## 4 Resonance Transition Periodic Families in the CRTBP Model
### The RTPFs Related with the 3:2 Resonance  
Even for very small \(\mu\
eq 0\), the _RTPFs_ already exist, although they are not shown in the above characteristic curves of periodic families. In this section, we take the family \(IB(3-4)\) as an example to show the process of family breakup and recombination with the _RTPFs_. This process can be viewed as the first-order resonances (here the 3:2 and the 4:3 ones) gradually being overlapped by the 1:1 resonance. In Figure 6, the family \(IB(3-4)\) is continued from the Sun-Earth system to the Sun-Jupiter system, and characteristic curves of some example mass parameters are given. Comparing the characteristic curves in  
Figure 4: Characteristic curves of the circular families and the first-order resonant periodic families in the CRTBP model of the Sun–Earth system (\(\mu_{SE}\)). (**a**): Interior circular family and the interior first-order resonant periodic families (3.2, 4:3, and 5:4 resonances). (**b**): Exterior circular family and the exterior first-order resonant periodic families (2:3, 3:4, and 4:5 resonances). The abscissa is half the orbital period (\(T/2\)) in the synodic frame and the ordinate is the Jacobi constant \(C\). Figure 5: The family \(IB(3-4)\) (**a**) and the family \(OB(3-4)\) (**b**) in the Sun–Earth system (\(\mu_{SE}\)), shown in the \(T-C\) plane. The abscissa is half the orbital period (\(T/2\)) in the synodic frame and the ordinate is the Jacobi constant \(C\). Figures 6a with 5a, we can see that the genealogy of family \(IB(3-4)\) becomes more complex and distorted with the increasing \(\mu\). Taking a closer look at the characteristic curves in Figure 6, we can see that there is no significant change of the family \(IB(3-4)\) in the continuation process starting from the Sun-Earth system (\(\mu_{SE}\)) to the system with \(\mu=10^{-4}\). However, as the mass parameter further increases to \(\mu=5\times 10^{-4}\), noticeable changes in the characteristic curve occur in the \(T-C\) plane. The changes are mainly reflected in two aspects: (1) for the branch near the 3:2 resonance, the characteristic curve bends as the Jacobi constant decreases, accompanied by the process of increasing, decreasing, and then increasing orbital period; (2) for the branch near the 4:3 resonance, the orbital period increases first, then decreases when it reaches a certain value, and finally continues to increase. The detailed characteristic curve of the family \(IB(3-4)\) in the system with \(\mu=5\times 10^{-4}\) is separately presented by the orange line in Figure 7a. As the \(\mu\) value further increases to \(\mu=\mu_{SJ}\), a closed characteristic curve of family \(IB(3-4)\) forms in the \(T-C\) plane, which is represented by the red line in Figures 6 and 7b. In the process of increasing \(\mu\) from \(\mu=5\times 10^{-4}\) to \(\mu_{SJ}\), most family members of family \(IB(3-4)\) in the system with \(\mu=5\times 10^{-4}\) still belong to the new family \(IB(3-4)\) in the Sun-Jupiter system (\(\mu_{SJ}\)). At the same time, small parts of the family \(IB(3-4)\) in the system with \(\mu=5\times 10^{-4}\), especially for those close to the 3:2 resonance and the 4:3 resonance, combine with the RTPFs to form the new RTPFs in the Sun-Jupiter system. The bifurcation and recombination details are as follows. In the system with \(\mu=5\times 10^{-4}\), two RTPFs exist (see the green and purple line in Figure 7a). The green family is denoted as RTPF-3:2(a), and the purple family is denoted as RTPF-3:2(b). Example orbits in these two families are shown in Figure 11a,b, respectively. The RTPF-3:2(a) combines with some family members in \"Branch-II\" of the 3:2 resonance to form a new RTPF-3:2(a) in the Sun-Jupiter system (see the green line in Figure 7b). Similarly, the RTPF-3:2(b) combines with some family members in \"Branch-I\" of the 4:3 resonance to form a new RTPF-3:2(b) in the Sun-Jupiter system (see the purple line in Figure 7b). The shapes of the orbits in RTPF-3:2(a) and RTPF-3:2(b) are similar.