## 3 Analytical Investigation and Autonomous Amplitude-Phase Equations  
This section is dedicated to obtaining an analytical solution for the twelve-pole system. Accordingly, the first-order approximate solution for Equations (8) and (9) can be sought utilizing the multiple-time scale perturbation procedures as follows [40]:  
\[x(t,\varepsilon)=x_{0}(T_{0},T_{1})+\varepsilon x_{1}(T_{0},T_{1}) \tag{10}\]  
\[y(t,\varepsilon)=y_{0}(T_{0},T_{1})+\varepsilon y_{1}(T_{0},T_{1}) \tag{11}\]  
where \(\varepsilon\) is an artificial parameter used as a book-keeping parameter only, \(T_{0}=t\) and \(T_{1}=\varepsilon t\) represent the fast and slow time scales, respectively. So, the ordinary derivatives \(\frac{d}{dt}\) and \(\frac{d^{2}}{dt^{2}}\) can be written using the chain rule for derivatives as:  
\[\frac{d}{dt}=D_{0}+\varepsilon D_{1},\ \frac{d^{2}}{dt^{2}}=D_{0}^{2}+2 \varepsilon D_{0}D_{1},\ D_{j}=\frac{\partial}{\partial T_{j}},\ \ \ \ j=0,1 \tag{12}\]  
Depending on the system nonlinearities, the coefficients of Equations (8) and (9) should be scaled as follows to execute the multiple time scales procedure:  
\[\mu=\varepsilon\dot{\mu},\ \alpha_{j}=\varepsilon\dot{\alpha}_{j},\ f=\varepsilon \dot{f}\,\ j=1,\ldots,7 \tag{13}\]  
Now, by inserting Equations (10)-(12) into Equations (8) and (9), and equating the coefficients that have the same power of \(\varepsilon\), we get  
\[O\Big{(}\varepsilon^{0}\Big{)}:\]  
\[\Big{(}D_{0}^{2}+\omega^{2}\Big{)}x_{0}=0 \tag{14}\]  
\[\Big{(}D_{0}^{2}+\omega^{2}\Big{)}y_{0}=0 \tag{15}\]  
\[O\Big{(}\varepsilon^{1}\Big{)}:\]  
\[ \begin{array}{rl}\big{(}D_{0}^{2}+\omega^{2}\big{)}x_{1}&=-2D_{0}D_{1}x_{0}-2 \hat{\mu}D_{0}x_{0}+\hat{\alpha}_{1}x_{0}^{3}+\hat{\alpha}_{2}x_{0}y_{0}^{2}+ \hat{\alpha}_{3}x_{0}^{2}D_{0}x_{0}\\ &+\hat{\alpha}_{4}y_{0}^{2}D_{0}x_{0}+\hat{\alpha}_{5}x_{0}(D_{0}y_{0})^{2}+ \hat{\alpha}_{6}x_{0}(D_{0}x_{0})^{2}+\hat{\alpha}_{7}x_{0}y_{0}D_{0}y_{0}\\ &+\hat{f}\Omega^{2}\cos(\Omega T_{0})\end{array}  \tag{16}\]\[ \begin{array}{ll}\big{(}D_{0}^{2}+\omega^{2}\big{)}y_{1}&=-2D_{0}D_{1}y_{0}-2\hat {\mu}D_{0}y_{0}+\hat{\alpha}_{1}y_{0}^{3}+\hat{\alpha}_{2}y_{0}x_{0}^{2}+\hat{ \alpha}_{3}y_{0}^{2}D_{0}y_{0}\\ &+\hat{\alpha}_{4}x_{0}^{2}D_{0}y_{0}+\hat{\alpha}_{5}y_{0}\big{(}D_{0}x_{0} \big{)}^{2}+\hat{\alpha}_{6}y_{0}\big{(}D_{0}y_{0}\big{)}^{2}+\hat{\alpha}_{7}y _{0}x_{0}D_{0}x_{0}\\ &+\hat{f}\Omega^{2}\sin(\Omega T_{0})\end{array}  \tag{17}\]  
The solutions of homogeneous differential equations given by Equations (14) and (15) are:  
\[x_{0}(T_{0},T_{1}) = A(T_{1})e^{i\omega T_{0}}+\overline{A}(T_{1})e^{-i\omega T_{0}} \tag{18}\] \[y_{0}(T_{0},T_{1}) = B(T_{1})e^{i\omega T_{0}}+\overline{B}(T_{1})e^{-i\omega T_{0}} \tag{19}\]  
where \(A(T_{1})\) and \(B(T_{1})\) are two unknowns that will be determined in the subsequent steps of analysis, and \(\overline{A}(T_{1})\) and \(\overline{B}(T_{1})\) are the conjugate functions of the unknowns \(A(T_{1})\) and \(B(T_{1})\). By inserting Equations (18) and (19) into Equations (16) and (17), yields  
\[ \begin{array}{ll}\big{(}D_{0}^{2}+\omega^{2}\big{)}x_{1}&=[-2i\omega(D_{1}A) -2i\hat{\mu}\omega A+3\hat{\alpha}_{1}A^{2}\overline{A}+2\hat{\alpha}_{2}AB \overline{B}+\hat{\alpha}_{2}\overline{A}B^{2}+i\hat{\alpha}_{3}\omega A^{2} \overline{A}+2i\hat{\alpha}_{4}\omega AB\overline{B}\\ &-i\hat{\alpha}_{4}\omega\overline{A}B^{2}+2\hat{\alpha}_{5}\omega^{2}AB \overline{B}-\hat{\alpha}_{5}\omega^{2}\overline{A}B^{2}+\hat{\alpha}_{6} \omega^{2}A^{2}\overline{A}+i\hat{\alpha}_{7}\omega\overline{A}B^{2}]e^{i \omega T_{0}}+\big{[}\hat{\alpha}_{1}A^{3}\\ &+\hat{\alpha}_{2}AB^{2}+i\hat{\alpha}_{3}\omega A^{3}+i\hat{\alpha}_{4} \omega AB^{2}-\hat{\alpha}_{5}\omega^{2}AB^{2}-\hat{\alpha}_{6}\omega^{2}A^{3 }+i\hat{\alpha}_{7}\omega AB^{2}\big{]}e^{3i\omega T_{0}}\\ &+\hat{\alpha}_{2}\Omega^{2}\hat{f}e^{i\Omega T_{0}}\end{array}  \tag{20}\]  
\[