Appendix A
(I): Calculating the eigenvalue \(\tilde{\lambda}\).
Since \(\tilde{\lambda}\) is the eigenvalue of \([G(\mathrm{i}\tilde{\omega};\tilde{\mu})J(\mu)]\), one has
$$\begin{aligned} \tilde{\lambda}^3+\frac{a}{\mathrm{i}\tilde{\omega}+\tilde{\mu }}\tilde{\lambda}^2+ \frac{b}{(\mathrm{i}\tilde{\omega}+\tilde {\mu})^2} \tilde{\lambda}+\frac{c}{(\mathrm{i}\tilde{\omega}+\tilde{\mu })^3}=0. \end{aligned}$$
(A.1)
Noticing that \(\tilde{\lambda}=(\hat{\lambda}\mathrm{i}\tilde{\omega};\tilde {\mu})\) is a real number and separating Eq. (A.1) into real and imaginary parts, we have
$$\begin{aligned} &\tilde{\lambda}\bigl(\tilde{\mu}^3-3\tilde{\omega}\tilde{\mu } \bigr)+a\tilde{\lambda}^2\bigl(\tilde{\mu}^2 -\tilde{ \omega}^2\bigr)+b\tilde{\lambda}\tilde{\mu}+c=0, \\ &\tilde{\lambda}\bigl(3\tilde{\mu}^2\tilde{\omega}-\tilde{\omega }^3\bigr)+2a\tilde{\lambda}^2(\tilde{\mu}\tilde{\omega} +b\tilde{\lambda}\tilde{\mu})=0. \end{aligned}$$
Then we have
$$\begin{aligned} &8\tilde{\lambda}\tilde{\mu}^3+8\tilde{\lambda}^2\tilde{ \mu }^2+\bigl(2\tilde{\lambda}b +2a^2\tilde{ \lambda}^3\bigr)\tilde{\mu}+ab\tilde{\lambda}^2-c=0, \end{aligned}$$
(A.2)
$$\begin{aligned} &\tilde{\omega}^2=3\tilde{\mu}^2+2a\tilde{\mu}+b. \end{aligned}$$
(A.3)
By (A.3), we can compute the eigenvalue \(\tilde{\lambda}\).
(II): Calculating the eigenvalue \((\frac{d\tilde{\lambda}}{d\mu} ) |_{\mu=\mu _{0}}\).
By using μ instead of \(\tilde{\mu}\) in Eq. (A.2) and by taking the derivative with respect to μ on both sides of Eq. (A.2), and letting \(\mu=\mu_{0}, \tilde{\lambda}=-1\), we obtain
$$\begin{aligned} \biggl(\frac{d\tilde{\lambda}}{d\mu} \biggr) \bigg|_{\mu=\mu_0} = \frac{12\mu_0^2-8\mu_0+a^2+b}{8\mu_0^3-11a\mu_0^2+2b\mu_0+6a^2\mu _0-2ab}. \end{aligned}$$
(A.4)
(III): Calculating the real and imaginary parts of \((\frac{d\tilde{\lambda}}{d\omega} ) |_{\omega =\omega_{0}}\).
