## 5 Steady State Solutions  
The major objective of the present section is to study the oscillations of the examined system in the case of steady state. From the equations of system (25), we can obtain both of the modified phases \(\theta_{j}\ (j=1,2,3)\) and amplitudes \(a_{j}\) in the steady state case. Alternatively, the zero values of the left-hand sides of the equations of this system are considered. Therefore, we consider \(\frac{da_{j}}{2\tau}=0\) and \(\frac{d\theta_{j}}{2\tau}=0\)[32], to obtain the next algebraic system of six equations of the functions \(\theta_{j}\) and \(a_{j}\). \[ \begin{array}{l}f_{1}\sin\theta_{1}-c_{1}a_{1}=0,\\ f_{1}\cos\theta_{1}+2a_{1}[\sigma_{1}-\alpha z_{\tau}-\frac{(1+w_{1}^{2})^{2}} {2(1-w_{1}^{2})}r^{2}\beta^{2}-\frac{3(1-w_{1}^{2})}{4(1-4\,w_{1}^{2})}w_{1}^{2 }a_{2}^{2}]=0,\\ m_{0}\sin\theta_{2}-w_{1}c_{2}a_{2}=0,\\ m_{0}\cos\theta_{2}+2\,w_{1}a_{2}\bigg{\{}\sigma_{2}-w_{1}(\alpha z_{\tau}^{2}- r^{2}\beta+\frac{(1-w_{1}^{2})}{8(1-4\,w_{1}^{2})}[12a_{1}^{2}-(1+8\,w_{1}^{2})a_{2}^{2 }]\\ -\frac{r^{2}\beta^{2}}{4(1-w_{1}^{2})\,w_{2}^{2}}[(14-3r^{2})\,w_{2}^{2}-2(4- w_{2}^{2})\,w_{1}^{4}+(8+3r^{2}w_{2}^{2})\,w_{1}^{2}])\bigg{\}}=0,\\ \frac{(1-w_{1}^{2})}{(1-4\,w_{1}^{2})}w_{1}^{2}a_{2}^{3}\sin\theta_{3}+w_{2}c_ {3}a_{3}=0,\\ \frac{(1-w_{1}^{2})}{(1-4\,w_{1}^{2})}w_{1}^{2}a_{2}^{3}\cos\theta_{3}-2\,w_{2 }a_{3}\bigg{\{}\sigma_{3}+3\,w_{1}\bigg{\{}\alpha z_{\tau}^{2}-r^{2}\beta+ \frac{(1-w_{1}^{2})}{8(1-4\,w_{1}^{2})}[12a_{1}^{2}\] \[-(1+8\,w_{1}^{2})a_{2}^{2}]-\frac{r^{2}\beta^{2}}{4(1-w_{1}^{2})\,w_{2}^ {2}}[(14-3r^{2})\,w_{2}^{2}-2(4-w_{2}^{2})\,w_{1}^{4}+(8+3r^{2}w_{2}^{2})\,w_{ 1}^{2}]\bigg{\}}\\ -\frac{3m_{0}}{2w_{1}a_{2}}\cos\theta_{2}+\frac{1}{2w_{2}}[\beta(w_{1}^{2}-w_ {2}^{2})+\frac{1}{2}w_{1}^{2}a_{2}^{2}+\frac{\beta^{2}}{4w_{2}^{2}}(4r^{2}w_{ 2}^{4}+4\,w_{2}^{2}(w_{1}^{2}-w_{2}^{2})\\ +(w_{1}^{2}-w_{2}^{2})^{2})]\big{\}}=0,\end{array}  \tag{26}\]  
Now, we can remove the adjusted phases \(\theta_{j}\) from the preceding system to produce the following three non-linear algebraic equations between longitudinal amplitude \(a_{1}\), the swing oscillations \(a_{2}\), and the frequency represented by the detuning parameters \(\sigma_{j}\) and the absorber's amplitude \(a_{3}\). \[ \begin{array}{l}f_{1}^{2}\hskip 28.452756 pt=a_{1}^{2}\Bigg{\{}c_{1}^{2}+4[ \sigma_{1}-\alpha z_{\tau}-\frac{(1+w_{1}^{2})^{2}}{2(1-w_{1}^{2})}r^{2}\beta^ {2}-\frac{3(1-w_{1}^{2})}{4(1-4\,w_{1}^{2})}w_{1}^{2}a_{2}^{2}]^{2}\Bigg{\}}, \\ m_{0}^{2}\hskip 28.452756 pt=w_{1}^{2}a_{2}^{2}\bigg{\{}c_{2}^{2}+4[ \sigma_{2}-w_{1}(\alpha z_{\tau}^{2}-r^{2}\beta+\frac{(1-w_{1}^{2})}{8(1-4\,w_ {1}^{2})}[12a_{1}^{2}-(1+8\,w_{1}^{2})a_{2}^{2}] \tag{27}\] \[-\frac{r^{2}\beta^{2}}{4(1-w_{1}^{2})w_{2}^{2}}[(14-3r^{2})\,w_{ 2}^{2}-2(4-w_{2}^{2})\,w_{1}^{4}+(8+3r^{2}w_{2}^{2})\,w_{1}^{2}])]^{2}\bigg{\}},\\ \frac{(1-w_{1}^{2})^{2}}{(1-4\,w_{1}^{2})^{2}}w_{1}^{4}\hskip 28.452756 pt=w_{2}^{2}a_{3} ^{2}\bigg{\{}c_{3}^{2}+4[\sigma_{3}+3\sigma_{2}+\frac{1}{8w_{2}^{2}}(\beta^{2} w_{1}^{4}+2(2\beta+\beta^{2}+a_{2}^{2})w_{1}^{2}w_{2}^{2}\] \[-\beta[4+(3-4r^{2})\beta]w_{2}^{4})]^{2}\big{\}}.\end{array}  \tag{28}\]  
Stability testing is considered a crucial aspect of the vibrations in the steady state case. To explore such a circumstance, the behavior of the system in a domain relatively near to fixed points is investigated. Therefore, the substitutions listed below are employed in (25) to achieve this purpose. \[ \begin{array}{l}a_{1}=a_{10}+a_{11},\hskip 28.452756 pt\theta_{1}=\theta_{10}+ \theta_{11},\\ a_{2}=a_{20}+a_{21},\hskip 28.452756 pt\theta_{2}=\theta_{20}+\theta_{21},\\ a_{3}=a_{30}+a_{31},\hskip 28.452756 pt\theta_{3}=\theta_{30}+\theta_{31}.\end{array}  \tag{29}\]  
Here, \(a_{j0}\) and \(\theta_{j0}\ (j=1,2,3)\) denote the steady state solutions, whereas \(a_{j1}\) and \(\theta_{j1}\) represent relatively small disturbances in relation to \(a_{j0}\) and \(\theta_{j0}\).