Appendix A
$$\begin{aligned} \kappa _{11}&= -\frac{2a_1^2 \left( {a_1^2 +\pi ^{2}} \right) ^{2}+\pi ^{4}\sin ^{2}a_1 }{8a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i}\end{aligned}$$
(106)
$$\begin{aligned} \kappa _{12}&= \frac{\omega _1 \pi ^{4}\sin ^{2}a_1 }{4a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(107)
$$\begin{aligned} \mu _{121}&= -\left\{ 9\left( {9a_1^6 -73a_1^4 \pi ^{2}-73a_1^2 \pi ^{4}+9\pi ^{6}} \right) ^{2}\right. \nonumber \\&\quad +1280a_1^2 \pi ^{4}\left( {73a_1^4-18a_1^2 \pi ^{2}+9\pi ^{4}} \right) \nonumber \\&\quad +\left. 128a_1^2 \pi ^{4}\left[ 9\left( {9a_1^2 -\pi ^{2}} \right) ^{2}\cos a_1\right. \right. \nonumber \\&\quad \left. \left. +\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\cos 3a_1 \right] \right\} \nonumber \\&\quad \Big /{\left[ {9\left( {9a_1^4 -82a_1^2 \pi ^{2}+9\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) } \right] }k_1^2 \hbox {i} \nonumber \\ \end{aligned}$$
(108)
$$\begin{aligned} \mu _{122}=\frac{512a_1^2 \pi ^{4}\omega _1 \cos ^{2}0.5a_1 \left[ {120\left( {\pi ^{4}-a_1^4 } \right) +\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\left( {\cos 2a_1 -2\cos a_1 } \right) } \right] }{3\left( {9a_1^4 -82a_1^2 \pi ^{2}+9\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(109)
$$\begin{aligned} \mu _{131}&= -9\left\{ 8\left( {4a_1^6 -13a_1^4 \pi ^{2}-13a_1^2 \pi ^{4}+4\pi ^{6}} \right) ^{2}\right. \nonumber \\&\quad +\,45a_1^2 \pi ^{4}\left( {13a_1^4 -8a_1^2\pi ^{2}+4\pi ^{4}} \right) \nonumber \\&\quad - 9a_1^2 \pi ^{4}\left[ 4\left( {4a_1^2 -\pi ^{2}} \right) ^{2}\cos 2a_1\right. \nonumber \\&\quad \left. \left. +\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\cos 4a_1 \right] \right\} \nonumber \\&\quad \Big /{\left[ {32\left( {4a_1^4 -17a_1^2 \pi ^{2}+4\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) } \right] }k_1^2 \hbox {i}\nonumber \\ \end{aligned}$$
(110)
$$\begin{aligned}&\mu _{132}\nonumber \\&\quad =\frac{81a_1^2 \pi ^{4}\omega _1 \sin ^{2}a_1 \left[ {15\left( {\pi ^{4}-a_1^4 } \right) +\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\cos 2a_1 } \right] }{2\left( {4a_1^4 -17a_1^2 \pi ^{2}+ 4\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \nonumber \\ \end{aligned}$$
(111)
$$\begin{aligned} \tau _{11}&= -\frac{27\pi ^{4}\sin ^{2}a_1 }{8\left( {4\pi ^{2}-a_1^2 } \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i}\end{aligned}$$
(112)
$$\begin{aligned} \tau _{12}&= \frac{9\pi ^{4}\omega _1 \sin ^{2}a_1 }{4\left( {4\pi ^{2}-a_1^2 } \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(113)
$$\begin{aligned} \varsigma _{11}&= 9\pi ^{4}\left[ 3\left( {2731a_1^4 -694a_1^2 \pi ^{2}+75\pi ^{4}} \right) \right. \nonumber \\&-\left( {a_1^4 -34a_1^2 \pi ^{2}+225\pi ^{4}} \right) \cos 4a_1 \nonumber \\&\quad \left. {+2048a_1^2 \left( {4a_1^2 -\pi ^{2}} \right) \cos a_1 } \right] \nonumber \\&\quad \Big / \left[ 16\left( {-4a_1^6 +137a_1^4 \pi ^{2}-934a_1^2 \pi ^{4}+225\pi ^{6}} \right) \right. \nonumber \\&\quad \times \left. \left( {\omega _1 +a_1 \gamma _0 } \right) \right] k_1^2 \hbox {i} \end{aligned}$$
(114)