Consider Eq. (A.1)
$$\lambda^3+\frac{a}{\mathrm{i}\tilde{\omega}+\tilde{\mu}}\lambda ^2+ \frac{b}{(\mathrm{i}\tilde{\omega}+\tilde{\mu})^2} \lambda+\frac{c}{(\mathrm{i}\tilde{\omega}+\tilde{\mu})^3}=0. $$
Fixing μ at \(\tilde{\mu}\) and taking the derivative with respect to ω on both sides of Eq. (A.1), we get
$$\begin{aligned} \biggl(\frac{d\tilde{\lambda}}{d\omega} \biggr) \bigg|_{\omega =\tilde{\omega}}= \frac{D_{11}(\tilde{\lambda},\tilde{\mu},\tilde{\omega })+iD_{12}(\tilde{\lambda},\tilde{\mu},\tilde{\omega})}{ D_{21}(\tilde{\lambda},\tilde{\mu},\tilde{\omega })+iD_{22}(\tilde{\lambda},\tilde{\mu},\tilde{\omega})}, \end{aligned}$$
(A.5)
where
$$\begin{aligned} D_{11}(\tilde{\lambda},\tilde{\mu},\tilde{\omega}) =&-2a\tilde { \omega}\tilde{\mu}\tilde{\lambda}^2 \bigl(\tilde{\mu}^2- \tilde{\omega}^2\bigr) +2b\tilde{\omega}\bigl(\tilde{\mu }^2-\tilde{\omega}^2\bigr)-4b\tilde{\omega} \tilde{ \lambda}\bigl(\tilde{\mu}^2-\tilde{\omega}^2 \bigr)^3 +6c\tilde{\omega}\bigl(\tilde{\mu}^2-\tilde{ \omega}^2\bigr)^3\\ &{}-6c\tilde {\omega}\bigl(\tilde{ \mu}^2-\tilde{\omega}^2\bigr)^2 \bigl(\tilde{ \mu}^2-3\tilde{\omega}^2\bigr), \\ D_{12}(\tilde{\lambda},\tilde{\mu},\tilde{\omega}) =& \bigl[a\bigl( \tilde{\mu}^2-\tilde{\omega}^2\bigr)+4a\tilde{\mu}\tilde{ \omega }^2\bigr]\bigl(\tilde{\mu}^2-\tilde{ \omega}^2\bigr)^4 \tilde{\lambda}^2 +3c\bigl( \tilde{\mu}^2-\tilde{\omega}^2\bigr)^4 +\bigl[2b\tilde{\mu}\bigl(\tilde{\mu}^2-\tilde{ \omega}^2\bigr)^2\\ &{}+8b\tilde {\mu}\tilde{ \omega}^2 \bigl(\tilde{\mu}^2-\tilde{\omega}^2 \bigr)\bigr]\bigl(\tilde{\mu}^2-\tilde{\omega }^2\bigr) \tilde{\lambda}, \\ D_{21}(\tilde{\lambda},\tilde{\mu},\tilde{\omega}) =&\bigl(\tilde{\mu }^2-\tilde{\omega}^2\bigr)^4 \bigl[3 \lambda^2\bigl(\tilde{\mu}^2-\tilde{\omega}^2 \bigr) +2\lambda{a}\tilde{\mu}\bigl(\tilde{\mu}^2-\tilde{ \omega}^2\bigr)+b\bigl(\tilde {\mu}^2-\tilde{ \omega}^2\bigr)\bigr], \\ D_{22}(\tilde{\lambda},\tilde{\mu},\tilde{\omega}) =&\bigl(\tilde{\mu }^2-\tilde{\omega}^2\bigr)^4 \bigl[2\tilde{ \lambda} {a}\tilde{\omega}\bigl(\tilde{\mu}^2-\tilde{\omega }^2\bigr)+2b\tilde{\mu}\tilde{\omega}\bigr]. \end{aligned}$$
Then we obtain
$$\begin{aligned} &\operatorname{Re} \biggl(\frac{d\tilde{\lambda}}{d\omega} \biggr) \bigg|_{\omega=\tilde{\omega}}= \frac{D_{11}(\tilde{\lambda},\tilde{\mu},\tilde{\omega })D_{21}(\tilde{\lambda},\tilde{\mu},\tilde{\omega}) +D_{12}(\tilde{\lambda},\tilde{\mu},\tilde{\omega})D_{22}(\tilde {\lambda},\tilde{\mu},\tilde{\omega})}{ D_{21}^2(\tilde{\lambda},\tilde{\mu},\tilde{\omega })-D_{22}^2(\tilde{\lambda},\tilde{\mu},\tilde{\omega})}, \end{aligned}$$
