## 6 The Stability Analysis  
In this section, we investigate the model's stability as well as its non-linear evolution using the Routh-Hurwitz non-linear stability approach. It must be remembered that a damped spring connected with a transverse absorber under the action of \(F(t)\) and \(M(t)\). Some factors have been revealed to play a substantial impact in the stability processes such as damping's constants \(c_{j}\)\((j=1,2,3)\), frequencies \(\omega_{k}\)\((k=1,2)\), and parameters of detuning \(\sigma_{j}\). To gain the stability plots of system (25), a specific process with various parameters of the system has been used. The adjusted amplitudes \(a_{j}\)\((j=1,2,3)\) plotted via time are for various parametrical regions, in addition to the graphical representations  
Figure 21: Curves of the planes \(\sigma_{2}a_{j}\ (j=1,2,3)\) at \(c_{1}=0.01,\ c_{3}=0.2,\ \omega_{1}=3.354,\ \omega_{2}=3.131\), and \(\sigma_{1}=0\): (**a–c**) when \(c_{2}\)(=0.018, 0.021, 0.024). Figure 23: Curves of the planes \(\sigma_{2}a_{j}\)\((j=1,2,3)\) at \(c_{1}=0.01\), \(c_{2}=0.02\), \(\omega_{1}=3.354\), \(\omega_{2}=3.131\), and \(\sigma_{1}=-0.001\): (**a–c**) when \(c_{3}\)(=0.19, 0.2, 0.21). Figure 25: Curves of the planes \(\sigma_{2}a_{j}\ (j=1,2,3)\) at \(c_{1}=0.01,\ c_{2}=0.02,\ \omega_{1}=3.354,\ \omega_{2}=3.131\), and \(\sigma_{1}=0.001\): (a–c) when \(c_{3}(=\)0.19, 0.2, 0.21). Figure 28: Curves of the planes \(\sigma_{2}a_{j}\)\((j=1,2,3)\) at \(\omega_{2}=3.131,\ c_{1}=0.01,\ c_{2}=0.02,\ c_{3}=0.2\), and \(\sigma_{1}=0.001\): (**a**) when \(\omega_{1}=3.316,\ (\textbf{b}\)) when \(\omega_{1}=3.354,\ (\textbf{c}\)) when \(\omega_{1}=3.391,\) and (**d**,**e**) when \(\omega_{1}(=\)3.316, 3.354, 3.391). Alternatively, Figures 20-22 explore the impact of the \(c_{2}\)(=0.018, 0.021, 0.024) values on \(a_{j}\) when \(c_{1}=0.01\), \(c_{3}=0.2\), \(\omega_{1}=3.354\), and \(\omega_{2}=3.131\). One critical fixed point has been found with the stability and instability areas \(-0.5\leq\sigma_{2}\leq-0.04\) and \(-0.04<\sigma_{2}\leq 0.5\), respectively. The location of the peak points has been shifted with the increasing of \(c_{2}\) values as seen in parts (a) and (b) of Figures 20-22, while it will be stationary as in part (c) of these figures. In addition, Figures 23-25 reveal the influence of \(c_{3}\)(=0.19, 0.2, 0.21) on the amplitudes \(a_{j}\) when \(c_{1}=0.01\), \(c_{2}=0.02\), \(\omega_{1}=3.354\), and \(\omega_{2}=3.131\), in which one peak and one critical point have been explored for each curve. The stable and unstable fixed points have been found in the ranges \(-0.5\leq\sigma_{2}\leq-0.04\) and \(-0.04<\sigma_{2}\leq 0.5\), respectively. The influence of the frequencies \(\omega_{1}(=\)3.316, 3.354, 3.391) at \(\omega_{2}=\) 3.131 and \(\omega_{2}(=\)3.212, 3.131, 3.084) at \(\omega_{1}=\) 3.354 has been examined through the curves of Figures 26-28, respectively. These figures are calculated when \(c_{1}=0.01\), \(c_{2}=0.02\), and \(c_{3}=0.2\) at \(\sigma_{1}=-0.001\), \(\sigma_{1}=0\), and \(\sigma_{1}=0.001\). Based on the included curves in Figures 26-28, we conclude that there exists one peak point with different locations, and each curve has just one essential fixed point. It is clear that \(\omega_{1}\) has a significant impact on the frequency response curves because the equations of system (25) depend directly on the frequencies parameters. Moreover, the stable and unstable regions of the fixed points are calculated as in Table 1.  
The above remarks can be applied to the curves of Figures 29-31 when the frequency values \(\omega_{2}(=\)3.212, 3.131, 3.084) are considered. The ranges of stable fixed points and unstable ones have been given in Table 2.