## 3 The Desired Solutions
\begin{array}{l}\textbf{Order of }\boldsymbol{\varepsilon}^{3}\\ \\ D_{0}^{2}\widetilde{\chi}_{3}+\widetilde{\chi}_{3}&=\widetilde{\gamma}_{1}\cos( p_{1}\tau_{0})-\widetilde{c}_{1}D_{0}\widetilde{\chi}_{1}+\widetilde{\chi}_{1} [(D_{0}^{2}\widetilde{\phi}_{1})^{2}-2\widetilde{\alpha}_{2}\widetilde{\alpha }_{2}]+\widetilde{\phi}_{1}[\frac{1}{2}\widetilde{\sigma}\sin(\varpi\tau_{0}) \\ -w_{1}^{2}\widetilde{\phi}_{2}-2\widetilde{\rho}D_{0}^{2}\widetilde{\gamma}_{1} ]-D_{1}^{2}\widetilde{\chi}_{1}+2D_{0}\widetilde{\phi}_{1}(D_{0}\widetilde{ \phi}_{2}+D_{1}\widetilde{\phi}_{1}-\widetilde{\rho}D_{0}^{2}\widetilde{\tau} _{1})\\ -2(D_{0}D_{2}\widetilde{\chi}_{1}+D_{0}D_{1}\widetilde{\chi}_{2}+r\widehat{ \rho}D_{0}D_{1}\widetilde{\phi}_{1})-\widetilde{\rho}(\widetilde{\gamma}_{1} D_{0}^{2}\widetilde{\phi}_{1}+rD_{0}^{2}\widetilde{\phi}_{2}),\\ D_{0}^{2}\widetilde{\sigma}_{0}+w_{1}^{2}\widetilde{\phi}_{3}&=\widetilde{m}_{ 0}\cos(p_{2}\tau_{0})+\frac{1}{2}\widetilde{\sigma}^{2}\cos(\varpi\tau_{0})( r\widetilde{\phi}-\widetilde{\phi}_{1})-\widetilde{c}_{2}D_{0} \widetilde{\phi}_{1}+\widetilde{\chi}_{1}[\frac{1}{2}\widetilde{\sigma}^{2} \\ \times\sin(\varpi\tau_{0})-w_{1}^{2}\widetilde{\phi}_{2}-2D_{0}\widetilde{ \chi}_{1}D_{0}\widetilde{\phi}_{1}-4D_{0}D_{1}\widetilde{\phi}_{1}-\widetilde{ \rho}D_{1}^{2}\widetilde{\gamma}_{1}-\widetilde{\chi}_{1}D_{0}^{2}\widetilde{ \phi}_{1}\\ -2D_{0}^{2}\widetilde{\phi}_{2}]-\widetilde{\phi}_{1}(w_{1}^{2}\widetilde{\chi} _{2}-1\widetilde{\rho}w_{1}^{2}\widetilde{\phi}_{1}-w_{1}^{2}\widetilde{\phi} _{1}^{2}-4\widetilde{\rho}D_{0}^{2}\widetilde{\phi}_{1})-\widetilde{\gamma}_{1 }(\widetilde{\rho}D_{0}^{2}\widetilde{\chi}_{1}\\ +2r\widehat{\rho}D_{0}^{2}\widetilde{\phi}_{1})-\widetilde{\rho}w_{1}^{2} \widetilde{\gamma}_{2}-D_{1}^{2}\widetilde{\phi}_{1}-2D_{0}\widetilde{\chi}_{ 1}(D_{1}\widetilde{\phi}_{1}+D_{0}\widetilde{\phi}_{2})-2D_{0}\widetilde{\phi} _{1}(D_{1}\widetilde{\chi}_{1}\\ +r\widehat{\rho}D_{0}\widetilde{\gamma}_{1}+D_{0}\widetilde{\chi}_{2}-r\widehat {\rho}D_{0}\widetilde{\phi}_{1})-2(D_{0}D_{2}\widetilde{\phi}_{1}+\widetilde{ \rho}D_{0}D_{1}\widetilde{\gamma}_{1}+r\widehat{\rho}D_{0}D_{1}\widetilde{ \chi}_{1}\\ +r\widehat{\rho}D_{0}D_{1}\widetilde{\phi}_{1}+D_{0}D_{1}\widetilde{\phi}_{2}+ \widetilde{\chi}_{2}D_{0}^{2}\widetilde{\phi}_{1})-\widetilde{\rho}(D_{0}^{2} \widetilde{\gamma}_{2}+rD_{0}^{2}\widetilde{\chi}_{2}+r^{2}\widetilde{\phi}_{ 2}),\\ D_{0}^{2}\widetilde{\gamma}_{3}+w_{2}^{2}\widetilde{\gamma}_{3}&=\widetilde{\phi }_{1}[\widetilde{\sigma}^{2}\cos(\varpi\tau_{0})+\frac{1}{w_{1}^{2}}\widetilde{ \sigma}^{2}+(2D_{0}\widetilde{\phi}_{1})^{2}-4D_{0}D_{1}\widetilde{\chi}_{1}-2D _{0}^{2}\widetilde{\chi}_{2}\\ +2\widetilde{\rho}_{1}D_{0}^{2}\widetilde{\phi}_{1}]-\widetilde{\chi}_{1}(2D_{0} D_{1}\widetilde{\phi}_{1}+D_{0}^{2}\widetilde{\phi}_{2})-\widetilde{c}_{3}D_{0} \widetilde{\gamma}_{1}-(D_{0}^{2}\widetilde{\phi}_{3}+w_{1}^{2}\widetilde{\phi} _{3})\\ -2D_{0}\widetilde{\phi}_{1}(D_{1}\widetilde{\chi}_{1}-rD_{1}\widetilde{\phi}_{1}+D _{0}\widetilde{\chi}_{2}-\frac{1}{2}\widetilde{\tau}_{1}D_{0}\widetilde{\phi} _{1}-rD_{0}\widetilde{\phi}_{2})-2D_{0}\widetilde{\chi}_{1}(D_{1}\widetilde{\phi} _{1}\\ +D_{0}\widetilde{\phi}_{2})-[D_{0}^{2}\widetilde{\gamma}_{1}+(1+\widetilde{\chi} _{2})D_{0}^{2}\widetilde{\phi}_{1}]-2(D_{0}D_{2}\widetilde{\gamma}_{1}+D_{0}D_{ 2}\widetilde{\phi}_{1}+D_{0}D_{1}\widetilde{\gamma}_{2}\\ +D_{0}D_{1}\widetilde{\phi}_{2}+\widetilde{\phi}_{2}D_{0}^{2}\widetilde{\chi}_{1}). \end{array}  \tag{12}\]  
Based on the preceding groups of Equations (10)-(12), we can solve them sequentially. Accordingly, we will start with the general solutions of the system of Equation (10) which take the following forms  
\[