# The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance
## 3 The Perturbation Technique
\begin{array}{ll}2i\omega_{1}\frac{\partial\Lambda_{1}}{\partial\tau_{2}}\ +\ i \omega_{1}\widetilde{c}_{1}\Lambda_{1}-\frac{\widetilde{\beta}^{2}\omega_{1}^{6} \Lambda_{1}}{4\big{(}\omega_{1}^{2}-\omega_{2}^{2}\big{)}^{3}}[-\omega_{1}^{2} \big{(}4\omega_{1}^{2}+11\big{)}+\omega_{2}^{2}\big{(}7-4\omega_{2}^{2}\big{)} \\ +8\omega_{1}^{2}\omega_{2}^{2}\big{]}-\frac{\widetilde{f}_{1}}{2}e^{i\widetilde {\omega}_{1}\tau_{1}}=0,\\ 2i\frac{\partial\Lambda_{2}}{\partial\tau_{2}}+i\widetilde{c}_{2}\Lambda_{2}- \frac{6\omega_{1}^{4}\big{(}\omega_{1}^{2}-2\omega_{1}^{2}+1\big{)}}{\big{(} \omega_{1}^{2}-4\big{)}\big{(}\omega_{1}^{2}-\omega_{2}^{2}\big{)}^{2}} \Lambda_{1}\Lambda_{2}\overline{\Lambda}_{1}-\frac{\Lambda_{2}\Lambda_{3} \overline{\Lambda}_{3}}{\big{(}\omega_{2}^{2}-4\big{)}\big{(}\omega_{2}^{2}- \omega_{1}^{2}\big{)}}\\ \times\{6\omega_{1}^{2}\big{(}\omega_{2}^{2}-1\big{)}+\omega_{2}^{2}\big{[}6 \big{(}1-\omega_{2}^{2}\big{)}+\widetilde{\beta}\omega_{2}\big{(}2\omega_{2}^{ 2}+3\omega_{2}-2\big{)}\big{]}\}-\frac{\frac{2\overline{\Lambda}_{2}}{2\omega_{ 2}^{2}\big{(}\omega_{2}^{2}-4\big{)}}}{2\omega_{2}^{2}\big{(}\omega_{2}^{2}-4 \big{)}}\\ \times\big{(}\omega_{2}^{4}-25\omega_{2}^{2}+24\big{)}-\frac{\widetilde{f}_{2}}{ 2}e^{i\widetilde{\omega}_{2}\tau_{1}}=0,\\ 2i\omega_{2}\frac{\partial\lambda_{1}}{\partial\tau_{2}}\ +i\widetilde{c}_{3}\omega_{2} \Lambda_{3}-\frac{\left(2\omega_{1}^{2}-1\right)}{2\big{(}\omega_{1}^{2}-\omega_ {2}^{2}\big{)}^{2}}\Lambda_{3}\widetilde{\beta}^{2}\omega_{2}^{6}-\frac{\left[ 3+2\omega_{2}(\omega_{2}+2)\right]}{\omega_{2}(\omega_{2}+2)}\\ \times\Lambda_{2}\Lambda_{3}\overline{\Lambda}_{2}-\frac{\widetilde{f}_{3}}{2}e^{i \widetilde{\omega}_{3}\tau_{1}}=0.\end{array}  \tag{43}\]It is obvious that these conditions constitute a system of three non-linear PDEs in terms of the functions \(\mathrm{A}_{j}\ (j=1,\,2,\,3)\), in which \(\mathrm{A}_{j}=\mathrm{A}_{j}(\tau_{1},\tau_{2})\). Therefore, we can write them in polar forms as follows:  
\[\mathrm{A}_{j}=\frac{\widetilde{h}_{j}(\tau_{1},\tau_{2})}{2}e^{i}\psi_{(\tau_ {1},\tau_{2})},\qquad h_{j}=\tilde{e}\widetilde{h}_{j}\,\,\,\,\,\,\,(j=1,2,3). \tag{44}\]  
Here, \(\widetilde{h}_{j}\) and \(\psi_{j}\) are real functions of the amplitudes and phases, respectively, for the functions \(\xi\), \(\varphi\) and \(\chi\), while \(h_{j}\) are the amplitudes of \(y\), \(\phi\), and \(x\), as presented in the assumptions of Equations (11) and (12). Making use of the following modified phases:  
\[\theta_{j}(\tau_{1},\tau_{2})=\tau_{j}\widetilde{\varphi}_{j}-\psi_{j}(\tau_ {1},\tau_{2}) \tag{45}\]  
and the time derivative operators (13), we can transform the PDE system (43) into another ODE system. The separation of the real and imaginary parts of the resulting system yields the following six ODEs from the first order that describe the modulation equations in terms of modified phases \(\theta_{j}\) and amplitudes \(h_{j}\) in the following forms:  
\[\begin{array}{l}h_{1}\frac{d\theta_{1}}{d\tau}=h_{1}\sigma_{1}+\frac{\tilde {\theta}^{2}\omega_{1}^{5}h_{1}}{4\left(\omega_{1}^{2}-4\omega_{2}^{2}\right) }\left[-\omega_{1}^{2}\left(4\omega_{1}^{2}+11\right)+\omega_{2}^{2}\left(7-4 \omega_{2}^{2}\right)+8\omega_{1}^{2}\omega_{2}^{2}\right]+\frac{f_{1}}{2 \omega_{1}}\cos\theta_{1},\\ \frac{d\eta_{1}}{d\tau}=-\frac{1}{2}h_{1}c_{1}+\frac{f_{1}}{2\omega_{1}}\sin \theta_{1},\\ h_{2}\frac{d\theta_{2}}{d\tau}=hFigures 5-7 illustrate the time histories of the attained approximate analytic solutions of \(Y\), \(X\), and \(\phi\) till the third order of approximation. These figures were drawn when the previously used data with various values of \(\overline{\omega}_{j}(j=1,2,3)\) were used. According to the plotted curves in these figures, we can conclude that these curves have periodic or quasi-periodic forms of wave packets, in which their amplitudes increase (see Figures 5 and 7) and decrease, as shown in Figure 6, with the passage of time until the waves behave in the form of stationary waves at the end of the examined period of time.