(A.6)
$$\begin{aligned} &\operatorname{Im} \biggl(\frac{d\tilde{\lambda}}{d\omega} \biggr) \bigg|_{\omega=\tilde{\omega}}= \frac{D_{12}(\tilde{\lambda},\tilde{\mu},\tilde{\omega })D_{21}(\tilde{\lambda},\tilde{\mu},\tilde{\omega}) -D_{11}(\tilde{\lambda},\tilde{\mu},\tilde{\omega})D_{22}(\tilde {\lambda},\tilde{\mu},\tilde{\omega})}{ D_{22}^2(\tilde{\lambda},\tilde{\mu},\tilde{\omega })-D_{22}^2(\tilde{\lambda},\tilde{\mu},\tilde{\omega})}. \end{aligned}$$
(A.7)
Using μ
0 instead of \(\tilde{\mu}\), ω
0 instead of \(\tilde{\omega}\), and −1 instead of \(\tilde{\lambda}\), we have
$$\begin{aligned} &\operatorname{Re} \biggl(\frac{d\tilde{\lambda}}{d\omega} \biggr) \bigg|_{\omega=\tilde{\omega}}= \frac{D_{11}(-1,\mu_0,\omega_0)D_{21}(-1,\mu_0,\omega_0) +D_{12}(-1,\mu_0,\omega_0)D_{22}(-1,\mu_0,\omega_0)}{ D_{21}^2(-1,\mu_0,\omega_0)-D_{22}^2(-1,\mu_0,\omega_0)}, \end{aligned}$$
(A.8)
$$\begin{aligned} &\operatorname{Im} \biggl(\frac{d\tilde{\lambda}}{d\omega} \biggr) \bigg|_{\omega=\tilde{\omega}}= \frac{D_{12}(-1,\mu_0,\omega_0)D_{21}(-1,\mu_0,\omega_0) -D_{11}(-1,\mu_0,\omega_0)D_{22}(-1,\mu_0,\omega_0)}{ D_{22}^2(-1,\mu_0,\omega_0)-D_{22}^2(-1,\mu_0,\omega_0)}, \end{aligned}$$
(A.9)
where
$$\begin{aligned} D_{11}(-1,\mu_0,\omega_0) =&-2a \omega_0\mu_0\bigl(\mu_0^2- \omega _0^2\bigr) +2b\omega_0\bigl( \mu_0^2-\omega_0^2\bigr) +4b \omega_0\bigl(\mu_0^2-\omega_0^2 \bigr)^3 +6c\omega_0\bigl(\mu_0^2- \omega_0^2\bigr)^3\\ &{}-6c\omega_0 \bigl(\mu_0^2-\omega _0^2 \bigr)^2\bigl(\mu_0^2-3\omega_0^2 \bigr), \\ D_{12}(-1,\mu_0,\omega_0) =& \bigl[a\bigl( \mu_0^2-\omega_0^2\bigr)+4a \mu_0\omega_0^2\bigr]\bigl( \mu_0^2-\omega _0^2 \bigr)^4+3c\bigl(\mu_0^2- \omega_0^2\bigr)^4 -\bigl[2b\mu_0\bigl(\mu_0^2- \omega_0^2\bigr)^2\\ &{}+8b\mu_0 \omega_0^2\bigl(\mu _0^2- \omega_0^2\bigr)\bigr]\bigl(\mu_0^2- \omega_0^2\bigr), \\ D_{21}(-1,\mu_0,\omega_0) =&\bigl( \mu_0^2-\omega_0^2 \bigr)^4\bigl[3\bigl(\mu _0^2- \omega_0^2\bigr)-2{a}\mu_0\bigl( \mu_0^2-\omega_0^2\bigr)+b\bigl( \mu_0^2-\omega _0^2\bigr)\bigr], \\ D_{22}(-1,\mu_0,\omega_0) =&\bigl( \mu_0^2-\omega_0^2 \bigr)^4\bigl[-2{a}\omega _0\bigl(\mu_0^2- \omega_0^2\bigr)+2b\mu_0 \omega_0\bigr]. \end{aligned}$$
Appendix B
