## 3 Analytical Investigation and Autonomous Amplitude-Phase Equations
\(\hat{\mu}=\frac{\mu}{\varepsilon},\hat{f}=\frac{\varepsilon}{\varepsilon}\), \(\hat{\lambda}_{j}=\frac{\alpha_{j}}{\varepsilon}\), \(j=1,\ldots,7\)), and then separating the real and imaginary parts of the resulting equations, we can obtain the following autonomous system of the first-order nonlinear differential equations:  
\[ \begin{array}{ll}\frac{da}{dt}=F_{1}(a,b,\varphi_{1},\varphi_{2})=&-\mu a+ \frac{1}{8}(\alpha_{3})a^{3}+\frac{1}{4}\alpha_{4}ab^{2}+\frac{1}{8}(-\alpha_ {4}+\alpha_{7})ab^{2}\cos(2\varphi_{2}-2\varphi_{1})\\ &-\frac{1}{8}(\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega)ab^{2}\sin(2\varphi_ {2}-2\varphi_{1})+\frac{1}{2\omega}(\omega+\sigma)^{2}f\sin\varphi_{1}\\ &\\ \frac{db}{dt}=F_{2}(a,b,\varphi_{1},\varphi_{2})=&-\mu b+\frac{1}{8}(\alpha_{3 })b^{3}+\frac{1}{4}\alpha_{4}ba^{2}+\frac{1}{8}(-\alpha_{4}+\alpha_{7})ba^{2} \cos(2\varphi_{2}-2\varphi_{1})\\ &+\frac{1}{8}(\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega)ba^{2}\sin(2\varphi_ {2}-2\varphi_{1})-\frac{1}{2\omega}(\omega+\sigma)^{2}f\cos\varphi_{2}\\ \frac{dp_{1}}{dt}=F_{3}(a,b,\varphi_{1},\varphi_{2})=&\sigma+\frac{1}{8}( \frac{3\alpha_{1}}{\omega}+\alpha_{6}\omega)a^{2}+\frac{1}{4}(\frac{\alpha_{2 }}{\omega}+\alpha_{5}\omega)b^{2}+\frac{1}{8}(\frac{\alpha_{2}}{\omega}- \alpha_{5}\omega)b^{2}\cos(2\varphi_{2}-2\varphi_{1})\\ &+\frac{1}{8}(-\alpha_{4}+\alpha_{7})b^{2}\sin(2\varphi_{2}-2\varphi_{1})+ \frac{1}{2\omega}(\omega+\sigma)^{2}f\cos\varphi_{1}\\ \frac{dp_{2}}{dt}=F_{4}(a,b,\varphi_{1},\varphi_{2})=&\sigma+\frac{1}{8}( \frac{3\alpha_{1}}{\omega}+\alpha_{6}\omega)b^{2}+\frac{1}{4}(\frac{\alpha_{2 }}{\omega}+\alpha_{5}\omega)a^{2}+\frac{1}{8}(\frac{\alpha_{2}}{\omega}- \alpha_{5}\omega)a^{2}\cos(2\varphi_{2}-2\varphi_{1})\\ &-\frac{1}{8}(-\alpha_{4}+\alpha_{7})a^{2}\sin(2\varphi_{2}-2\varphi_{1})+ \frac{1}{2\omega}(\omega+\sigma)^{2}f\sin\varphi_{2}\\ \end{array}  \tag{28}\]  
where \(\varphi_{1}=\sigma t-\theta_{1},\ \varphi_{2}=\sigma t-\theta_{2}\). Now, by substituting Equation (27) into Equations (18) and (19), and then inserting the resulting equations into Equations (10) and (11), yields  
\[x(t)=a(t)\cos(\Omega t-\varphi_{1}(t)) \tag{29}\]  
\[y(t)=b(t)\cos(\Omega t-\varphi_{2}(t)) \tag{30}\]  
Accordingly, Equations (32) and (33) represent the periodic solution of the twelve-pole system given by Equations (8) and (9), where the amplitudes (\(a(t)\) and \(b(t)\)) and the corresponding phase angles (\(\phi_{1}(t)\) and \(\phi_{2}(t)\)) are functions of time that evolve depending on the nonlinear dynamical system given by Equations (28)-(31). Therefore, at the steady-state periodic vibrations of the twelve-pole system, we have \(\frac{da}{dt}=\frac{db}{dt}=\frac{d\varphi_{1}}{dt}=\frac{d\varphi_{2}}{dt}=0\). Substituting \(\frac{da}{dt}=\frac{db}{dt}=\frac{d\varphi_{1}}{dt}=\frac{d\varphi_{2}}{dt}=0\) into Equations (28)-(31), we can obtain the following nonlinear coupled algebraic equations:  
\[ \begin{array}{ll}-\mu a+\frac{1}{8}(\alpha_{3})a^{3}+\frac{1}{4}\alpha_{4}ab ^{2}+\frac{1}{8}(-\alpha_{4}+\alpha_{7})ab^{2}\cos(2\varphi_{2}-2\varphi_{1} )-\frac{1}{8}(\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega)ab^{2}\sin(2\varphi_ {2}-2\varphi_{1})\\ +\frac{1}{2\omega}(\omega+\sigma)^{2}f\sin\varphi_{1}=0\\ -\mu b+\frac{1}{8}(\alpha_{3})b^{3}+\frac{1}{4}\alpha_{4}ba^{2}+\frac{1}{8}( -\alpha_{4}+\alpha_{7})ba^{2}\cos(2\varphi_{2}-2\varphi_{1})+\frac{1}{8}( \frac{\alpha_{2}}{\omega}-\alpha_{5}\omega)ba^{2}\sin(2\varphi_{2}-2\varphi_{1}) \\ -\frac{1}{2\omega}(\omega+\sigma)^{2}f\cos\varphi_{2}=0\\ \end{array}  \tag{31}\]  
\[ \begin{array}{ll}\sigma+\frac{1}{8}(\frac{3\alpha_{1}}{\omega}+\alpha_{6} \omega)a^{2}+\frac{1}{4}(\frac{\alpha_{2}}{\omega}+\alpha_{5}\omega)b^{2}+ \frac{1}{8}(\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega)b^{2}\cos(2\varphi_{2}- 2\varphi_{1})+\frac{1}{8}(-\alpha_{4}+\alpha_{7})b^{2}\\ \times\sin(2\varphi_{2}-2\varphi_{1})+\frac{1}{2\omega}(\omega+\sigma)^{2}f \cos\varphi_{1}=0\\ \end{array}  \tag{32}\]  
\[