APPENDICES
APPENDIX A
$$\begin{gathered} {{a}_{i}} = 4h_{i}^{2}\mu {{\omega }_{i}}( - 38{{l}^{2}}\mu \omega _{i}^{2} + 16{{l}^{2}}\mu l_{i}^{2} + 32{{l}^{2}}\omega _{i}^{2} - 13{{l}^{2}}l_{i}^{2} \\ + \;3\mu l_{i}^{2} - 6\mu \omega _{i}^{2} - 3l_{i}^{2} + 6\omega _{i}^{2}),\quad (i = 1,2), \\ \end{gathered} $$
$$\begin{gathered} {{a}_{{3 + i}}} = 4h_{i}^{2}\mu {{\omega }_{i}}(38{{l}^{2}}\mu \omega _{i}^{2} + 16{{l}^{2}}\mu l_{i}^{2} - 104{{l}^{2}}\omega _{i}^{4} \\ - \;32{{l}^{2}}\omega _{i}^{2} - 40{{l}^{2}}l_{i}^{2}\omega _{i}^{2} + 104{{l}^{2}}{{l}_{i}}\omega _{i}^{2} - 13{{l}^{2}}l_{i}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;3\mu l_{i}^{2} - 12l_{i}^{2}\omega _{i}^{2} + 6\mu \omega _{i}^{2} + 24{{l}_{i}}\omega _{i}^{2} - 3l_{i}^{2} \\ - \;24\omega _{i}^{4} - 6\omega _{i}^{2}),\quad (i = 1,2), \\ \end{gathered} $$
$$\begin{gathered} {{a}_{7}} = 8{{h}_{1}}{{h}_{2}}\mu \sqrt {{{\omega }_{1}}} \sqrt {{{\omega }_{2}}} (38{{l}^{2}}\mu {{\omega }_{2}}{{\omega }_{1}} + 16{{l}^{2}}\mu {{l}_{1}}{{l}_{2}} \\ - \;26{{l}^{2}}{{\omega }_{2}}\omega _{1}^{3} + 52{{l}^{2}}\omega _{2}^{2}\omega _{1}^{2} + 26{{l}^{2}}{{l}_{2}}\omega _{1}^{2} - 10{{l}^{2}}{{l}_{1}}{{l}_{2}}\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} - \;26{{l}^{2}}\omega _{2}^{3}{{\omega }_{1}} - 32{{l}^{2}}{{\omega }_{2}}{{\omega }_{1}} + 26{{l}^{2}}{{l}_{1}}{{\omega }_{2}}{{\omega }_{1}} - 26{{l}^{2}}{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} \\ + \;20{{l}^{2}}{{l}_{1}}{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} - 26{{l}^{2}}{{l}_{1}}\omega _{2}^{2} - 10{{l}^{2}}{{l}_{1}}{{l}_{2}}\omega _{2}^{2} - 13{{l}^{2}}{{l}_{1}}{{l}_{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3\mu {{l}_{1}}{{l}_{2}} - 3{{l}_{1}}{{l}_{2}}\omega _{1}^{2} + 6{{l}_{2}}\omega _{1}^{2} + 6\mu {{\omega }_{2}}{{\omega }_{1}} + 6{{l}_{1}}{{\omega }_{2}}{{\omega }_{1}} \\ + \;6{{l}_{1}}{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} - 6{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} - 6{{l}_{1}}\omega _{2}^{2} - 3{{l}_{1}}{{l}_{2}}\omega _{2}^{2} - 3{{l}_{1}}{{l}_{2}} \\ - \;6{{\omega }_{2}}\omega _{1}^{3} + 12\omega _{2}^{2}\omega _{1}^{2} - 6\omega _{2}^{3}{{\omega }_{1}} - 6{{\omega }_{2}}{{\omega }_{1}}), \\ \end{gathered} $$
