# The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance
## 3 The Perturbation Technique
The substitution of the first-order solutions of Equations (25)-(27) into Equations (19)-(21) yields secular terms. The elimination of these terms demands that:  
\[2i\omega_{1}\frac{\partial\text{A}_{1}}{\partial\tau_{1}}-\frac{\widetilde{ \beta}\omega_{1}^{4}}{(\omega_{1}^{2}-\omega_{2}^{2})}\text{A}_{1}=0, \tag{28}\]  
\[\frac{\partial\text{A}_{2}}{\partial\tau_{1}}=0. \tag{29}\]  
According to these conditions, the second-order solutions have the following forms:  
\[\xi_{2}=\frac{\widetilde{\beta}\omega_{2}^{2}}{(\omega_{1}^{2}-\omega_{2}^{2} )}\text{A}_{3}e^{i\omega_{2}\tau_{0}}+CC., \tag{30}\]  
\[ \begin{array}{l}\phi_{2}=-\frac{\omega_{1}\text{A}_{2}}{(\omega_{1}^{2}- \omega_{2}^{2})}\{\frac{\{(1+2\omega_{1})-(\omega_{1}^{2}-\omega_{2}^{2})\}} {(\omega_{1}+2)}\text{A}_{1}e^{i(1+\omega_{1})\tau_{0}}+\frac{\{(1-2\omega_{ 1})-(\omega_{1}^{2}-\omega_{2}^{2})\}}{(\omega_{1}-2)}\}\\ \times\overline{\text{A}}_{1}e^{i(1-\omega_{1})\tau_{0}}\}-\frac{\Delta_{2}}{ \omega_{2}}\frac{(1+2\omega_{2})}{(\omega_{2}+2)}\text{A}_{3}e^{i(1+\omega_{ 2})\tau_{0}}+\frac{(1-2\omega_{2})}{(\omega_{2}-2)}\overline{\text{A}}_{3}e^ {i(1-\omega_{2})\tau_{0}}]+CC.\end{array}  \tag{31}\]  
Here, \(CC.\) represents the complex conjugates of the preceding terms. After grasping the above procedure, we can obtain another condition for eliminating the secular terms by substituting the solutions from Equations (25)-(27), (30) and (31) into Equation (21), in the following form:  
\[2i\omega_{2}\frac{\partial\text{A}_{3}}{\partial\tau_{1}}-\frac{\widetilde{ \beta}\,\omega_{2}^{4}}{\omega_{1}^{2}-\omega_{2}^{2}}\text{A}_{3}=0. \tag{32}\]  
Therefore, the solution \(\chi_{2}\) has the form:  
\[\chi_{2}=-\frac{3\text{A}_{2}\overline{\text{A}}_{2}}{\omega_{2}^{2}}\,-\frac{ 3\,\text{A}_{2}^{2}}{2(\omega_{2}^{4}-4)}\,e^{2i\,\tau_{0}}+\frac{\widetilde{ \beta}\,\omega_{1}^{4}\omega_{2}^{2}}{(\omega_{1}^{2}-\omega_{2}^{2})^{3}}\, \text{A}_{1}e^{i\omega_{1}\tau_{0}}+CC. \tag{33}\]  
Making use of the previous solutions to Equations (25)-(27), (30), (31) and (33) and the third-order approximate Equations (22)-(24), secular terms are produced. The eliminating conditions of these terms required that:  
\[2i\omega_{1}\frac{\partial\text{A}_{1}}{\partial\tau_{2}}\,\,+i\omega_{1} \widetilde{\varepsilon}_{1}\text{A}_{1}-\frac{\widetilde{\beta}^{2}\omega_{1 }^{6}\text{A}_{1}}{4(\omega_{1}^{2}-\omega_{2}^{2})^{3}}[-\omega_{1}^{2}(4 \omega_{1}^{2}+11)+\omega_{2}^{2}(7-4\omega_{2}^{2})+8\omega_{1}^{2}\omega_{2 }^{2}]\,\,\,=0, \tag{34}\]  
\[ \begin{array}{l}2i\frac{\partial\text{A}_{2}}{\partial\tau_{2}}\,\,+i\, \widetilde{\varepsilon}_{2}\text{A}_{2}-\frac{6\omega_{1}^{4}(\omega_{2}^{2} -2\omega_{1}^{2}+1)}{(\omega_{1}^{2}-4)(\omega_{1}^{2}-\omega_{2}^{2})^{3}} \text{A}_{1}\text{A}_{2}\overline{\text{A}}_{1}-\frac{\text{A}_{1}^{2} \overline{\text{A}}_{2}}{2\omega_{2}^{4}(\omega_{2}^{4}-4)}(\omega_{2}^{4}-25 \omega_{2}^{2}+24)\\ -\frac{\text{A}_{2}\text{A}_{2}\overline{\text{A}}_{3}}{(\omega_{2}^{2}-4)( \omega_{2}^{2}-\omega_{1}^{2})}\{6\omega_{1}^{2}(\omega_{2}^{2}-1)+\omega_{2 }^{2}[6(1-\omega_{2}^{2})+\widetilde{\beta}\omega_{2}(2\omega_{2}^{2}+3 \omega_{2}-2)]\}=0,\end{array}  \tag{35}\]  
\[ \begin{array}{l}2i\omega_{2}\frac{\partial\text{A}_{3}}{\partial\tau_{2}}\,\,+i \,\widetilde{\varepsilon}_{3}\omega_{2}\text{A}_{3}+\frac{\widetilde{\beta}^{2 }\omega_{2}^{6}\text{A}_{3}}{2(\omega_{1}^{2}-\omega_{2}^{2})^{2}}(1-2\,\omega_ {1}^{2})-\frac{3+2\omega_{2}(\omega_{2}+2)}{\omega_{2}(\omega_{2}+2)}\text{A}_{ 2}\text{A}_{3}\overline{\text{A}}_{2}=0.\end{array}  \tag{36}\]  
Based on the previous conditions of Equations (34)-(36), the approximate third-order solutions turn into:\[