$$\begin{aligned} \varsigma _{12}&= -9\pi ^{4}\omega _1 \cos ^{2}0.5a_1 \left[ \left( {a_1^4 -34a_1^2 \pi ^{2}+225\pi ^{4}} \right) \right. \nonumber \\&\quad \times \left( {2\cos 2a_1 -3\cos a_1 -\cos 3a_1 } \right) \nonumber \\&\quad \left. +2\left( {225\pi ^{4}+94a_1^2 \pi ^{2}-511a_1^4 } \right) \right] \nonumber \\&\quad \Big /\left[ \left( {-4a_1^6 +137a_1^4 \pi ^{2}-934a_1^2 \pi ^{4}+225\pi ^{6}} \right) \right. \nonumber \\&\quad \times \left. \left( {\omega _1 +a_1 \gamma _0 } \right) \right] k_1^2 \end{aligned}$$
(115)
$$\begin{aligned} \kappa _{21}&= -\frac{8a_1^2 \left( {a_1^2 +\pi ^{2}} \right) ^{2}+\pi ^{4}\sin ^{2}2a_1 }{4a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i}\end{aligned}$$
(116)
$$\begin{aligned} \kappa _{22}&= \frac{\omega _1 \pi ^{4}\sin ^{2}2a_1 }{a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(117)
$$\begin{aligned}&\mu _{211} =\mu _{121} /2\nonumber \\&\quad =-\left\{ 9\left( {9a_1^6 -73a_1^4 \pi ^{2}-73a_1^2 \pi ^{4}+9\pi ^{6}} \right) ^{2}\right. \nonumber \\&\quad +1280a_1^2 \pi ^{4}\left( {73a_1^4 -18a_1^2 \pi ^{2}+9\pi ^{4}} \right) \nonumber \\&\quad +128a_1^2 \pi ^{4}\left[ 9\left( {9a_1^2 -\pi ^{2}} \right) ^{2}\cos a_1\right. \nonumber \\&\quad +\left. \left. \left( {a_1^2 -9\pi ^{2}} \right) ^{2}\cos 3a_1 \right] \right\} \nonumber \\&\quad \Bigg /{\left[ {18\left( {9a_1^4 -82a_1^2 \pi ^{2}+9\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) } \right] }k_1^2 i\nonumber \\ \end{aligned}$$
(118)
$$\begin{aligned} \mu _{212}&= \frac{256a_1^2 \pi ^{4}\omega _1 \cos ^{2}0.5a_1 \left[ {\left( {123a_1^4 -54a_1^2 \pi ^{2}+123\pi ^{4}} \right) +\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\left( {\cos 2a_1 -2\cos a_1 } \right) } \right] }{3\left( {9a_1^4 -82a_1^2 \pi ^{2}+9\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(119)
$$\begin{aligned} \mu _{231}&= -9i\left\{ 25\left( {25a_1^6 -601a_1^4 \pi ^{2}-601a_1^2 \pi ^{4}+25\pi ^{6}} \right) ^{2}\right. \nonumber \\&+29952a_1^2 \pi ^{4}\left( {601a_1^4 -50a_1^2 \pi ^{2}+25\pi ^{4}} \right) \nonumber \\&\quad +1152a_1^2 \pi ^{4}\left[ 25\left( {25a_1^2 -\pi ^{2}} \right) ^{2}\cos a_1\right. \nonumber \\&\quad \left. \left. +\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\cos 5a_1 \right] \right\} \nonumber \\&\quad /\left[ 50\left( {25a_1^4 -626a_1^2 \pi ^{2}+25\pi ^{4}} \right) ^{2}\right. \nonumber \\&\quad \times \left. \left( {\omega _1 +a_1 \gamma _0 } \right) \right] k_1^2\nonumber \\ \end{aligned}$$
(120)
$$\begin{aligned} \mu _{232}&= \frac{5184a_1^2 \pi ^{4}\omega _1 \left[ {4\left( {155\pi ^{4}+50a_1^2 \pi ^{2}-781a_1^4 } \right) -5\left( {25a_1^2 -\pi ^{2}} \right) ^{2}\cos a_1 +\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\cos 5a_1 } \right] }{5\left( {25a_1^4 -626a_1^2 \pi ^{2}+25\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(121)
$$\begin{aligned} \xi _{21}&= -\frac{18\pi ^{4}\left[ {\left( {a_1^4 -34a_1^2 \pi ^{2}+225\pi ^{4}} \right) \cos 4a_1 -2048a_1^2 \left( {4a_1^2 -\pi ^{2}} \right) \cos a_1 -3\left( {2731a_1^4 -694a_1^2 \pi ^{2}+75\pi ^{4}} \right) } \right] }{32\left( {-4a_1^6 +137a_1^4 \pi ^{2}-934a_1^2 \pi ^{4}+225\pi ^{6}} \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i}\nonumber \\ \end{aligned}$$