$$\begin{gathered} {{a}_{8}} = - 8{{h}_{1}}{{h}_{2}}\mu \sqrt {{{\omega }_{1}}} \sqrt {{{\omega }_{2}}} ( - 38{{l}^{2}}\mu {{\omega }_{2}}{{\omega }_{1}} + 16{{l}^{2}}\mu {{l}_{1}}{{l}_{2}} \\ + \;26{{l}^{2}}{{\omega }_{2}}\omega _{1}^{3} + 52{{l}^{2}}\omega _{2}^{2}\omega _{1}^{2} + 26{{l}^{2}}{{l}_{2}}\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} - \;10{{l}^{2}}{{l}_{1}}{{l}_{2}}\omega _{1}^{2} + 26{{l}^{2}}\omega _{2}^{3}{{\omega }_{1}} + 32{{l}^{2}}{{\omega }_{2}}{{\omega }_{1}} \\ - \;26{{l}^{2}}{{l}_{1}}{{\omega }_{2}}{{\omega }_{1}} + 26{{l}^{2}}{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} - 20{{l}^{2}}{{l}_{1}}{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} \\ \end{gathered} $$
$$\begin{gathered} - \;26{{l}^{2}}{{l}_{1}}\omega _{2}^{2} - 10{{l}^{2}}{{l}_{1}}{{l}_{2}}\omega _{2}^{2} - 13{{l}^{2}}{{l}_{1}}{{l}_{2}} + 3\mu {{l}_{1}}{{l}_{2}} \\ - \;3{{l}_{1}}{{l}_{2}}\omega _{1}^{2} + 6{{l}_{2}}\omega _{1}^{2} - 6\mu {{\omega }_{2}}{{\omega }_{1}} - 6{{l}_{1}}{{\omega }_{2}}{{\omega }_{1}} \\ \end{gathered} $$
$$\begin{gathered} - \;6{{l}_{1}}{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} + 6{{l}_{2}}{{\omega }_{2}}{{\omega }_{1}} - 6{{l}_{1}}\omega _{2}^{2} - 3{{l}_{1}}{{l}_{2}}\omega _{2}^{2} - 3{{l}_{1}}{{l}_{2}} \\ + \;6{{\omega }_{2}}\omega _{1}^{3} + 12\omega _{2}^{2}\omega _{1}^{2} + 6\omega _{2}^{3}{{\omega }_{1}} + 6{{\omega }_{2}}{{\omega }_{1}}), \\ \end{gathered} $$
$$\begin{gathered} {{b}_{1}} = - 4h_{1}^{2}\mu \omega _{1}^{2}(70\mu {{l}_{1}}{{l}^{2}} + 92{{l}_{1}}{{l}^{2}}\omega _{1}^{2} - 116{{l}^{2}}\omega _{1}^{2} - 46l_{1}^{2}{{l}^{2}} \\ + \;29{{l}_{1}}{{l}^{2}} + 12\mu {{l}_{1}} + 24{{l}_{1}}\omega _{1}^{2} - 12l_{1}^{2} + 6{{l}_{1}} - 24\omega _{1}^{2}), \\ \end{gathered} $$
$$\begin{gathered} {{b}_{2}} = - 4h_{2}^{2}\mu \omega _{2}^{2}(70\mu {{l}_{2}}{{l}^{2}} + 92{{l}_{2}}{{l}^{2}}\omega _{2}^{2} + 116{{l}^{2}}\omega _{2}^{2} + 46l_{2}^{2}{{l}^{2}} \\ + \;29{{l}_{2}}{{l}^{2}} + 12\mu {{l}_{2}} + 24{{l}_{2}}\omega _{2}^{2} + 12l_{2}^{2} + 6{{l}_{2}} + 24\omega _{2}^{2}), \\ \end{gathered} $$
