## Appendix B
\right] \tag{19}\]  
\[k_{1}=k_{SDD}(\lambda=i\omega_{c}),k_{2}=k_{SDD}(\lambda=-i\omega_{c}) \tag{20}\]  
\[Q_{2}=i\omega_{c}k_{1}(1-\rho k_{1})\left(e^{-i\omega_{c}\tau_{n0}}-e^{-2i \omega_{c}\tau_{n0}}\right) \tag{21}\]  
\[Q_{0}=i\omega_{c}k_{1}\Big{(}1-e^{i\omega_{c}\tau_{n0}}\Big{)}(1-\rho k_{1}) \Big{(}e^{-i\omega_{c}\tau_{n0}}-e^{-2i\omega_{c}\tau_{n0}}\Big{)} \tag{22}\]  
\[\overline{Q_{2}}=-i\omega_{c}k_{2}(1-\rho k_{2})\Big{(}e^{i\omega_{c}\tau_{n0 }}-e^{2i\omega_{c}\tau_{n0}}\Big{)}. \tag{23}\]  
The particular solution of Equation (18) can be obtained in the form:  
\[\mathbf{x}^{(1)}=\mathbf{P}_{2}^{(1)}A_{1}^{2}e^{2i\omega_{c}\tau_{0}}+\mathbf{P}_{0}^{(1)} A_{1}A_{2}+\overline{\mathbf{P}_{2}^{(1)}}A_{2}^{2}e^{-2i\omega_{c}\tau_{0}}, \tag{24}\]  
where  
\[\mathbf{P}_{n}^{(1)}=\big{(}s_{n}\mathbf{I}-\mathbf{A}_{CS}-e^{-s_{n}\tau_{n0}}\mathbf{B}_{ \mathbf{CS}}\big{)}^{-1}\mathbf{Q}_{n}^{F_{1}},s_{n}=in\omega_{c},(\mathrm{n}=-2,0,2). \tag{25}\]  
**Appendix B.3**: Coefficient Vector \(Q_{1}^{F2}\), \(\overline{Q_{1}^{F2}}\), \(Q_{3}^{F2}\), \(\overline{Q_{1}^{F2}}\) in Equation (53)  
\[\mathbf{Q}_{1}^{F_{2}}=\left[ \begin{array}{c}\psi_{CS}k_{r}v^{-2}\tau_{n0}\alpha \Big{(}G_{1}-\frac{1}{2}\tau_{n0}\alpha G_{2}\Big{)}\\ 0\\ \psi_{CS}\tau_{n0}\alpha\Big{(}G_{1}-\frac{1}{2}\tau_{n0}\alpha G_{2}\Big{)}\\ 0\end{array} \right],\overline{\mathbf{Q}_{1}^{F_{2}}}=\left[ \begin{array}{c}\psi_ {CS}k_{r}v^{-2}\tau_{n0}\alpha\Big{(}\overline{G_{1}}-\frac{1}{2}\tau_{n0} \alpha\overline{G_{2}}\Big{)}\\ 0\\ \psi_{CS}\tau_{n0}\alpha\Big{(}\overline{G_{1}}-\frac{1}{2}\tau_{n0}\alpha\overline {G_{2}}\Big{)}\\ 0\end{array} \right] \tag{26}\]\[ \begin{array}{ll}G_{1}&=\Big{(}P_{01}^{(1)}k_{1}-\rho P_{03}^{(1)}k_{1}-\omega_{ c}^{2}k_{1}^{2}\tau_{n0}\alpha-\omega_{c}^{2}k_{1}k_{2}\tau_{n0}\alpha+2\omega_{c}^{2 }k_{1}^{2}k_{2}\rho\tau_{n0}\alpha\Big{)}\left(1-e^{-i\omega_{c}\tau_{n0}}\right) \\ &-i\omega_{c}P_{24}^{(1)}(1-k_{2}\rho)\big{(}e^{i\omega_{c}\tau_{n0}}-e^{-i \omega_{c}\tau_{n0}}\big{)}-\\ &\Big{(}P_{21}^{(1)}k_{2}-\rho k_{2}P_{23}^{(1)}+\omega_{c}^{2}k_{1}k_{2}\tau_ {n0}\alpha+\omega_{c}^{2}k_{1}^{2}k_{2}\rho\tau_{n0}\alpha\Big{)}\left(e^{-i \omega_{c}\tau_{n0}}-e^{-2i\omega_{c}\tau_{n0}}\right)\\ \overline{G_{1}}&=\Big{(}\overline{P_{01}^{(1)}}k_{2}-\rho k_{2}\overline{P_ {03}^{(1)}}-\omega_{c}^{2}k_{2}^{2}\tau_{n0}\alpha-\omega_{c}^{2}k_{1}k_{2} \tau_{n0}\alpha+2\omega_{c}^{2}k_{2}^{2}k_{1}\rho\tau_{n0}\alpha\Big{)}\left(1 -e^{i\omega_{c}\tau_{n0}}\right)\\ &-i\omega_{c}\overline{P_{24}^{(1)}}(1-k_{1}\rho)\big{(}e^{i\omega_{c}\tau_{n0 }}-e^{-i\omega_{c}\tau_{n0}}\big{)}-\\ &\left(\overline{P_{21}^{(1)}}k_{1}-\rho k_{1}\overline{P_{23}^{(1)}}+\omega_{ c}^{2}k_{1}k_{2}\tau_{n0}\alpha-\omega_{c}^{2}k_{1}^{2}k_{2}\rho\tau_{n0}\alpha \right)\left(e^{i\omega_{c}\tau_{n0}}-e^{2i\omega_{c}\tau_{n0}}\right)\\ \end{array}  \tag{103}\]  
\[ \begin{array}{ll}H_{1}&=i\omega_{c}P_{24}^{(1)}(1-k_{1}\rho)\big{(}e^{-i \omega_{c}\tau_{n0}}-e^{-3i\omega_{c}\tau_{n0}}\big{)}\\ &+\Big{(}P_{21}^{(1)}k_{1}-P_{23}^{(1)}k_{1}\rho+\omega_{c}^{2}k_{1}^{2}\tau_ {n0}\alpha-\omega_{c}^{2}k_{1}^{3}\tau_{n0}\alpha\rho\Big{)}\left(e^{-2i\omega _{c}\tau_{n0}}-e^{-3i\omega_{c}\tau_{n0}}\right)\\ \overline{H_{1}}&=-i\omega_{c}\overline{P_{24}^{(1)}}(1-k_{2}\rho)\big{(}e^{i \omega_{c}\tau_{n0}}-e^{3i\omega_{c}\tau_{n0}}\big{)}\\ &+\Big{(}\overline{P_{21}^{(1)}}k_{2}-\overline{P_{23}^{(1)}}k_{2}\rho+\omega_ {c}^{2}k_{2}^{2}\tau_{n0}\alpha-\omega_{c}^{2}k_{2}^{3}\tau_{n0}\alpha\rho \Big{)}\left(e^{2i\omega_{c}\tau_{n0}}-e^{3i\omega_{c}\tau_{n0}}\right)\\ &H_{2}=\omega_{c}^{2}k_{1}^{2}(1-k_{1}\rho)\big{(}e^{-i\omega_{c}\tau_{n0}}-2 e^{-2i\omega_{c}\tau_{n0}}+e^{-3i\omega_{c}\tau_{n0}}\big{)}\\ \end{array}  \tag{104}\]  
_Appendix B.4. \(c_{r},c_{i}\) in Equation (54)_  
Taking the forcing vector associated with the secular components including \(e^{i\omega_{c}\tau_{0}}\) (Equation (53)) gives:  
\[