(122)
$$\begin{aligned} \xi _{22}&= \frac{9\pi ^{4}\omega _1 \sin ^{2}2a_1 }{2\left( {4a_1^2 -\pi ^{2}} \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(123)
$$\begin{aligned} \kappa _{31}&= -\frac{3\left[ {18a_1^2 \left( {a_1^2 +\pi ^{2}} \right) ^{2}+\pi ^{4}\sin ^{2}3a_1 } \right] }{8a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i} \end{aligned}$$
(124)
$$\begin{aligned} \kappa _{32}&= \frac{9\omega _1 \pi ^{4}\sin ^{2}3a_1 }{4a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(125)
$$\begin{aligned}&\mu _{311}=-3\left\{ 8\left( 4a_1^6 -13a_1^4 \pi ^{2}-13a_1^2 \pi ^{4}\right. +4\pi ^{6}\right) ^{2}\nonumber \\&\qquad +45a_1^2 \pi ^{4}\left( {13a_1^4 -8a_1^2 \pi ^{2}+4\pi ^{4}} \right) \nonumber \\&\qquad -9a_1^2 \pi ^{4}\left[ 4\left( {4a_1^2 -\pi ^{2}} \right) ^{2}\cos 2a_1\right. \nonumber \\&\qquad \left. \left. +\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\cos 4a_1 \right] \right\} \nonumber \\&\qquad \Big /\left[ 32\left( {4a_1^4 -17a_1^2 \pi ^{2}+4\pi ^{4}} \right) ^{2}\right. \nonumber \\&\qquad \left. \times \left( {\omega _1 +a_1 \gamma _0} \right) \right] k_1^2 \hbox {i} \end{aligned}$$
(126)
$$\begin{aligned} \mu _{312}&= \frac{27a_1^2 \pi ^{4}\omega _1 \sin ^{2}a_1 \left[ {17a_1^4 -16a_1^2 \pi ^{2}+17\pi ^{4}+\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\cos 2a_1 } \right] }{2\left( {4a_1^4 -17a_1^2 \pi ^{2}+4\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(127)
$$\begin{aligned}&\mu _{321} ={2\mu _{231} }/3\nonumber \\&\quad =-3\left\{ 25\left( {25a_1^6 -601a_1^4 \pi ^{2}-601a_1^2 \pi ^{4}+25\pi ^{6}} \right) ^{2}\right. \nonumber \\&\qquad +29952a_1^2 \pi ^{4}\left( {601a_1^4 -50a_1^2 \pi ^{2}+25\pi ^{4}} \right) \nonumber \\&\qquad +1152a_1^2 \pi ^{4}\left[ 25\left( {25a_1^2 -\pi ^{2}} \right) ^{2}\cos a_1\right. \nonumber \\&\qquad \left. \left. +\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\cos 5a_1 \right] \right\} \nonumber \\&\qquad \Big /\left[ 25\left( {25a_1^4 -626a_1^2 \pi ^{2}+25\pi ^{4}} \right) ^{2}\right. \nonumber \\&\qquad \left. \times \left( {\omega _1 +a_1 \gamma _0 } \right) \right] k_1^2 \hbox {i} \end{aligned}$$
(128)
$$\begin{aligned} \mu _{322}&= \frac{3456a_1^2 \pi ^{4}\omega _1 \left[ {6\left( {521a_1^4 -50a_1^2 \pi ^{2}+105\pi ^{4}} \right) +5\left( {25a_1^2 -\pi ^{2}} \right) ^{2}\cos a_1 \!+\!\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\cos 5a_1 } \right] }{5\left( {25a_1^4 -626a_1^2 \pi ^{2}\!+\!25\pi ^{4}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(129)
$$\begin{aligned} \upsilon _{31}&= -\frac{3\pi ^{4}\sin ^{2}a_1 }{8\left( {4\pi ^{2}-a_1^2 } \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i}\end{aligned}$$
(130)
$$\begin{aligned} \upsilon _{32}&= \frac{3\pi ^{4}\omega _1 \sin ^{2}a_1 }{4\left( {4\pi ^{2}-a_1^2 } \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \end{aligned}$$
(131)
