## 3 The Desired Solutions  
In this part of the present research work, we use the AMS to acquire the approximate analytic solutions of the EOM (4), categorize the different cases of resonance, and extract both the solvability criteria and the ME. Then, we look at the oscillations of the system close to the static equilibrium position [31]. To accomplish this target, we approximate the functions \(\sin\Phi\) and \(\cos\Phi\) up to the third-order as follows  
\[\sin\Phi=\Phi-\frac{\Phi^{3}}{6},\qquad\qquad\cos\Phi=1-\frac{\Phi^{2}}{2}. \tag{5}\]  
The damping coefficients, force's amplitudes, moment, elliptical semi-axes, and other parameters can then be represented in terms of a small parameter \(0<\varepsilon<<1\) as follows  
\[ \begin{array}{llll}c_{1}=\varepsilon^{2}\widetilde{c}_{1},&c_{2}= \varepsilon^{2}\widetilde{c}_{2},&c_{3}=\varepsilon^{2}\widetilde{c}_{3},& \beta=\varepsilon\widetilde{\beta},\\ f_{1}=\varepsilon^{3}\widetilde{f}_{1},&m_{0}=\varepsilon^{3}\widetilde{m}_ {0},&a=\varepsilon^{2}\widetilde{a},&b=\varepsilon^{2}\widetilde{b},&\alpha =\varepsilon^{2}\widetilde{a}.\end{array}  \tag{6}\]  
As a result, we can express the functions \(z,\Phi\), and \(\eta\) in terms of \(\varepsilon\) and the new functions \(\widetilde{z},\widetilde{\Phi}\), and \(\widetilde{\eta}\) as follows  
\[z=\varepsilon\widetilde{z},\quad\Phi=\varepsilon\,\widetilde{\Phi},\quad\quad \eta=\varepsilon\widetilde{\eta}. \tag{7}\]  
According to the procedure of AMS, we can write these functions as follows  
\[ \begin{array}{llll}\widetilde{z}(\tau,\varepsilon)=\sum\limits_{i=1}^{3} \varepsilon^{i-1}\widetilde{\chi}_{i}(\tau_{0},\tau_{1},\tau_{2})+O( \varepsilon^{3}),\\ \widetilde{\Phi}(\tau,\varepsilon)=\sum\limits_{i=1}^{3}\varepsilon^{i-1} \widetilde{\psi}_{i}(\tau_{0},\tau_{1},\tau_{2})+O(\varepsilon^{3}),\\ \widetilde{\eta}(\tau,\varepsilon)=\sum\limits_{i=1}^{3}\varepsilon^{i-1} \widetilde{\gamma}_{i}(\tau_{0},\tau_{1},\tau_{2})+O(\varepsilon^{3}),\end{array}  \tag{8}\]  
where \(\tau_{n}=\varepsilon^{n}\tau\ \left(n=0,1,2\right)\) denote new time dependent scales on \(\tau\), in which \(\tau_{0}\) and \(\tau_{k}\ \left(k=1,2\right)\) denote a fast and slow time scales, respectively. Because of the smallness of \(\varepsilon\), the orders \(\varepsilon^{3}\) and higher have been excluded. In light of the supposed solutions (8), we need to transform the time derivatives regarding \(\tau\) into additional ones with respect to the scales \(\tau_{0},\tau_{1}\), and \(\tau_{2}\). Therefore, we consider the following differential operators  
\[ \begin{array}{llll}\frac{d}{d\tau}=D_{0}+\varepsilon D_{1}+\varepsilon^{2} D_{2},\\ \frac{d^{2}}{d\tau^{2}}=D_{0}^{2}+2\varepsilon D_{0}D_{1}+\varepsilon^{2}(D_{1 }^{2}+2D_{0}D_{2}),\end{array}  \tag{9}\]  
where \(D_{n}=\frac{\partial}{\partial\tau_{n}};\ n=1,2\). When (5)-(9) are substituted into (4), a family of partial differential equations (PDE) regarding \(\varepsilon\) arises. Equating the coefficients of the like powers of \(\varepsilon\) on both sides of each one of the families of PDE to gain the below groups of PDE gives as a result  
**Order of \(\varepsilon\)**  
\[ \begin{array}{llll}D_{0}^{2}\widetilde{\chi}_{1}+\widetilde{\chi}_{1}=0,\\ D_{0}^{2}\widetilde{\psi}_{1}+w_{1}^{2}\widetilde{\psi}_{1}=-r\widetilde{ \psi}w_{1}^{2},\\ D_{0}^{2}\widetilde{\gamma}_{1}+w_{2}^{2}\widetilde{\gamma}_{1}=-(D_{0}^{2} \widetilde{\psi}_{1}+w_{1}^{2}\widetilde{\psi}_{1}),\end{array}  \tag{10}\]\[