$$\begin{aligned} k_{11} =&\frac{1}{1+a_{11}} \biggl\{ 1-\frac {a_{21}a_{13}}{a_{13}a_{21}-a_{23}(1+a_{11})}- \frac {a_{21}}{1+a_{11}}y_1 \\ &{}-\frac {a_{31}[(1+a_{11})(1+a_{12})-a_{12}a_{21}]-a_{21}[a_{12}a_{31}-a_{32}(1+a_{11})]}{ (1+a_{11})[(1+a_{11})(1+a_{12})-a_{12}a_{21}]}x_1y_1 \biggr\} , \\ k_{12} =&\frac{1}{1+a_{11}} \biggl[\frac {a_{13}(1+a_{11})}{a_{13}a_{21}-a_{23}(1+a_{11})}+ \frac {a_{12}a_{31}-a_{32}(1+a_{11})}{(1+a_{11})(1+a_{12})-a_{12}a_{21}}x_1y_1 \biggr], \\ k_{13} =&\frac{1}{1+a_{11}}x_1y_1, \\ k_{21} =&-\frac{1+a_{11}}{(1+a_{11})(1+a_{12})-a_{12}a_{21}} \biggl\{ \frac{a_{21}}{1+a_{11}} \\ &{}+\frac {a_{31}[(1+a_{11})(1+a_{12})-a_{12}a_{21}]-a_{21}[a_{12}a_{31}-a_{32}(1+a_{11})]}{ (1+a_{11})[(1+a_{11})(1+a_{12})-a_{12}a_{21}]}x_1 \biggr\} , \\ k_{22} =&\frac{1+a_{11}}{(1+a_{11})(1+a_{12})-a_{12}a_{21}}\frac {a_{12}a_{31}-a_{32}(1+a_{11})}{ (1+a_{11})(1+a_{12})-a_{12}a_{21}}x_1, \\ k_{23} =&\frac{1+a_{11}}{(1+a_{11})(1+a_{12})-a_{12}a_{21}}x_1, \\ k_{31} =&-\frac{1}{z_1} \biggl\{ \frac {a_{31}[(1+a_{11})(1+a_{12})-a_{12}a_{21}]-a_{21}[a_{12}a_{31}-a_{32}(1+a_{11})]}{ (1+a_{11})[(1+a_{11})(1+a_{12})-a_{12}a_{21}]}x_1 \biggr\} , \\ k_{32} =&\frac{1}{z_1}\frac{a_{12}a_{31}-a_{32}(1+a_{11})}{ (1+a_{11})(1+a_{12})-a_{12}a_{21}}, \\ k_{33} =&\frac{1}{z_1}, \\ x_1 =&-\frac {[(1+a_{11})(1+a_{12})-a_{12}a_{21}][a_{13}a_{21}-a_{23}(1+a_{11})]}{m_1+n_1}, \\ m_1 =&\bigl[(1+a_{11}) (1+a_{33})-a_{13}a_{31} \bigr] \bigl[(1+a_{11}) (1+a_{12})-a_{12}a_{21} \bigr], \\ n_1 =&\bigl[a_{23}(1+a_{11})-a_{13}a_{21} \bigr] \bigl[a_{12}a_{31}-a_{32}(1+a_{11}) \bigr], \\ y_1 =&-\frac{\{ a_{12}[a_{13}a_{21}-a_{23}(1+a_{11})]+a_{33}[(1+a_{11})(1+a_{12})-a_{12}a_{21}]\} (1+a_{11})}{ [a_{13}a_{21}-a_{23}(1+a_{11})][(1+a_{11})(1+a_{12})-a_{12}a_{21}]}, \\ z_1 =&\frac{m_1+n_1}{(1+a_{11})[(1+a_{11})(1+a_{12})-a_{12}a_{21}]}, \end{aligned}$$
where
$$\begin{aligned} &a_{11}=-\frac{f_{11}'(0)}{\mu}, \quad a_{12}=-\frac{f_{12}'(0)}{\mu },\quad a_{13}=- \frac{f_{13}'(0)}{\mu}, \\ & a_{21}=-\frac{f_{21}'(0)}{\mu},\quad a_{22}=-\frac{f_{22}'(0)}{\mu },\quad a_{23}=- \frac{f_{23}'(0)}{\mu}, \\ &a_{31}=-\frac{f_{31}'(0)}{\mu},\quad a_{32}=-\frac{f_{32}'(0)}{\mu },\quad a_{33}=- \frac{f_{33}'(0)}{\mu}. \end{aligned}$$
Appendix C