$$\begin{gathered} {{b}_{4}} = 4{{h}_{1}}{{h}_{2}}\mu \sqrt {{{\omega }_{1}}} \sqrt {{{\omega }_{2}}} (70{{l}^{2}}\mu {{l}_{2}}{{\omega }_{1}} + 70{{l}^{2}}\mu {{l}_{1}}{{\omega }_{2}} \\ + \;23{{l}^{2}}{{l}_{2}}\omega _{1}^{3} - 116{{l}^{2}}{{\omega }_{2}}\omega _{1}^{2} + 23{{l}^{2}}{{l}_{1}}{{\omega }_{2}}\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} - \;46{{l}^{2}}{{l}_{2}}{{\omega }_{2}}\omega _{1}^{2} + 116{{l}^{2}}\omega _{2}^{2}{{\omega }_{1}} - 46{{l}^{2}}{{l}_{1}}\omega _{2}^{2}{{\omega }_{1}} \\ + \;23{{l}^{2}}{{l}_{2}}\omega _{2}^{2}{{\omega }_{1}} + 29{{l}^{2}}{{l}_{2}}{{\omega }_{1}} - 46{{l}^{2}}{{l}_{1}}{{l}_{2}}{{\omega }_{1}} \\ \end{gathered} $$
$$\begin{gathered} + \;23{{l}^{2}}{{l}_{1}}\omega _{2}^{3} + 29{{l}^{2}}{{l}_{1}}{{\omega }_{2}} + 46{{l}^{2}}{{l}_{1}}{{l}_{2}}{{\omega }_{2}} + 12\mu {{l}_{2}}{{\omega }_{1}} \\ + \;12\mu {{l}_{1}}{{\omega }_{2}} + 6{{l}_{2}}\omega _{1}^{3} + 6{{l}_{1}}{{\omega }_{2}}\omega _{1}^{2} - 12{{l}_{2}}{{\omega }_{2}}\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} - 12{{l}_{1}}\omega _{2}^{2}{{\omega }_{1}} + 6{{l}_{2}}\omega _{2}^{2}{{\omega }_{1}} - 12{{l}_{1}}{{l}_{2}}{{\omega }_{1}} + 6{{l}_{2}}{{\omega }_{1}} + 6{{l}_{1}}\omega _{2}^{3} \\ + \;6{{l}_{1}}{{\omega }_{2}} + 12{{l}_{1}}{{l}_{2}}{{\omega }_{2}} - 24{{\omega }_{2}}\omega _{1}^{2} + 24\omega _{2}^{2}{{\omega }_{1}}), \\ \end{gathered} $$
$$\begin{gathered} {{b}_{5}} = 4{{h}_{1}}{{h}_{2}}\mu \sqrt {{{\omega }_{1}}} \sqrt {{{\omega }_{2}}} (70{{l}^{2}}\mu {{l}_{2}}{{\omega }_{1}} - 70{{l}^{2}}\mu {{l}_{1}}{{\omega }_{2}} \\ + \;23{{l}^{2}}{{l}_{2}}\omega _{1}^{3} + 116{{l}^{2}}{{\omega }_{2}}\omega _{1}^{2} - 23{{l}^{2}}{{l}_{1}}{{\omega }_{2}}\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;46{{l}^{2}}{{l}_{2}}{{\omega }_{2}}\omega _{1}^{2} + 116{{l}^{2}}\omega _{2}^{2}{{\omega }_{1}} - 46{{l}^{2}}{{l}_{1}}\omega _{2}^{2}{{\omega }_{1}} \\ + \;23{{l}^{2}}{{l}_{2}}\omega _{2}^{2}{{\omega }_{1}} + 29{{l}^{2}}{{l}_{2}}{{\omega }_{1}} - 46{{l}^{2}}{{l}_{1}}{{l}_{2}}{{\omega }_{1}} \\ \end{gathered} $$