$$\begin{aligned} \varsigma _{31}&= \frac{6\pi ^{4}\left[ {3\left( {2731a_1^4 -694a_1^2 \pi ^{2}+75\pi ^{4}} \right) +2048a_1^2 \left( {4a_1^2 -\pi ^{2}} \right) \cos a_1 -\left( {a_1^4 -34a_1^2 \pi ^{2}+225\pi ^{4}} \right) \cos 4a_1 } \right] }{32\left( {-4a_1^6 +137a_1^4 \pi ^{2}-934a_1^2 \pi ^{4}+225\pi ^{6}} \right) \left( {\omega _1 +a_1 \gamma _0 } \right) }k_1^2 \hbox {i}\nonumber \\ \end{aligned}$$
(132)
$$\begin{aligned} \varsigma _{32}&= -3\pi ^{4}\omega _1 \cos ^{2}0.5a_1 \left[ -\left( {a_1^4 - 34a_1^2 \pi ^{2}+225\pi ^{4}} \right) \right. \nonumber \\&\quad \times \left( {3\cos a_1 -2\cos 2a_1 +\cos 3a_1 }\right) \nonumber \\&\quad \left. +18\left( {57a_1^4 -18a_1^2 \pi ^{2}+25\pi ^{4}} \right) \right] \nonumber \\&\quad \Big /\left[ \left( {-4a_1^6 +137a_1^4 \pi ^{2}-934a_1^2 \pi ^{4}+225\pi ^{6}} \right) \right. \nonumber \\&\quad \times \left. \left( {\omega _1 +a_1 \gamma _0 } \right) \right] k_1^2 \end{aligned}$$
(133)
Appendix B
$$\begin{aligned}&\delta _2^\mathrm{R}=4\pi ^{2}\nonumber \\&\quad \times \frac{a_1 \omega _1 \left( {a_1^2 -9\pi ^{2}} \right) +\sin a_1 \left[ {\omega _1 \left( {a_1^2 +9\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -9\pi ^{2}} \right) } \right] }{\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(134)
$$\begin{aligned}&\delta _2^\mathrm{I}=-\frac{4\pi ^{2}\left( {1+\cos a_1 } \right) \left[ {\omega _1 \left( {a_1^2 +9\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -9\pi ^{2}} \right) } \right] }{\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(135)
$$\begin{aligned}&\chi _1^\mathrm{R}=-2\pi ^{2}\nonumber \\&\quad \times \frac{a_1 \omega _1 \left( {a_1^2 -9\pi ^{2}} \right) +\sin a_1 \left[ {\omega _1 \left( {a_1^2 +9\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -9\pi ^{2}} \right) } \right] }{\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(136)
$$\begin{aligned}&\chi _1^\mathrm{I}\nonumber \\&\quad =-\frac{2\pi ^{2}\left( {1+\cos a_1 } \right) \left[ {\omega _1 \left( {a_1^2 +9\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -9\pi ^{2}} \right) } \right] }{\left( {a_1^2 -9\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(137)
$$\begin{aligned}&\delta _3^\mathrm{R}=18\pi ^{2}\nonumber \\&\quad \times \frac{a_1 \omega _1 \left( {a_1^2 -25\pi ^{2}} \right) +\sin a_1 \left[ {\omega _1 \left( {a_1^2 +25\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -25\pi ^{2}} \right) } \right] }{\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(138)
$$\begin{aligned}&\delta _3^\mathrm{I}\nonumber \\&\quad =-\frac{18\pi ^{2}\left( {1+\cos a_1 } \right) \left[ {\omega _1 \left( {a_1^2 +25\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -25\pi ^{2}} \right) } \right] }{\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(139)
$$\begin{aligned}&\chi _2^\mathrm{R}=-12\pi ^{2}\nonumber \\&\quad \times \frac{a_1 \omega _1 \left( {a_1^2 -25\pi ^{2}} \right) +\sin a_1 \left[ {\omega _1 \left( {a_1^2 +25\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -25\pi ^{2}} \right) } \right] }{\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(140)
$$\begin{aligned}&\chi _2^\mathrm{I}\nonumber \\&\quad =-\frac{24\pi ^{2}\cos ^{2}0.5a_1 \left[ {\omega _1 \left( {a_1^2 +25\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -25\pi ^{2}} \right) } \right] }{\left( {a_1^2 -25\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) } \end{aligned}$$
