## 5 Computation of the Symmetric RTPOs
### Results  
Similar to the computation of the resonant periodic families in the above sections, once a _RTPO_ is found, the whole family can be generated by the predict-correct algorithm. In this section, we limit ourselves to the computation of specific _RTPO_s, but not the families. Some example orbits are given below.  
Examples of the _RTPO_s related to \(L_{1}\) as well as \(L_{2}\) are given in Figures 15 and 16, respectively. In Figure 15, six _RTPO_s related to \(L_{1}\) are presented, including (1) a resonance transition between the 1:1 resonance and other resonances (Orbit-1, Orbit-2, and Orbit-4); (2) a resonance transition between the interior and exterior resonances by the 1:1 resonance (Orbit-3); (3) a resonance transition between two interior resonances with the aid of 1:1 resonance (Orbit-5); (4) a resonance transition between two exterior resonances by the 1:1 resonance (Orbit-6). Similarly, six examples of the _RTPO_s corresponding to \(L_{2}\) are displayed in Figure 16, which correspond to (1) a resonance transition between the 1:1 and other resonances (Orbit-1, Orbit-2, and Orbit-3); (2) a resonance transition between the interior and exterior resonances by the 1:1 resonance (Orbit-4); (3) a resonance transition between two exterior resonances with the aid of 1:1 resonance (Orbit-5 and Orbit-6).  
The _RTPO_s can stay around \(L_{1}\) or \(L_{2}\) for one or more loops before leaving to other resonances. This feature shows that the _RTPO_s are suitable for missions requiring long-term continuous observations of the second primary. The transition between different resonances by the 1:1 resonance indicates that the _RTPO_s can be applied to trajectory design for tour missions of the interior and exterior regions of the system. Various _RTPO_s emanating  
Figure 14: Curves of the velocity along the \(x\)-direction when the trajectory intersects with the \(x\)-axis for the first (**a**), second (**b**), third (**c**), and fourth (**d**) time. The abscissa is the initial velocity along the \(y\)-axis. The reference orbit is a planar Lyapunov orbit around the \(L_{2}\) point with a Jacobi constant of \(C=2.99747681824366\). The purple dots indicate a positive \(\Delta\
u\) and the red points indicate a negative \(\Delta v\).  
from the same generating orbit of \(L_{1}\) or \(L_{2}\) indicate that the spacecraft can transfer from one \(RTPO\) to another by performing a small maneuver in the vicinity of the generating orbit, which may significantly increase the flexibility of the mission design [18]. In our study, the \(RTPO\)s are unstable; therefore, station-keeping maneuvers are required if they are applied to practical missions.  
Figure 16: _Cont._  
Figure 15: Example \(RTPO\)s passing through the \(L_{1}\) region in the Sun–Jupiter system. (**a**) Orbit-1: initials generated by the second intersection of the trajectory with the \(x\)-axis (see Figure 13b). (**b**) Orbit-2 and (**c**) Orbit-3: initials generated by the third intersection of the trajectory with the \(x\)-axis (see Figure 13c). (**d**) Orbit-4, (**e**) Orbit-5 and (**f**) Orbit-6: initials generated by the fourth intersection of the trajectory with the \(x\)-axis (see Figure 13d). The larger and the smaller blue dots represent the positions of \(P_{1}\) and \(P_{2}\), respectively.