## 3 Perturbation Analyses  
For a weak nonlinear system, the perturbation methods, such as the method of multiple scales, are valid [37; 38], and these can transform the nonlinear equation into a set of linear ordinary differential equations. Considering the method of multiple scales, and, assuming that \(\text{T}_{\text{n}}=\varepsilon^{\text{n}}\text{t}\), one obtains  
\[\frac{\text{d}}{\text{dt}}=\frac{\partial}{\partial\text{T}_{0}}+\varepsilon \frac{\partial}{\partial\text{T}_{1}}+\cdots=\text{D}_{0}+\varepsilon\text{D} _{1}+\cdots \tag{13}\]\[\frac{\mathrm{d}^{2}}{\mathrm{dt}^{2}}=\frac{\mathrm{d}}{\mathrm{dt}}\bigg{(} \frac{\partial}{\partial T_{0}}+\varepsilon\frac{\partial}{\partial T_{1}}+ \cdots\bigg{)}=\mathrm{D}_{0}^{2}+2\varepsilon\mathrm{D}_{0}\mathrm{D}_{1}+\cdots \tag{14}\]  
In the present work, we only adopt the first-order approximation, which is sought in the form  
\[\mathrm{s}(\mathrm{t};\varepsilon)=\mathrm{s}_{0}(\mathrm{T}_{0},\mathrm{T}_ {1})+\varepsilon\mathrm{s}_{1}(\mathrm{T}_{0},\mathrm{T}_{1})+\cdots, \tag{15}\]  
where \(\mathrm{T}_{0}\) is a fast time scale characterizing motion occurring at the natural frequencies of the corresponding linear vibration system, \(\mathrm{T}_{1}\) is a slow time scale characterizing the modulation of the amplitudes and phases. The combination of Equations (12)-(15) yields  
\[ \begin{split}&\omega_{0}^{2}\Big{(}\mathrm{D}_{0}^{2}+2 \varepsilon\mathrm{D}_{0}\mathrm{D}_{1}\Big{)}(\mathrm{s}_{0}+\varepsilon \mathrm{s}_{1})-\varepsilon(\mathrm{s}_{0}+\varepsilon\mathrm{s}_{1})^{3}+ \omega_{0}^{2}(\mathrm{s}_{0}+\varepsilon\mathrm{s}_{1})\\ &-\omega_{0}^{2}\pi^{2}\tau^{2}(\mathrm{s}_{0}+\varepsilon \mathrm{s}_{1})+\varepsilon\kappa(\mathrm{D}_{0}+\varepsilon\mathrm{D}_{1})( \mathrm{s}_{0}+\varepsilon\mathrm{s}_{1})=\varepsilon\mathrm{a}_{\mathrm{m} }\sin(\omega\mathrm{T}_{0})\end{split}  \tag{16}\]  
Considering the excitation is \(\mathrm{O}(\varepsilon)\), we assume \(\omega-\omega_{0}\) is \(\mathrm{O}(\varepsilon)\) for consistency. Hence, we introduce a relation \(\omega=\omega_{0}+\varepsilon\upsilon\), where \(\upsilon\) is a detuning parameter (i.e., the frequency of disturbing force, \(\upsilon=\mathrm{O}(1)\)) that is a notation for the mathematical solution of a nonlinear vibration with cubic nonlinearity [38]. Equating the coefficients of \(\varepsilon^{\mathrm{n}}\) with \(\mathrm{n}=0\) and \(1\) on both sides in the expansion of Equation (16) yields  
\[\omega_{0}^{2}\mathrm{D}_{0}^{2}\mathrm{s}_{0}+\Big{(}\omega_{0}^{2}-\omega_ {0}^{2}\pi^{2}\tau^{2}\Big{)}\mathrm{s}_{0}=0 \tag{17}\]  
\[\omega_{0}^{2}\mathrm{D}_{0}^{2}\mathrm{s}_{1}+\Big{(}\omega_{0}^{2}-\omega_ {0}^{2}\pi^{2}\tau^{2}\Big{)}\mathrm{s}_{1}=\mathrm{s}_{0}^{3}-2\omega_{0}^{ 2}\mathrm{D}_{0}\mathrm{D}_{1}\mathrm{s}_{0}-\kappa\mathrm{D}_{0}\mathrm{s}_{ 0}+\mathrm{a}_{\mathrm{m}}\sin(\omega_{0}\mathrm{T}_{0}+\upsilon\mathrm{T}_{1}) \tag{18}\]  
From Equation (17) we obtain  
\[\mathrm{s}_{0}=\mathrm{B}(\mathrm{T}_{1})\mathrm{e}^{\mathrm{i}\sqrt{1-\pi^{2} \tau^{2}}\mathrm{T}_{0}}+\mathrm{cc} \tag{19}\]  
where \(\mathrm{B}\) is a complex function with respect to \(\mathrm{T}_{1}\) and \(\mathrm{cc}\) is the conjugate complex number of its left terms, similarly hereinafter. It can be determined that \(\mathrm{cc}=\overline{\mathrm{B}}(\mathrm{T}_{1})\mathrm{e}^{-\mathrm{i}\sqrt {1-\pi^{2}\tau^{2}}\mathrm{T}_{0}}\) in Equation (19), and \(\overline{\mathrm{B}}\) is the conjugate function of \(\mathrm{B}\). Substituting Equation (19) into (18) yields  
\[