$$\begin{gathered} - \;23{{l}^{2}}{{l}_{1}}\omega _{2}^{3} - 29{{l}^{2}}{{l}_{1}}{{\omega }_{2}} - 46{{l}^{2}}{{l}_{1}}{{l}_{2}}{{\omega }_{2}} + 12\mu {{l}_{2}}{{\omega }_{1}} \\ - \;12\mu {{l}_{1}}{{\omega }_{2}} + 6{{l}_{2}}\omega _{1}^{3} - 6{{l}_{1}}{{\omega }_{2}}\omega _{1}^{2} + 12{{l}_{2}}{{\omega }_{2}}\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} - \;12{{l}_{1}}\omega _{2}^{2}{{\omega }_{1}} + 6{{l}_{2}}\omega _{2}^{2}{{\omega }_{1}} - 12{{l}_{1}}{{l}_{2}}{{\omega }_{1}} + 6{{l}_{2}}{{\omega }_{1}} - 6{{l}_{1}}\omega _{2}^{3} \\ - \;6{{l}_{1}}{{\omega }_{2}} - 12{{l}_{1}}{{l}_{2}}{{\omega }_{2}} + 24{{\omega }_{2}}\omega _{1}^{2} + 24\omega _{2}^{2}{{\omega }_{1}}). \\ \end{gathered} $$
APPENDIX B
$${{f}_{{2,0}}} = \frac{{ - 2h_{1}^{2}\mu }}{{\omega _{1}^{2}\omega _{2}^{2}(4\omega _{1}^{2} - \omega _{2}^{2})({{l}_{1}}(2\mu {{l}^{2}} + {{l}^{2}} + \mu - \omega _{1}^{2} + 1) - 2((3\mu - 2){{l}^{2}} + ({{l}^{2}} + 1)\omega _{1}^{2} + \mu - 1))}}$$
$$\begin{gathered} \times \;[4\omega _{1}^{4}(\omega _{1}^{2}(2(26{{l}^{2}} + 3)\mu \omega _{2}^{2} + 24\mu ((29 - 32\mu ){{l}^{2}} \\ - \;3\mu + 3) + 9(7{{l}^{2}} + 1)\omega _{2}^{4}) + 7\mu \omega _{2}^{2}((32\mu - 29){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3(\mu - 1)) - 36(7{{l}^{2}} + 1)\omega _{2}^{2}\omega _{1}^{4}) \\ + \;2l_{1}^{2}\omega _{1}^{2}(\omega _{2}^{2}(2(23{{l}^{2}} + 3)\mu \omega _{1}^{2} + {{l}^{2}}\mu (29\mu + 52) \\ \end{gathered} $$
$$\begin{gathered} + \;24(6{{l}^{2}} + 1)\omega _{1}^{4} + 3\mu (\mu + 2)) \\ + \;24\mu \omega _{1}^{2}((29\mu - 26){{l}^{2}} + 3(\mu - 1)) \\ \end{gathered} $$
$$\begin{gathered} - \;6(6{{l}^{2}} + 1)\omega _{1}^{2}\omega _{2}^{4}) + 2{{l}_{1}}\omega _{1}^{2}( - 6\omega _{1}^{2}\omega _{2}^{4}((7 - 8\mu ){{l}^{2}} \\ + \;(6{{l}^{2}} + 1)\omega _{1}^{2} - \mu + 1) + \omega _{2}^{2}(8\omega _{1}^{4}(3 - {{l}^{2}}(\mu - 21)) \\ \end{gathered} $$
$$\begin{gathered} + \;\mu \omega _{1}^{2}(13(8 - 29\mu ){{l}^{2}} - 39\mu + 12) \\ + \;\mu ((\mu (175\mu - 416) + 232){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3(\mu - 1)(5\mu - 8)) + 24(6{{l}^{2}} + 1)\omega _{1}^{6}) \\ + \;24\mu \omega _{1}^{2}(\omega _{1}^{2}((29\mu - 26){{l}^{2}} + 3(\mu - 1)) \\ \end{gathered} $$
$$\begin{gathered} - \;(\mu - 1)((35\mu - 29){{l}^{2}} + 3(\mu - 1)))) \\ + \;l_{1}^{3}(6\omega _{1}^{2}\omega _{2}^{4}((6 - 7\mu ){{l}^{2}} + (5{{l}^{2}} + 1)\omega _{1}^{2} - \mu + 1) \\ \end{gathered} $$
