## 4 Resonance Transition Periodic Families in the CRTBP Model
### The RTPFs Related with the 4:3 Resonance  
Similar to the family \(IB(3-4)\) which connects the 3:2 resonance to the 4:3 resonance, the family \(IB(4-5)\) also has three components: (1) \"Branch-II\" of the 4:3 resonance, (2) \"Branch-I\" of the 5:4 resonance, and (3) the branch of the near-circular family. We continue the family \(IB(4-5)\) from the Sun-Earth system to the Sun-Jupiter system. A similar breakup and recombination process happens during the process of increasing \(\mu\). Characteristic curves of some mass parameters during the process of increasing \(\mu\) from the Sun-Earth system (\(\mu_{SE}\)) to the Sun-Jupiter system (\(\mu_{SJ}\)) are displayed in Figure 8. In the \(T-C\) as well as the \(x_{0}-C\) plane, the characteristic curves of the family \(IB(4-5)\) have no significant change from the Sun-Earth system to the system with \(\mu=5\times 10^{-5}\). 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 4:3 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 5:4 resonance, the orbital period increases first, then decreases when it reaches a certain value, and finally continues to increase. The characteristic curve of the family \(IB(4-5)\) for the system with \(\mu=10^{-4}\) is separately represented by the yellow line in Figure 9a. Similarly, the branches near the 4:3 resonance and the 5:4 resonance come closer to each other with the increasing \(\mu\). Different from the case of the family \(IB(3-4)\), a closed characteristic curve of the family \(IB(4-5)\) already forms in the \(T-C\) plane as the \(\mu\) value increases to \(\mu=5\times 10^{-4}\). This means the breakup and recombination of the family \(IB(4-5)\) happens for smaller \(\mu\) values when compared with the family \(IB(3-4)\). When the mass parameter further increases to \(\mu_{SJ}\), the size of the closed curve becomes smaller, which is represented by the red line in Figure 9b. Figure 7: Characteristic curves of the family \(IB(3-4)\) and two _RTPFs_ in the system for \(\mu=5\times 10^{-4}\) (**a**) and \(\mu=\mu_{SJ}\) (**b**). In the process of mass parameter continuation from \(\mu=10^{-4}\) to \(\mu_{SJ}\), most family members of the family \(IB(4-5)\) in the system with \(\mu=10^{-4}\) still belong to the new family \(IB(4-5)\) in the Sun-Jupiter system (\(\mu_{SJ}\)). At the same time, small parts of the family \(IB(4-5)\) in the system with \(\mu=5\times 10^{-4}\), especially for those close to the 4:3 resonance and the 5:4 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-4:3(a)_ and _RTPF-4:3(b)_ (see the green and purple line in Figure 9a). The _RTPF-4:3(a)_ combines with some family members in \"Branch-II\" of the 4:3 resonance to form a new _RTPF-4:3(a)_ in the Sun-Jupiter system (see the green line in Figure 9b). Similarly, the _RTPF-4:3(b)_ combines with some family members in \"Branch-I\" of the 5:4 resonance to form a new _RTPF-4:3(b)_) in the Sun-Jupiter system (see the purple line in Figure 9b). Example orbits from the _RTPF-4:3_(a)_ and the _RTPF-4:3(b)_ are presented in Figure 11c,d, respectively. The shapes of the orbits in _RTPF-4:3(a)_ and the _RTPF-4:3_(b)_ are similar. Both of them have five apogees in the synodic frame, two of which are near the collinear libration point \(L_{1}\). Part of the orbits close to the collinear libration point resembles the planar Lyapunov periodic orbits around the \(L_{1}\) point and the remaining part of the orbits resembles the 4:3 resonant periodic orbits (see the Type-II in Figure 2b).