# The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance
## 3 The Perturbation Technique  
The main objective of the present section is to obtain the approximate analytic solutions of the EOM (5)-(7) using the PTMS to address the cases of resonance, obtain the conditions of solvability, and achieve the equations of the modulations. Therefore, we can approximate the functions \(\sin\phi\) and \(\cos\phi\) in Equations (5)-(7) using a Taylor series according to their first terms, for which these approximations are valid in a small neighborhood of the position of static equilibrium [33]. Then, Equations (5)-(7) become:  
\[ \begin{array}{l}\ddot{y}(\tau)+\omega_{1}^{2}y-2\beta\dot{x}(\tau)\dot{\phi }(\tau)(\phi-\frac{\phi^{3}}{6})-\beta(1+x)\dot{\phi}^{2}(\tau)(1-\frac{\phi^ {2}}{2})\\ +\beta\ddot{x}(\tau)(1-\frac{\phi^{2}}{2})-\beta(1+x)\dot{\phi}(\tau)(\phi- \frac{\phi^{3}}{6})+c_{1}\dot{y}(\tau)-2=f_{1}\cos(\Omega_{1}\tau),\\ \\ (1+x)^{2}\ddot{\phi}(\tau)-(1+x)\ddot{y}(\tau)(\phi-\frac{\phi^{3}}{6})+2\dot{ x}(\tau)\dot{\phi}(\tau)+2x\dot{x}(\tau)\dot{\phi}(\tau)\\ +(1+x)(\phi-\frac{\phi^{3}}{6})+c_{2}\dot{\phi}(\tau)=f_{2}\cos(\Omega_{2}\tau),\\ \\ \ddot{x}(\tau)+\omega_{2}^{2}x+(1+x)\dot{\phi}^{2}(\tau)+(1-\frac{\phi^{2}}{2} )\ddot{y}(\tau)+\frac{\phi^{2}}{2}+c_{3}\dot{x}(\tau)=f_{3}\cos(\Omega_{3}\tau).\end{array}  \tag{8}\]  
Based on the motion in a small neighborhood of the static equilibrium position, we consider a small parameter \(0<\varepsilon<<1\), in which the parameters \(y,\dot{\phi}\), and \(x\) can be rewritten in terms of new variables, \(\zeta\), \(\varphi\), and \(\chi\), respectively, according to the following forms:  
\[y(\tau)=\varepsilon\,\xi(\tau;\varepsilon),\qquad\dot{\phi}(\tau)=\varepsilon \,\varphi(\tau;\varepsilon),\qquad x(\tau)=\varepsilon\,\chi(\tau;\varepsilon). \tag{11}\]  
According to PTMS, we can seek the approximate solutions of the variables \(\xi,\varphi\), and \(\chi\) in the following forms [34]:  
\[ \begin{array}{l}\xi=\sum\limits_{k=0}^{2}\,\xi^{k}\xi_{k+1}(\tau_{0},\tau_{ 1},\tau_{2})+O(\varepsilon^{3}),\\ \varphi=\sum\limits_{k=0}^{2}\,\varepsilon^{k}\varphi_{k+1}(\tau_{0},\tau_{ 1},\tau_{2})+O(\varepsilon^{3}),\\ \chi=\sum\limits_{k=0}^{2}\,\varepsilon^{k}\chi_{k+1}(\tau_{0},\tau_{ 1},\tau_{2})+O(\varepsilon^{3}),\end{array}  \tag{12}\]  
where \(\tau_{n}=\varepsilon^{n}\tau\ (n=0,\,1,\,2)\) are various time scales, in which \(\tau_{0}\) and \(\tau_{1},\tau_{2}\) are defined as fast and slow time scales, respectively. Therefore, the time derivatives regarding \(\tau\) can be transformed according to the time scales \(\tau_{n}\) as the following chain rule:  
\[ \begin{array}{l}\frac{d}{d\tau}=\frac{\partial}{\partial\tau_{1}}+ \varepsilon\frac{\partial}{\partial\tau_{1}}+\varepsilon^{2}\frac{\partial}{ \partial\tau_{2}}+O(\varepsilon^{3}),\\ \frac{d^{2}}{d\tau^{2}}=\frac{\partial^{2}}{\partial\tau_{0}^{2}}+2\varepsilon \frac{\partial^{2}}{\partial\tau_{0}\partial\tau_{1}}+\varepsilon^{2}(\frac{ \partial^{2}}{\partial\tau_{1}^{2}}+2\frac{\partial^{2}}{\partial\tau_{0} \partial\tau_{2}})+O(\varepsilon^{3}),\end{array}  \tag{13}\]  
where terms of \(O(\varepsilon^{3})\) and higher are neglected due to their minuteness. As before, we consider the minuteness of the amplitudes of external forces and moments \(f_{j}\ (j=1,2,3)\), as well as the coefficients of damping \(c_{i}\). Therefore, we can write:  
\[f_{j}=\varepsilon^{3}\widetilde{f}_{j},\qquad c_{j}=\varepsilon^{2}\widetilde{ c}_{j}\qquad(j=1,2,3). \tag{14}\]  
The substitution of Equation (25) into Equation (18) yields a nonhomogeneous equation, and its solution can be written in the form:  
\[\chi_{1}=\text{A}_{3}e^{i\omega_{2}\tau_{0}}+\overline{\text{A}}_{3}e^{-i\omega_ {2}\tau_{0}}+\frac{\omega_{1}^{2}}{\omega_{1}^{2}-\omega_{2}^{2}}(\text{A}_{1}e ^{i\omega_{1}\tau_{0}}+\overline{\text{A}}_{1}e^{-i\omega_{1}\tau_{0}}), \tag{27}\]  
where A\({}_{j}\) (\(j=1\), 2, 3) represent unknown complex functions of the slow time scales \(\tau_{1}\), \(\tau_{2}\), while \(\overline{\text{A}}_{j}\) are the complex conjugates of these functions.