$$\begin{gathered} - \;\omega _{2}^{2}( - 4\omega _{1}^{4}((62\mu - 36){{l}^{2}} + 9\mu - 6) \\ + \;\mu \omega _{1}^{2}({{l}^{2}}(115 - 26\mu ) - 3(\mu - 5)) \\ \end{gathered} $$
$$\begin{gathered} + \;5(\mu - 1)\mu ((32\mu - 26){{l}^{2}} + 3(\mu - 1)) \\ + \;24(5{{l}^{2}} + 1)\omega _{1}^{6}) + 24\mu \omega _{1}^{2}(\omega _{1}^{2}((23 - 26\mu ){{l}^{2}} - 3\mu \\ + \;3) + (\mu - 1)((32\mu - 26){{l}^{2}} + 3(\mu - 1))))]. \\ \end{gathered} $$
$${{f}_{{0,2}}} = \frac{{ - 12\mu h_{2}^{2}}}{{\omega _{1}^{3}({{l}_{1}}(2\mu {{l}^{2}} + {{l}^{2}} - \omega _{1}^{2} + \mu + 1) - 2((3\mu - 2){{l}^{2}} + ({{l}^{2}} + 1)\omega _{1}^{2} + \mu - 1))}}$$
$$ \times \;\frac{1}{{(4\omega _{2}^{5} - 17\omega _{1}^{2}\omega _{2}^{3} + 4\omega _{1}^{4}{{\omega }_{2}})}}[2( - 24(7{{l}^{2}} + 1)\omega _{1}^{2}\omega _{2}^{8}$$
$$\begin{gathered} + \;2(51(7{{l}^{2}} + 1)\omega _{1}^{4} - ( - 4(6{{l}^{2}} + 1)l_{2}^{2} \\ + \;3(23{{l}^{2}} + 3)\mu {{l}_{2}} + 6(26{{l}^{2}} + 3)\mu )\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;3\mu ((29\mu - 26){{l}^{2}} + 3(\mu - 1)){{l}_{2}})\omega _{2}^{6} \\ - \;(24(7{{l}^{2}} + 1)\omega _{1}^{6} - 2( - 17(6{{l}^{2}} + 1)l_{2}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;9(23{{l}^{2}} + 3)\mu {{l}_{2}} + 4(26{{l}^{2}} + 3)\mu )\omega _{1}^{4} \\ + \;\mu (5(23{{l}^{2}} + 3)l_{2}^{2} + 6((87\mu + 52){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;9\mu + 6){{l}_{2}} - 42((32\mu - 29){{l}^{2}} + 3(\mu - 1)))\omega _{1}^{2} \\ + \;2\mu {{l}_{2}}((87 - 96\mu ){{l}^{2}} - 9\mu - 4((29\mu - 26){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3(\mu - 1)){{l}_{2}} + 9))\omega _{2}^{4} + \omega _{1}^{2}(2{{l}_{2}}(4(6{{l}^{2}} + 1){{l}_{2}} \\ - \;3(23{{l}^{2}} + 3)\mu )\omega _{1}^{4} + \mu (5(23{{l}^{2}} + 3)l_{2}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;2((87\mu + 52){{l}^{2}} + 9\mu + 6){{l}_{2}} - 8((32\mu - 29){{l}^{2}} \\ + \;3(\mu - 1)))\omega _{1}^{2} + 3\mu {{l}_{2}}((64\mu - 58){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;6(\mu - 1) + ((156 - 319\mu ){{l}^{2}} - 33\mu + 18){{l}_{2}}))\omega _{2}^{2} \\ + \;4\mu ((29\mu - 26){{l}^{2}} + 3(\mu - 1))l_{2}^{2}\omega _{1}^{4})\omega _{1}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;{{l}_{1}}(16\omega _{1}^{2}((7 - 8\mu ){{l}^{2}} + (6{{l}^{2}} + 1)\omega _{1}^{2} - \mu + 1)\omega _{2}^{8} \\ - \;2(34(6{{l}^{2}} + 1)\omega _{1}^{6} + ((238 - 249\mu ){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;4(5{{l}^{2}} + 1)l_{2}^{2} - 31\mu + 34)\omega _{1}^{4} + ( - 4((7\mu - 6){{l}^{2}} \hfill \\ + \;\mu - 1)l_{2}^{2} + 6(23{{l}^{2}} + 3)\mu {{l}_{2}} + \mu ((338 - 145\mu ){{l}^{2}} \hfill \\ \end{gathered} $$
$$\begin{gathered} - \;15\mu + 39))\omega _{1}^{2} + 4(\mu - 1)\mu ((35\mu - 29){{l}^{2}} \\ + \;3(\mu - 1)))\omega _{2}^{6} + 2(8(6{{l}^{2}} + 1)\omega _{1}^{8} \\ \end{gathered} $$
$$\begin{gathered} + \;((56 - 87\mu ){{l}^{2}} + 17(5{{l}^{2}} + 1)l_{2}^{2} - 11\mu + 8)\omega _{1}^{6} \\ + \;(((102 - 139\mu ){{l}^{2}} - 20\mu + 17)l_{2}^{2} + 6(23{{l}^{2}} + 3)\mu {{l}_{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;\mu (26(23 - 29\mu ){{l}^{2}} - 78\mu + 69))\omega _{1}^{4} \\ + \;\mu (3(5\mu - 2)(63\mu - 58){{l}^{2}} - ((26\mu + 69){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3(\mu + 3))l_{2}^{2} + 9(\mu - 1)(9\mu - 4) + 6((58\mu - 78){{l}^{2}} \\ + \;6\mu - 9){{l}_{2}})\omega _{1}^{2} + 2(\mu - 1)\mu ((32\mu - 26){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3(\mu - 1))l_{2}^{2})\omega _{2}^{4} - \omega _{1}^{2}(8(5{{l}^{2}} + 1)l_{2}^{2}\omega _{1}^{6} \\ - \;2((8(\mu - 3){{l}^{2}} + \mu - 4)l_{2}^{2} + 2(23{{l}^{2}} + 3)\mu {{l}_{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;4\mu ((29\mu - 26){{l}^{2}} + 3(\mu - 1)))\omega _{1}^{4} \\ + \;\mu (8(\mu - 1)((35\mu - 29){{l}^{2}} + 3(\mu - 1)) \\ \end{gathered} $$
$$\begin{gathered} + \;{{l}_{2}}(8(58\mu - 13){{l}^{2}} + 48\mu + ((391 - 754\mu ){{l}^{2}} \\ - \;87\mu + 51){{l}_{2}} - 12))\omega _{1}^{2} + 27(\mu - 1)\mu ((32\mu - 26){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} + \;3(\mu - 1))l_{2}^{2})\omega _{2}^{2} + 4l_{2}^{2}\omega _{1}^{4}(\mu ((23 - 26\mu ){{l}^{2}} - 3\mu \\ + \;3)\omega _{1}^{2} + (\mu - 1)\mu ((32\mu - 26){{l}^{2}} + 3(\mu - 1))))]. \\ \end{gathered} $$
$${{g}_{{0,2}}} = \frac{{ - 2h_{2}^{2}\mu }}{{\omega _{1}^{2}\omega _{2}^{2}(\omega _{1}^{2} - 4\omega _{2}^{2})({{l}_{2}}((3 - 2\mu ){{l}^{2}} - \mu + \omega _{2}^{2} + 3)}}$$
