# Resonance in the Cart-Pendulum System--An Asymptotic Approach
## 3 The Announced Method  
In this section, the MMS perturbation technique will be utilized to solve the governing EOM (11) and (12) up to the second order of approximation. Therefore, we need only three time scales, having the forms \(T_{0}=\tau,\ T_{1}=\varepsilon\tau\), and \(T_{2}=\varepsilon^{2}\tau\). According to the procedure of MMS, we express \(\xi\) and \(\theta\) in powers of \(\varepsilon\) as  
\[\begin{array}{l}\xi=\xi_{0}(T_{0},T_{1},T_{2})+\varepsilon\xi_{1}(T_{0},T_{1 },T_{2})+\varepsilon^{2}\xi_{2}(T_{0},T_{1},T_{2})+\ldots,\\ \theta=\theta_{0}(T_{0},T_{1},T_{2})+\varepsilon\theta_{1}(T_{0},T_{1},T_{2}) +\varepsilon^{2}\theta_{2}(T_{0},T_{1},T_{2})+\ldots.\end{\[ \begin{array}{c}\textbf{Order of }\varepsilon^{0}\\ D_{0}^{2}\xi_{0}+\zeta_{0}=0,\\ D_{0}^{2}\theta_{0}+K^{2}\theta_{0}=0.\end{array}  \tag{15}\]  
**Order of \(\varepsilon\)**  
\[ \begin{array}{c}D_{0}^{2}\xi_{1}+\xi_{1}=\hat{\mu}\theta_{0}(D_{0}\theta_{0} )^{2}-2D_{0}D_{1}\xi_{0}-\hat{c}_{1}D_{0}\xi_{0}-\hat{\mu}D_{0}^{2}\theta_{0}, \\ D_{0}^{2}\theta_{1}+K^{2}\theta_{1}=-2D_{0}D_{1}\theta_{0}-\hat{c}_{2}D_{0} \theta_{0}-\alpha\hat{\mu}D_{0}^{2}\xi_{0}.\end{array}  \tag{16}\]  
**Order of \(\varepsilon^{2}\)**  
\[ \begin{array}{c}D_{0}^{2}\xi_{2}+\xi_{2}=-2D_{0}D_{1}\xi_{1}-(2D_{0}D_{2}+D_{ 1}^{2})\xi_{0}-\hat{c}_{1}(D_{0}\xi_{1}+D_{1}\xi_{0})\\ -\hat{\mu}(D_{0}^{2}\theta_{1}+2D_{0}D_{1}\theta_{0})+\hat{\mu}[2\theta_{0}D_{ 0}\theta_{0}D_{0}\theta_{1}+2\theta_{0}D_{0}\theta_{0}D_{1}\theta_{0}\\ +\theta_{1}(D_{0}\theta_{0})^{2}]+\hat{f}\cos(pT_{0}),\\ D_{0}^{2}\theta_{2}+K^{2}\theta_{2}=-2D_{0}D_{1}\theta_{1}-(2D_{0}D_{2}+D_{ 1}^{2})\theta_{0}-\hat{c}_{2}(D_{0}\theta_{1}+D_{1}\theta_{0})\\ -\alpha\hat{\mu}(D_{0}^{2}\xi_{1}+2D_{0}D_{1}\xi_{0})+\alpha\hat{f}\cos(pT_{0 }).\end{array}  \tag{17}\]  
It is worthy to notice that the system in Equations (11) and (12) can be approximated by the above systems in Equations (15)-(17) of PDEs, which can be solved successively. Solutions of these equations starting from the zero approximation equations can be inserted into the next higher orders of approximation. Equations of system (15) are mutually independent homogenous equations. Consequently, their solutions are harmonic and can be expressed in their exponential forms as  
\[\xi_{0}=A(T_{1},T_{2})e^{iT_{0}}+\overline{A}(T_{1},T_{2})e^{-iT_{0}}, \tag{18}\]  
\[\theta_{0}=B(T_{1},T_{2})e^{iKT_{0}}+\overline{B}(T_{1},T_{2})e^{-iKT_{0}}, \tag{19}\]  
where \(A,B\) are determinable complex functions from the elimination of the secular terms and \(\overline{A},\overline{B}\) are the corresponding complex conjugate. It is obvious that the solutions of (16) and (17) depend on the solutions of (18) and (19) to some extent. Therefore, substituting (18) and (19) into the system of Equation (16), we remove the produced secular terms to gain the removable conditions in the forms  
\[2D_{1}A+\hat{c}_{1}A=0, \tag{20}\]  
\[2D_{1}B+\hat{c}_{2}B=0. \tag{21}\]  
Therefore, the first-order approximations of the solutions have the forms  
\[\xi_{1}=\frac{\hat{\mu}B^{3}K^{2}}{(9K^{2}-1)}e^{3 iKT_{0}}-\frac{\hat{\mu}K^ {2}B(B\overline{B}+1)}{(K^{2}-1)}e^{iKT_{0}}+CC, \tag{22}\]  
\[\theta_{1}=\frac{\alpha\hat{\mu}A}{(K^{2}-1)}e^{iT_{0}}+CC. \tag{23}\]  
where \(CC\) stands for the complex conjugate of the preceding terms. This symbol allows presentation of the long term and thus will be used frequently. Substituting the solutions (18), (19), (22), and (23) into the system of Equation (17) and eliminating terms leads to secular ones, yielding  
\[2 iD_{2}A+D_{1}^{2}A+\hat{c}_{1}D_{1}A-\frac{\hat{\mu}^{2}\alpha A}{(K^{2}-1)}( 1+2B\overline{B}K^{2})=0, \tag{24}\]  
\[2 iKD_{2}B+D_{1}^{2}B+\hat{c}_{2}D_{1}B+\frac{\hat{\mu}^{2}K^{4}\alpha B}{(K^{2} -1)}(1+B\overline{B})=0. \tag{25}\]Based on the above, the second-order approximations of the solutions become  
\[