$$\begin{aligned} l_{11} =&\frac{1}{1+d_{11}} \biggl\{ 1-\frac {d_{21}d_{13}}{d_{13}d_{21}-d_{23}(1+d_{11})}- \frac {d_{21}}{1+d_{11}}y_2 \\ &{}-\frac {d_{31}[(1+d_{11})(1+d_{12})-d_{12}d_{21}]-d_{21}[d_{12}d_{31}-d_{32}(1+d_{11})]}{ (1+d_{11})[(1+d_{11})(1+d_{12})-d_{12}d_{21}]}x_2y_2 \biggr\} , \\ l_{12} =&\frac{1}{1+d_{11}} \biggl[\frac {d_{13}(1+d_{11})}{d_{13}d_{21}-d_{23}(1+d_{11})}+ \frac {d_{12}d_{31}-d_{32}(1+d_{11})}{(1+d_{11})(1+d_{12})-d_{12}d_{21}}x_2y_2 \biggr], \\ l_{13} =&\frac{1}{1+d_{11}}x_2y_2, \\ l_{21} =&-\frac{1+d_{11}}{(1+d_{11})(1+d_{12})-d_{12}d_{21}} \biggl\{ \frac{d_{21}}{1+d_{11}} \\ &{}+\frac {d_{31}[(1+d_{11})(1+d_{12})-d_{12}d_{21}]-d_{21}[d_{12}d_{31}-d_{32}(1+d_{11})]}{ (1+d_{11})[(1+d_{11})(1+d_{12})-d_{12}d_{21}]}x_2 \biggr\} , \\ l_{22} =&\frac{1+d_{11}}{(1+d_{11})(1+d_{12})-d_{12}d_{21}}\frac {d_{12}d_{31}-d_{32}(1+d_{11})}{ (1+d_{11})(1+d_{12})-d_{12}d_{21}}x_2, \\ l_{23} =&\frac{1+d_{11}}{(1+d_{11})(1+d_{12})-d_{12}d_{21}}x_2, \\ l_{31} =&-\frac{1}{z_2} \biggl\{ \frac {d_{31}[(1+d_{11})(1+d_{12})-d_{12}d_{21}]-d_{21}[d_{12}d_{31}-d_{32}(1+d_{11})]}{ (1+d_{11})[(1+d_{11})(1+d_{12})-d_{12}d_{21}]}x_2 \biggr\} , \\ l_{32} =&\frac{1}{z_2}\frac{d_{12}d_{31}-d_{32}(1+d_{11})}{ (1+d_{11})(1+d_{12})-d_{12}d_{21}}, \\ l_{33} =&\frac{1}{z_2}, \\ x_2 =&-\frac {[(1+d_{11})(1+d_{12})-d_{12}d_{21}][d_{13}d_{21}-d_{23}(1+d_{11})]}{m_2+n_2}, \\ m_2 =&\bigl[(1+d_{11}) (1+d_{33})-d_{13}d_{31} \bigr] \bigl[(1+d_{11}) (1+d_{12})-d_{12}d_{21} \bigr], \\ n_2 =&\bigl[d_{23}(1+d_{11})-d_{13}d_{21} \bigr] \bigl[d_{12}d_{31}-d_{32}(1+d_{11}) \bigr], \\ y_2 =&-\frac{\{ d_{12}[d_{13}d_{21}-d_{23}(1+d_{11})]+d_{33}[(1+d_{11})(1+d_{12})-d_{12}d_{21}]\} (1+d_{11})}{ [d_{13}d_{21}-d_{23}(1+d_{11})][(1+d_{11})(1+d_{12})-d_{12}d_{21}]}, \\ z_2 =&\frac{m_2+n_2}{(1+d_{11})[(1+d_{11})(1+d_{12})-d_{12}d_{21}]}, \end{aligned}$$
where
$$\begin{aligned} &d_{11}=-\frac{f_{11}'(0)}{2\mathrm{i}\tilde{\omega}+\mu}, \quad d_{12}=-\frac{f_{12}'(0)}{2\mathrm{i}\tilde{\omega}+\mu },\quad d_{13}=- \frac{f_{13}'(0)}{2\mathrm{i}\tilde{\omega}+\mu}, \\ & d_{21}=-\frac{f_{21}'(0)}{2\mathrm{i}\tilde{\omega}+\mu}, \quad d_{22}=-\frac{f_{22}'(0)}{2\mathrm{i}\tilde{\omega}+\mu}, \quad d_{23}=-\frac{f_{23}'(0)}{\mu}, \\ &d_{31}=-\frac{f_{31}'(0)}{2\mathrm{i}\tilde{\omega}+\mu}, \quad d_{32}=-\frac{f_{32}'(0)}{2\mathrm{i}\tilde{\omega}+\mu },\qquad d_{33}=- \frac{f_{33}'(0)}{2\mathrm{i}\tilde{\omega}+\mu}. \end{aligned}$$
Appendix D
