## 3 The Desired Solutions
An inspection of the curves of these figures shows that we have spirals curves that are directed to one point, which gives an impression of the steady motion of these amplitudes and phases. Figure 7: Phase planes \(a_{j}\theta_{j}\)\((j=1,2,3)\) at \(c_{2}=0.02,\ c_{3}=0.2,\ \omega_{1}=3.354\), and \(\omega_{2}=3.131\): (**a–c**) when \(c_{1}(=0.01,0.015,0.02)\). Figure 9: Phase planes \(a_{j}\theta_{j}\;(j=1,2,3)\) when \(c_{1}=0.01\), \(c_{2}=0.02\), \(\omega_{1}=3.354\) and \(\omega_{2}=3.131\): (**a**,**b**) when \(c_{3}(=0.19,0.2,0.21)\), (**c**) when \(c_{3}=0.19\), (**d**) when \(c_{3}=0.2\), and (**e**) when \(c_{3}=0.21\). Figure 10: Phase planes \(a_{j}\theta_{j}\;(j=1,2,3)\) at \(\omega_{2}=3.131\), \(c_{1}=0.01\), \(c_{2}=0.02\), and \(c_{3}=0.2\): (**a**,**b**) when \(\omega_{1}(=3.316,3.354,3.391)\), (**c**) when \(\omega_{1}=3.316\), (**d**) when \(\omega_{1}=3.354\), and (**e**) when \(\omega_{1}=3.354\). Parts of Figures 7-9 are drawn when \(c_{1},c_{2}\), and \(c_{3}\) have different values, respectively, to reveal the variation of curves of the phase planes \(a_{j}\theta_{j}\ \ (j=1,2,3)\) with these values, while Figures 10 and 11 describe the change of these planes that happened at different values of \(\omega_{1}\) and \(\omega_{2}\), respectively. According to the curves of these figures and the system of equations (25), we observe that the plane \(a_{1}\theta_{1}\) is impacted by the various values of \(c_{1}\), as seen in Figure 7a, while there is no variation of the curves of planes \(a_{2}\theta_{2}\) and \(a_{3}\theta_{3}\) as indicated in Figure 7b,c. The curves of the phase planes \(a_{2}\theta_{2}\) and \(a_{3}\theta_{3}\) have been impacted with the variation of \(c_{2}\) values of as seen in Figure 8b,c. On the other hand, there is no observed variation of the curves drawn in plane \(a_{1}\theta_{1}\) when \(c_{2}\) changes, as noticed in Figure 8a. According to the plotted curves in Figure 9, we can see that the curves shown in parts (a) and (b) that describe the phase planes \(a_{1}\theta_{1}\) and \(a_{2}\theta_{2}\) respectively, have no variation with various values of \(c_{3}\). The good impact of the values of \(c_{3}\) is observed in parts (c), (d), and (e) of Figure 9 for the phase plane \(a_{3}\theta_{3}\). The influence of the frequencies \(\omega_{1}(=\)3.316, 3.354, 3.391) and \(\omega_{2}(=\)3.212, 3.131, 3.084) on the phase plane diagrams \(a_{j}\theta_{j}\ \ (j=1,2,3)\) is observed from the curves of Figures 10 and 11, respectively. Therefore, we can say that these curves have a spiral form from the outside to the inside, and it is directed toward a single point for each curve, which means that all values of \(\omega_{k}\ (k=1,2)\) have a significant impact on the curves of these planes. The reason goes back to the equations of system (25) that depend explicitly on \(\omega_{k}\ (k=1,2)\). It is important to remember that the obtained approximate solutions \(z,\Phi\), and \(\eta\) describe the spring's elongation, the rotation angle at the point \(O_{1}\), and the elongation of the transverse absorber, respectively. Figures 12-16 depict the graphical representations of these solutions over time \(\tau\), revealing their behaviors across the analyzed time interval as presented in parts (a), (b), and (c) when \(\tau\in[0,100]\) and in parts (d), (e), and (f) when \(\tau\in[0,1000]\) of these figures. Figures 12-14 are performed at \(\omega_{1}=3.354\) and \(\omega_{2}=3.131\) at different values of \(c_{1}(=0.01,0.015,0.02)\), \(c_{2}(=0.018,0.021,0.024)\), and \(c_{3}(=0.19,0.2,0.21)\) respectively. When these figures are compared with each other, we can see that the variations with different values of \(c_{j}\ (j=1,2,3)\) seem to be slight. This is because  
Figure 13: Change of \(z,\Phi\), and \(\eta\) via \(\tau\) when \(c_{1}=0.01,\ c_{2}(=0.018,0.021,0.024),\ c_{3}=0.2,\)\(\omega_{1}=3.354\) and \(\omega_{2}=3.131\): (**a**–**c**) at \(\tau\in[0,100]\) and (**d**–**f**) at \(\tau\in[0,1000]\).