# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 4 Stability of Motion Near Equilibrium Points
### Stability at Collinear Points  
The stability of the collinear equilibrium point of R2+2 BP can be approximated by Equation (23), with condition \(y_{1}=y_{2}=0\), i.e.,  
\[U_{x_{1}x_{1}}^{0} = \mu_{1}\bigg{(}n+\frac{2(1-\mu)}{|x_{1}-\mu|^{3}}+\frac{2\mu}{|x_ {1}-\mu+1|^{3}}+\frac{\mu_{2}}{|x_{1}-x_{2}|^{3}}\bigg{)}+\frac{3M_{b}x_{1}^{ 2}}{\big{(}T^{2}+x_{1}^{2}\big{)}^{\frac{3}{2}}} \tag{26}\] \[-\frac{M_{b}}{\big{(}T^{2}+x_{1}^{2}\big{)}^{\frac{3}{2}}},\] \[U_{y_{1}y_{1}}^{0} = \mu_{1}\bigg{(}n-\frac{(1-\mu)}{|x_{1}-\mu|^{3}}-\frac{\mu}{|x_{1} -\mu+1|^{3}}-\frac{\mu_{2}}{|x_{1}-x_{2}|^{3}}\bigg{)}-\frac{M_{b}}{\big{(}T^{ 2}+x_{1}^{2}\big{)}^{\frac{3}{2}}}, \tag{27}\]  
and  
\[U_{x_{1}y_{1}}^{0}=0, \tag{28}\]  
for \(i=1\). Consequently, \(b^{2}-4c>0\), and hence, the characteristic equation yields at least one positive real root. This results in the instability at the collinear points. Tables 4-6 provide the collinear equilibrium points for the following systems: Sun-Saturn with the Kuiper belt, Sun-Mars with an asteroid belt, and the Proxima Centauri system with a dust disc. With the help of these collinear equilibrium points, we calculated the stability of these systems using Equations (23)-(28). We found that real nonzero eigenvalues occur with opposite signs and purely imaginary eigenvalues with opposite signs. Thus, the collinear equilibrium points have a saddle\(\times\)center behavior [25], i.e., all collinear equilibrium points are unstable. Again, as shown in Figure 7, we selected the Sun-Saturn system with the Kuiper belt with \(\mu_{1}=10^{-7}\), and plotted the eigenvalues against \(\mu_{2}\in(10^{-10},10^{-7})\). Continuous lines indicate the nonzero real parts of eigenvalues, denoted as \(\alpha\), and the dashed lines represent the nonzero imaginary parts of eigenvalues, denoted as \(\beta\). Figure 7a,b presents the stability near \(L_{1}\), Figure  
The characteristic Equation (23) provides the eigenvalues to study the stability of the noncollinear equilibrium points in the presence of the planetesimal belt effect. The kinds of roots of Equation (23) depend on the parameters \(\mu\), \(\mu_{1}\), \(\mu_{2}\), \(M_{b}\), and \(T\). The noncollinear equilibrium points are shown in Tables 11-14 for different planetary systems. We analyzed the stability of th noncollinear equilibrium points using Equations (23)-(28), and we found that the real nonzero eigenvalues occur with opposite signs and purely imaginary eigenvalues occur with opposite signs. Thus, the noncollinear equilibrium points have a saddle\(\times\)center behavior. In Figures 8 and 9, for the Sun-Saturn system with the Kuiper belt with \(\mu_{1}=10^{-7}\), we plot the eigenvalues against \(\mu_{2}\in(10^{-10},10^{-7})\). Figure 8 describes the stability near \(L_{4}\). Figure 8a,b presents the eigenvalues of the perpendicular equilibrium points, and Figure 8c,d depicts the eigenvalues of the inline equilibrium points near \(L_{4}\). It can be observed in Figure 8a-d that real nonzero positive eigenvalues exist; thus, all noncollinear equilibrium points near \(L_{4}\) are unstable in the considered system. Similarly, in Figure 9, we plot the eigenvalues of the equilibrium positions near \(L_{5}\). Figure 9a,b shows the eigenvalues of the perpendicular equilibrium points, whereas Figure 9c,d depicts the eigenvalues of the inline equilibrium points. Clearly, it can be observed that