$$\begin{gathered} + \;(4 - 6\mu ){{l}^{2}} + 6({{l}^{2}} + 1)\omega _{2}^{2} - 2\mu + 2) \\ \times \;[4\omega _{2}^{4}(\omega _{1}^{2}(2(26{{l}^{2}} + 3)\mu \omega _{2}^{2} \\ \end{gathered} $$
$$\begin{gathered} + \;7\mu ((32\mu - 29){{l}^{2}} + 3(\mu - 1)) - 36(7{{l}^{2}} + 1)\omega _{2}^{4}) \\ + \;24\mu \omega _{2}^{2}((29 - 32\mu ){{l}^{2}} - 3\mu + 3) \\ \end{gathered} $$
$$\begin{gathered} + \;9(7{{l}^{2}} + 1)\omega _{2}^{2}\omega _{1}^{4}) + 2l_{2}^{2}\omega _{2}^{2}(\omega _{1}^{2}(2(23{{l}^{2}} + 3)\mu \omega _{2}^{2} \\ + \;\mu ((29\mu + 52){{l}^{2}} + 3\mu + 6) + 24(6{{l}^{2}} + 1)\omega _{2}^{4}) \\ \end{gathered} $$
$$\begin{gathered} + 24\mu \omega _{2}^{2}((29\mu - 26){{l}^{2}} + 3\mu - 3) - 6(6{{l}^{2}} + 1)\omega _{2}^{2}\omega _{1}^{4}) \\ - \;2{{l}_{2}}\omega _{2}^{2}( - 6\omega _{2}^{2}\omega _{1}^{4}({{l}^{2}}( - 8\mu + 6\omega _{2}^{2} + 7) - \mu \\ \end{gathered} $$
$$\begin{gathered} + \;\omega _{2}^{2} + 1) + \omega _{1}^{2}(8\omega _{2}^{4}(3 - {{l}^{2}}(\mu - 21)) \\ + \;\mu \omega _{2}^{2}(13(8 - 29\mu ){{l}^{2}} - 39\mu + 12) \\ \end{gathered} $$
$$\begin{gathered} + \;\mu ((\mu (175\mu - 416) + 232){{l}^{2}} + 3(\mu - 1)(5\mu - 8)) \\ + \;24(6{{l}^{2}} + 1)\omega _{2}^{6}) - 24\mu \omega _{2}^{2}(\omega _{2}^{2}((26 - 29\mu ){{l}^{2}} \\ \end{gathered} $$
$$\begin{gathered} - \;3\mu + 3) + (\mu - 1)((35\mu - 29){{l}^{2}} + 3(\mu - 1)))) \\ + \;l_{2}^{3}( - 6\omega _{2}^{2}\omega _{1}^{4}({{l}^{2}}( - 7\mu + 5\omega _{2}^{2} + 6) \\ \end{gathered} $$
$$\begin{gathered} - \;\mu + \omega _{2}^{2} + 1) + \omega _{1}^{2}( - 4\omega _{2}^{4}((62\mu - 36){{l}^{2}} \\ + \;9\mu - 6) + \mu \omega _{2}^{2}({{l}^{2}}(115 - 26\mu ) - 3(\mu - 5)) \\ \end{gathered} $$
$$\begin{gathered} + \;5(\mu - 1)\mu ((32\mu - 26){{l}^{2}} + 3(\mu - 1)) \\ + \;24(5{{l}^{2}} + 1)\omega _{2}^{6}) - 24\mu \omega _{2}^{2}(\omega _{2}^{2}((23 - 26\mu ){{l}^{2}} \\ - \;3\mu + 3) + (\mu - 1)((32\mu - 26){{l}^{2}} + 3(\mu - 1))))]. \\ \end{gathered} $$
APPENDIX C
The graphs (see Fig. 6) of \(f(\mu ,l)\) and \(g(\mu ,l)\) in the linear stability interval \(\left( {8(1 - {{l}^{2}}){\text{/}}9,\;1 - {{l}^{2}}} \right)\) for different values of \(l \in [0,1]\).
