## 4 Resonance Transition Periodic Families in the CRTBP Model
### The RTPF's Related with the 2:3 Resonance  
The exterior first-order resonant periodic families have similar breakup and recombination processes as the interior resonances during the process of increasing \(\mu\). Taking the family \(OB(3-4)\) as an example, the breakup and recombination process with the _RTPF_s are as follows. The family \(OB(3-4)\) in the CRTBP model has three components: (1) \"Branch-I\" of the 2:3 resonance, (2) \"Branch-II\" of the 3:4 resonance, and (3) the branch of the near-circular family (see Figure 5b). We continue the family \(OB(3-4)\) from the Sun-Earth system to the Sun-Jupiter system. Characteristic curves of some mass parameters during the process of increasing \(\mu\) from the Sun-Earth system (\(\mu_{SE}\)) to the Sun-Jupiter system (\(\mu_{SI}\)) are displayed in the \(T-C\) plane (see Figure 10a). Similarly, when the mass parameter is small (from the Sun-Earth system to the system with \(\mu=5\times 10^{-5}\)), there are no significant changes in the characteristic curves. However, when the mass parameter further increases to \(\mu=10^{-4}\), noticeable changes of the characteristic curve occur in the \(T-C\) plane in two ways: (1) for the branch near the 2:3 resonance, the characteristic curve bends as the Jacobi constant decreases, accompanied with the process of increasing, decreasing, and then increasing orbital period; (2) for the branch near the 3:4 resonance, the orbital period increases first, then decreases when it reaches a certain value, and finally continues to increase. A closed characteristic curve of family \(OB(3-4)\) forms when \(\mu=5\times 10^{-4}\) (see the orange line in Figure 10a). When the mass parameter further increases to \(\mu_{SI}\), the size of the closed curve becomes smaller, which is represented by the red line in Figure 10.  
With the increasing mass parameter \(\mu\), most family members of the family \(OB(3-4)\) in the system with \(\mu=10^{-4}\) still belong to the new family \(OB(3-4)\) in the Sun-Jupiter system (\(\mu_{SI}\)). At the same time, small parts of the family \(OB(3-4)\) in the system with \(\mu=10^{-4}\), especially for those close to the 2:3 resonance and the 3:4 resonance, combine with the _RTPF_s to form the new _RTPF_s in the Sun-Jupiter system. Two _RTPF_s work in the bifurcation and recombination process, which are denoted as _RTPF-2:3(a)_ and _RTPF-2:3(b)_. _RTPF-2:3(a)_ combines with some family members in \"Branch-I\" of the 2:3 resonance to form a new _RTPF-2:3(a)_ in the Sun-Jupiter system (see the green line in Figure 10b). Similarly, _RTPF-2:3(b)_ combines with some family members in \"Branch-II\" of the 3:4 resonance to  
Figure 10: (**a**) Characteristic curves of the family \(OB(3-4)\) continued from the Sun–Earth system (\(\mu_{SE}\)) to the Sun–Jupiter system (\(\mu_{SI}\)). Different colors indicate different mass parameters. (**b**) Characteristic curves of the family \(OB(3-4)\) and two _RTPF_s in the system with \(\mu_{SI}\). \(T/2\) is half the orbital period in the synodic frame. \(C\) is the Jacobi constant. \(x_{0}\) is the initial state of the periodic orbits on the \(x\)-axis. form a new _RTPF-2:3(b)_) in the Sun-Jupiter system (see the purple line in Figure 10b). Example orbits from _RTPF-2:3(a)_ and _RTPF-2:3(b)_ in the Sun-Jupiter system are presented in Figure 11e,f, respectively. The shapes of the orbits in _RTPF-2:3(a)_ and the _RTPF-2:3(b)_ are similar. Both of them have three perigees in the synodic frame, two of which are near the collinear libration point \(L_{2}\) (see Figure 1b). Part of the orbits close to the collinear libration point resembles the planar Lyapunov periodic orbits around the \(L_{2}\) point and the remaining part of the orbits resembles the 2:3 resonant periodic orbits (see Figure 2d (_Type-I_)). In addition, characteristic curves of the two _RTPFs_ in Figure 10b (see the green and purple line) indicate that the difference in Jacobi constants between these two _RTPFs_ is not significant.