(141)
Appendix C
$$\begin{aligned}&\delta _3^\mathrm{R}=9\pi ^{2}\nonumber \\&\quad \times \frac{2a_1 \omega _1 \left( {a_1^2 -4\pi ^{2}} \right) -\sin 2a_1 \left[ {\omega _1 \left( {a_1^2 +4\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -4\pi ^{2}} \right) } \right] }{8\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(142)
$$\begin{aligned}&\delta _3^\mathrm{I} =-\frac{9\pi ^{2}\sin ^{2}a_1 \left[ {\omega _1 \left( {a_1^2 +4\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -4\pi ^{2}} \right) } \right] }{4\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(143)
$$\begin{aligned}&\bar{{\delta }}_1^\mathrm{R} =\pi ^{2}\frac{2a_1 \omega _1 -\sin 2a_1 \left( {\omega _1 -2a_1 \kappa \gamma _0 } \right) }{8a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(144)
$$\begin{aligned}&\bar{{\delta }}_1^\mathrm{I} =\frac{\pi ^{2}\sin ^{2}a_1 \left( {\omega _1 -2a_1 \kappa \gamma _0 } \right) }{4a_1^2 \left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(145)
$$\begin{aligned}&\chi _1^\mathrm{R} =-3\pi ^{2}\nonumber \\&\quad \times \frac{2a_1 \omega _1 \left( {a_1^2 -4\pi ^{2}} \right) -\sin 2a_1 \left[ {\omega _1 \left( {a_1^2 +4\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -4\pi ^{2}} \right) } \right] }{8\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(146)
$$\begin{aligned}&\chi _1^\mathrm{I} =-\frac{3\pi ^{2}\sin ^{2}a_1 \left[ {\omega _1 \left( {a_1^2 +4\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {a_1^2 -4\pi ^{2}} \right) } \right] }{4\left( {a_1^2 -4\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) } \end{aligned}$$
(147)
Appendix D
$$\begin{aligned}&\bar{{\delta }}_2^\mathrm{R} =4\pi ^{2}\nonumber \\&\quad \times \frac{3a_1 \omega _1 \left( {9a_1^2 -\pi ^{2}} \right) +\sin 3a_1 \left[ {\omega _1 \left( {9a_1^2 +\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {9a_1^2 -\pi ^{2}} \right) } \right] }{3\left( {9a_1^2 -\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(148)
$$\begin{aligned}&\bar{{\delta }}_2^\mathrm{I} =\frac{8\pi ^{2}\cos ^{2}\left( {{3a_1 }/2} \right) \left[ {\omega _1 \left( {9a_1^2 +\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {9a_1^2 -\pi ^{2}} \right) } \right] }{3\left( {9a_1^2 -\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(149)
$$\begin{aligned}&\bar{{\delta }}_1^\mathrm{R} =2\pi ^{2}\nonumber \\&\quad \times \frac{3a_1 \omega _1 \left( {9a_1^2 -\pi ^{2}} \right) +\sin 3a_1 \left[ {\omega _1 \left( {9a_1^2 +\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {9a_1^2 -\pi ^{2}} \right) } \right] }{3\left( {9a_1^2 -\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) }\end{aligned}$$
(150)
$$\begin{aligned}&\bar{{\delta }}_1^\mathrm{I} =\frac{4\pi ^{2}\cos ^{2}\left( {{3a_1 }/2} \right) \left[ {\omega _1 \left( {9a_1^2 +\pi ^{2}} \right) -2a_1 \kappa \gamma _0 \left( {9a_1^2 -\pi ^{2}} \right) } \right] }{3\left( {9a_1^2 -\pi ^{2}} \right) ^{2}\left( {\omega _1 +a_1 \gamma _0 } \right) } \end{aligned}$$
(151)
