) = e_i,$$ where $e_i$ represents the roots of the cubic equation . It is known [@abramowitz_handbook_1964 eq. 18.3.1] that $\wp_\xi\left(\omega_i\right) = e_i$, where $$\begin{aligned}
\omega_1 &= \omega,\\
\omega_2 &= \omega + \omega^\prime,\\
\omega_3 &= \omega^\prime,\end{aligned}$$ which implies that the global minimum of $\wp_\xi\left(\tau\right)$ is in correspondence of $\tau = \omega_R$. We can then conclude that the condition for bound motion is $$e_R > \frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{0}^{2}}{2}\right),$$ where we have denoted with $e_R$ the root of the cubic equation for which $\wp_\xi\left(\omega_R\right) = e_R$. Figure \[fig:plot\_3d\] displays the evolution of two representative bound orbits in the three-dimensional space.

![image](plot_3d){width="100.00000%"}

Figure \[fig:bound\_vs\_unbound\] displays the evolution in $\tau$ of the parabolic coordinates and of the real time $t$ in a bound and an unbound case. It is interesting to note that in the unbound case only $\xi$ and $t$ present vertical asymptotes, whereas $\eta$ and $\phi$ assume finite values when $\xi$ and $t$ go to infinity. With respect to the evolution in real time $t$, this means that $\eta$ 