## 5 Computation of the Symmetric RTPOs
### Methodology  
* Firstly, we restrict the starting position on the Lyapunov orbit around the collinear \(L_{1}\) or \(L_{2}\) point in the Sun-Jupiter system and extend the method in the previous study [35] to the Sun-Jupiter system. The amplitude of the Lyapunov orbit is defined as the distance between its right intersection point (see the yellow dot in Figure 12) with the \(x\)-axis and the \(L_{1}\) or the \(L_{2}\) point. In our work, it is fixed at 0.05 (dimensionless unit) of the Sun-Jupiter distance. One remark is that a different value can be chosen. Initial states of the planar Lyapunov orbit are denoted as \((x_{0},y_{0},0,\dot{x_{0}},y_{0},0)\). The Lyapunov orbit is symmetric about the \(x\)-axis. The initial position is placed on the positive \(x\)-axis and the initial velocity is exactly along the positive \(y\)-direction. Therefore, the initial state of the Lyapunov orbit is \((x_{0},0,0,0,y_{0},0)\);
* We add a tiny perturbation \(\Delta v\) along the direction of initial velocity \(\dot{y}_{0}\) on the Lyapunov orbit. The velocity change \(\Delta v\) is in the range of \([-0.01,0.01]\) in this study, i.e., the departure velocity \(\dot{y}_{d}\) at the initial position is restricted to be within \([-0.01+\dot{y}_{0},0.01+\dot{y}_{0}]\), where 0.01 is a dimensionless velocity in the Sun-Jupiter system. Due to the linear stability of the collinear libration point, the orbit can leave the vicinity of the collinear libration point with this tiny perturbation to the initial velocity. The schematic diagram of generating initials of the RTPOs is displayed in Figure 12;
* By integration, the trajectory leaves the initial position on the \(x\)-axis. If the initial velocity \(\dot{y}_{d}\) is appropriate, it crosses the \(x\)-axis again one or more times after some integration time. Generally, at the new intersection point (see the green point in Figure 12), the velocity is not perpendicular to the \(x\)-axis. We record the \(x\) component of the velocity at the new intersection point and denote it as \(\dot{x}_{  
CRTBP model \((x,y,z,\dot{x},\dot{y},\dot{z},t)\leftarrow\rightarrow(x,-y,z,-\dot{x},\dot{y},- \dot{z},-t)\)[26], if \(\dot{x}_{T/2}=0\), the trajectory from \(t=0\) to \(t=T/2\) is half of a periodic orbit. From Figures 13 and 14, we can see that there exist some dots satisfying \(\dot{x}_{T/2}\approx 0\) within a certain tolerance, which means that we can use these initial states \((x_{0},0,0,0,\dot{y}_{d},0)\) where \(\dot{x}_{T/2}\approx 0\) to generate the periodic orbits, especially for the _RTPOs_. After finding these initial states \((x_{0},0,0,0,\dot{y}_{d},0)\), we can use the Newton iteration method to find the exact initial velocity \(\dot{y}_{d}\) that makes \(\dot{x}_{T/2}=0\). The same algorithm has been used to generate the symmetric horseshoe periodic orbits in the CRTBP model [33; 36].  
Figure 12: Schematic diagram of generation of the initials of _RTPOs_ in the Sun–Jupiter system.  
Figure 13: 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 \(L_{1}\) point with a Jacobi constant of \(C=3.00391283019118\). The purple dots indicate a positive \(\Delta v\) and the red points indicate a negative \(\Delta v\).