$$\begin{aligned} &t_1=\frac{k_{11}f_{11}^{\prime\prime}(0)+k_{12}f_{21}^{\prime \prime}(0)+k_{13}f_{31}^{\prime\prime}(0)}{\mu},\qquad t_2=\frac{k_{11}f_{12}^{\prime\prime}(0)+k_{12}f_{22}^{\prime \prime}(0)+k_{13}f_{32}^{\prime\prime}(0)}{\mu}, \\ &t_3=\frac{k_{11}f_{13}^{\prime\prime}(0)+k_{12}f_{23}^{\prime \prime}(0)+k_{13}f_{33}^{\prime\prime}(0)}{\mu}, \qquad t_4=\frac{k_{21}f_{11}^{\prime\prime}(0)+k_{22}f_{21}^{\prime \prime}(0)+k_{23}f_{31}^{\prime\prime}(0)}{\mu}, \\ &t_5=\frac{k_{21}f_{12}^{\prime\prime}(0)+k_{22}f_{22}^{\prime \prime}(0)+k_{23}f_{32}^{\prime\prime}(0)}{\mu},\qquad t_6=\frac{k_{21}f_{13}^{\prime\prime}(0)+k_{22}f_{23}^{\prime \prime}(0)+k_{23}f_{33}^{\prime\prime}(0)}{\mu}, \\ &t_7=\frac{k_{31}f_{11}^{\prime\prime}(0)+k_{32}f_{21}^{\prime \prime}(0)+k_{33}f_{31}^{\prime\prime}(0)}{\mu}, \qquad t_8=\frac{k_{31}f_{12}^{\prime\prime}(0)+k_{32}f_{22}^{\prime \prime}(0)+k_{33}f_{32}^{\prime\prime}(0)}{\mu}, \\ &t_9=\frac{k_{31}f_{13}^{\prime\prime}(0)+k_{32}f_{23}^{\prime \prime}(0)+k_{33}f_{33}^{\prime\prime}(0)}{\mu},\\ &r_1=\frac{[f_{13}'(0)(f_{22}'(0)+\tilde{\lambda}\tilde{\mu })-f_{12}'(0)f_{23}'(0)]^2 +(\tilde{\lambda}\tilde{\mu})^2}{[(f_{11}'(0)+\tilde{\lambda }\tilde{\mu}) (f_{22}'(0)+\tilde{\lambda}\tilde{\mu})-f_{12}'(0)f_{21}'(0)]^2+ [\tilde{\lambda}\tilde{\mu}(f_{11}'(0)+2\tilde{\lambda}\tilde {\mu}+f_{22}'(0))]^2}, \\ &r_2=\frac{(f_{11}'(0)f_{23}'(0)+\tilde{\lambda}\tilde{\mu }f_{23}'(0)-f_{13}'(0)f_{21}'(0))^2 +(\tilde{\lambda}\tilde{\mu}f_{23}'(0))^2}{ [(f_{11}'(0)+\tilde{\lambda}\tilde{\mu}) (f_{22}'(0)+\tilde{\lambda}\tilde{\mu})-f_{12}'(0)f_{21}'(0)]^2+ [\tilde{\lambda}\tilde{\mu}(f_{11}'(0)+2\tilde{\lambda}\tilde {\mu}+f_{22}'(0))]^2}. \end{aligned}$$
Appendix E
$$\begin{aligned} &n_1=\frac{l_{11}f_{11}^{\prime\prime}(0)+l_{12}f_{21}^{\prime \prime}(0)+l_{13}f_{31}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}},\qquad n_2=\frac{l_{11}f_{12}^{\prime\prime}(0)+l_{12}f_{22}^{\prime \prime}(0)+l_{13}f_{32}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \\ &n_3=\frac{l_{11}f_{13}^{\prime\prime}(0)+l_{12}f_{23}^{\prime \prime}(0)+l_{13}f_{33}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \qquad n_4=\frac{l_{21}f_{11}^{\prime\prime}(0)+l_{22}f_{21}^{\prime \prime}(0)+l_{23}f_{31}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \\ &n_5=\frac{l_{21}f_{12}^{\prime\prime}(0)+l_{22}f_{22}^{\prime \prime}(0)+l_{23}f_{32}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \qquad