## 3 Frequency Response
When considering the frequency response of the system, the external force can be expressed as the following function:  
\[q_{m}=\widehat{q}_{m}e^{i\Omega T_{0}}=\widehat{q}_{m}e^{i(\omega_{m}+x^{2} \sigma)T_{0}}=\widehat{q}_{m}e^{i\omega_{m}T_{0}}e^{i\sigma T_{1}} \tag{35}\]  
\[\{2\omega_{1}\sigma B_{1}-\frac{1}{Q_{1}}[3B_{1}\overline{B_{1}}B_{1}( \int_{0}^{1}\phi_{1}\phi^{\prime\prime}{}_{3}^{3}dx+\int_{0}^{1}\phi_{1}\phi_{1}^ {\prime i}\phi_{1}^{\prime\prime}{}_{1}^{2}dx+\int_{0}^{1}4\phi_{1}\phi^{\prime }{}_{1}\phi^{\prime\prime}{}_{1}\phi^{\prime\prime\prime}{}_{1}dx) \tag{39}\] \[+2B_{1}\overline{B_{2}}B_{2}(\int_{0}^{1}3\phi^{\prime\prime}{}_{ 1}\phi^{\prime\prime}{}_{2}^{2}\phi_{1}dx+\int_{0}^{1}\phi_{1}\phi_{1}^{\prime i }\phi_{1}^{\prime}{}_{2}^{2}dx+\int_{0}^{1}2\phi_{1}\phi_{2}^{\prime i}\phi_{1 }^{\prime}{}_{1}\phi^{\prime}{}_{2}dx+\int_{0}^{1}4\phi^{\prime}{}_{1}\phi^{ \prime\prime}{}_{2}\phi^{\prime\prime\prime}{}_{2}\phi_{1}dx\] \[+\int_{0}^{1}4\phi^{\prime}{}_{2}\phi^{\prime\prime}{}_{2}^{2} \phi^{\prime\prime\prime}{}_{1}dx+\int_{0}^{1}4\phi^{\prime}{}_{2}^{2}\phi^{ \prime\prime}{}_{1}^{2}dx)\] \[+2B_{1}\overline{B_{3}}B_{3}(\int_{0}^{1}3\phi^{\prime\prime}{}_{ 1}\phi^{\prime\prime}{}_{2}^{2}\phi_{1}dx+\int_{0}^{1}\phi_{1}^{\prime i}\phi _{1}{}_{1}\phi^{\prime}{}_{3}^{2}dx+\int_{0}^{1}2\phi_{1}\phi_{3}^{\prime i} \phi_{1}^{\prime}{}_{1}\phi^{\prime\prime}{}_{3}dx+\int_{0}^{1}4\phi^{\prime}{} _{1}\phi^{\prime\prime}{}_{3}^{3}\phi_{1}dx\] \[+\int_{0}^{1}4\phi^{\prime}{}_{3}\phi^{\prime\prime}{}_{3}^{3} \phi^{\prime\prime\prime}{}_{1}\phi_{1}dx+\int_{0}^{1}4\phi^{\prime}{}_{3}^{ 3}\phi^{\prime\prime}{}_{1}^{\prime\prime\prime}{}_{3}^{3}\phi_{1}dx)]\] \[-\frac{1}{2Q_{1}}\{-4\omega_{1}^{2}B_{1}\overline{B_{1}}B_{1}[ \int_{0}^{1}\phi^{\prime\prime}{}_{1}\phi_{1}(\int_{1}^{x}\int_{0}^{1}\phi^{ \prime}{}_{1}\phi^{\prime}{}_{1}dxdx)dx+\int_{0}^{1}\phi^{\prime}{}_{1}\phi_{1 }(\int_{1}^{x}\int_{0}^{x}\phi^{\prime}{}_{1}\phi^{\prime}{}_{1}dxdx)^{\prime}dx]\] \[+4(-\omega_{1}^{2}-\omega_{2}^{2})B_{1}\overline{B_{2}}B_{2}[ \int_{0}^{1}\phi^{\prime\prime}{}_{2}\phi_{1}(\int_{1}^{x}\int_{0}^{x}\phi^{ \prime}{}_{1}\phi^{\prime}{}_{2}dxdx)dx+\int_{0}^{1}\phi^{\prime}{}_{2}\phi_{1 }(\int_{1}^{x}\int_{0}^{x}\phi^{\prime}{}_{1}\phi^{\prime}{}_{2}dxdx)^{\prime}dx]\] \[+4(-\omega_{1}^{2}-\omega_{3}^{2})B_{1}\overline{B_{3}}B_{3}[ \int_{0}^{1}\phi^{\prime\prime}{}_{3}\phi_{1}(\int_{1}^{x}\int_{0}^{x}\phi^{ \prime}{}_{1}\phi^{\prime}{}_{3}dxdx)dx]+\int_{0}^{1}\phi^{\prime}{}_{3}\phi_{1 }(\int_{1}^{x}\int_{0}^{x}\phi^{\prime}{}_{1}\phi^{\prime}{}_{3}dxdx)^{\prime} ]\}^{2}\] \[+(-\mu\omega_{1}B_{1}-C_{G}\frac{1}{e^{B_{2}^{2}T}}\int_{0}^{ \omega}\omega_{1}B_{1}e^{Rx}d\tau)=\vec{q}_{1}^{2}\]  
Similarly, multiplying the secular terms for the second mode by \(e^{i\xi_{2}}\) and the secular terms for the third mode by \(e^{i\xi_{3}}\), we obtain the equations for the real and imaginary parts of the second and third modes. Utilizing the IMSL subroutine \"NEQNF\", employing the Levenberg-Marquardt method, allows us to solve the solvability conditions. Consequently, we can derive the dimensionless amplitudes \(B_{1}\), \(B_{2}\), and \(B_{3}\) for this system and generate tuned-frequency response diagrams for each mode, known as fixed-point plots. Applying the same approach, we can obtain tuned-frequency response diagrams for the second and third modes. Figure 7 represents the fixed-point plots for the first, second, and third modes when the first mode is excited. Figure 8 depicts the fixed-point plots for the second mode when the second mode is excited. Figure 9 illustrates the fixed-point plots for the first, second, and third modes when the third mode is excited. Notably, Figure 7 demonstrates that when the first mode is excited, the amplitude of the first mode significantly surpasses that of the second and third modes. This observation is consistent with other modes.