## 2 Model Framework
### Periodic Orbits in the CRTBP Model  
The CRTBP model is an autonomous dynamical system. In general, equations of motion (see Equation (2)) describing an autonomous system can be written as  
\[\dot{\mathbf{X}}=F(\mathbf{X}) \tag{5}\]  
where \(\mathbf{X}\) denotes the state vector of the small particle \(P\). The solution to this equation is  
\[\mathbf{X}(t)=\Phi(\mathbf{X}_{0},t) \tag{6}\]  
where \(\mathbf{X}_{0}\) is the initial state vector at the initial time \(t_{0}=0\). The condition that a periodic orbit needs to satisfy is  
\[\mathbf{X}(\mathbf{X}_{0},T)-\mathbf{X}_{0}=0 \tag{7}\]  
where \(T\) is the orbital period. To find a periodic orbit is to determine the initial state vector \(\mathbf{X}_{0}\) and the period \(T\) that satisfy Equation (7). In total, Equation (7) has seven variables (\(\mathbf{X}\) and \(T\)), which can be solved by the Newton-Raphson iteration method. Because of the existence of the Jacobi integral (see Equation (4)), only five out of the six equations in Equation (7) are independent. Therefore, we need to fix two variables out of the seven in Equation (7) when solving this equation. Once a solution, i.e., a periodic orbit of Equation (7), is obtained, the whole family of periodic orbits can be generated by the well-known predictor-corrector algorithm [27].  
In this work, we focus on two types of periodic orbits in the CRTBP model. The first type of periodic orbit is the planar Lyapunov periodic orbit around the collinear libration points \(L_{1}\) and \(L_{2}\). Families of planar Lyapunov periodic orbits around \(L_{1}\) and \(L_{2}\) in the Sun-Earth system are shown as examples in Figure 1a,b, which have been extensively used by previous researchers. Details to compute these periodic orbits can be found in reference [26]. The second type of periodic orbit is the resonant periodic orbit. Orbital resonance is defined by the ratio \(p\):\(q\) between the orbital periods, where \(p\) and \(q\) are incommensurable positive integers. Neglect the secondary's gravity, i.e., treat the massless particle's orbit around the primary as an unperturbed Keplerian orbit. We denote the orbital period of the massless particle \(P\) as \(T_{p}\) and the secondary's orbital period as \(T_{s}\). A \(p\):\(q\) resonant periodic orbit in the synodic frame should strictly satisfy [26; 28]  
\[\frac{T_{s}}{T_{p}}=\frac{p}{q} \tag{8}\]  
When \(p>q\) (\(p<q\)), the resonance is interior (exterior) resonance. Denote \(k\) as  
\[k=|p-q| \tag{9}\]  
\begin{table}
\begin{tabular}{c c} \hline
**System/Name** & **Values** \\ \hline Two-body system (\(\mu_{0}\)) & 0 \\ \hline Sun–Earth system (\(\mu_{SE}\)) & 3.003480575402412 \(\times\)\(10^{-6}\) \\ \hline Sun–Jupiter system (\(\mu_{SJ}\)) & 9.538811803631013 \(\times\)\(10^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 1: The mass parameter values of the unperturbed two-body system (\(\mu_{0}\)), the Sun–Earth system (\(\mu_{SE}\)), and the Sun–Jupiter (\(\mu_{SJ}\)) system.  
in which \(k\geq 0\), and \(k\) is the resonance order. In this work, \(k=1\). One remark is that, in the CRTBP model, the orbital period of the massless particle does not exactly satisfy Equation (8) due to the secondary's perturbation. Example orbits of the first-order interior resonant periodic orbits (3:2, 4:3, and 5:4 resonances), as well as the exterior resonant periodic orbits (2:3, 3:4, and 4:5 resonances) in the Sun-Earth system are presented in Figure 2.