n_6=\frac{l_{21}f_{13}^{\prime\prime}(0)+l_{22}f_{23}^{\prime \prime}(0)+l_{23}f_{33}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \\ &n_7=\frac{l_{31}f_{11}^{\prime\prime}(0)+l_{32}f_{21}^{\prime \prime}(0)+l_{33}f_{31}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \qquad n_8=\frac{l_{31}f_{12}^{\prime\prime}(0)+l_{32}f_{22}^{\prime \prime}(0)+l_{33}f_{32}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}, \\ &n_9=\frac{l_{31}f_{13}^{\prime\prime}(0)+l_{32}f_{23}^{\prime \prime}(0)+l_{33}f_{33}^{\prime\prime}(0)}{2\mathrm{i}\tilde {\omega}+\tilde{\mu}}. \end{aligned}$$
Appendix F
$$\begin{aligned} p_1^{(1)} =&-\frac{1}{4}f_{11}^{\prime\prime }(0) \biggl[(t_1r_1+t_2r_2+t_3)v_1 + \frac {1}{2}\bigl(n_1v_1^2+n_2v_2^2+n_3 \bigr)\overline{v_1}\biggr] \\ &{}-\frac{1}{4}f_{12}^{\prime\prime }(0)\biggl[(t_4r_1+t_5r_2+t_6)v_2 + \frac {1}{2}\bigl(n_4v_1^2+n_5v_2^2+n_6 \bigr)\overline{v_2}\biggr] \\ &{}-\frac{1}{4}f_{13}^{\prime\prime }(0)\biggl[(t_7r_1+t_8r_2+t_9)v_2 + \frac{1}{2}\bigl(n_7v_1^2+n_8v_2^2+n_9 \bigr)\biggr] \\ &{}-\frac{1}{8}\bigl[f_{111}^{\prime\prime\prime}(0)v_1^2 \overline {v_1}+f_{122}^{\prime\prime\prime}(0)v_2^2 \overline {v_2}-f_{333}^{\prime\prime\prime}(0)\bigr], \\ p_1^{(2)} =&-\frac{1}{4}f_{21}^{\prime\prime }(0) \biggl[(t_1r_1+t_2r_2+t_3)v_1 + \frac {1}{2}\bigl(n_1v_1^2+n_2v_2^2+n_3 \bigr)\overline{v_1}\biggr] \\ &{}-\frac{1}{4}f_{22}^{\prime\prime }(0)\biggl[(t_4r_1+t_5r_2+t_6)v_2 + \frac {1}{2}\bigl(n_4v_1^2+n_5v_2^2+n_6 \bigr)\overline{v_2}\biggr] \\ &{}-\frac{1}{4}f_{23}^{\prime\prime }(0)\biggl[(t_7r_1+t_8r_2+t_9)v_2+ \frac{1}{2}\bigl(n_7v_1^2+n_8v_2^2+n_9 \bigr)\biggr] \\ &{}-\frac{1}{8}\bigl[f_{211}^{\prime\prime\prime}(0)v_1^2 \overline {v_1}+f_{222}^{\prime\prime\prime}(0)v_2^2 \overline {v_2}-f_{223}^{\prime\prime\prime}(0)\bigr], \\ p_1^{(3)} =&-\frac{1}{4}f_{31}^{\prime\prime }(0) \biggl[(t_1r_1+t_2r_2+t_3)v_1+ \frac {1}{2}\bigl(n_1v_1^2+n_2v_2^2+n_3 \bigr)\overline{v_1}\biggr] \\ &{}+\frac{1}{4}f_{32}^{\prime\prime }(0)\biggl[(t_4r_1+t_5r_2+t_6)v_2+ \frac {1}{2}\bigl(n_4v_1^2+n_5v_2^2+n_6 \bigr)\overline{v_2}\biggr] \\ &{}+\frac{1}{4}f_{33}^{\prime\prime }(0)\biggl[(t_7r_1+t_8r_2+t_9)v_2+ \frac{1}{2}\bigl(n_7v_1^2+n_8v_2^2+n_9 \bigr)\biggr] \\ &{}-\frac{1}{8}\bigl[f_{311}^{\prime\prime\prime}(0)v_1^2 \overline {v_1}+f_{322}^{\prime\prime\prime}(0)v_2^2 \overline {v_2}-f_{333}^{\prime\prime\prime}(0)\bigr]. \end{aligned}$$
