## 3 Analytical Investigations
 \tag{60}\]  
Substituting Equation (60) into Equations (56)-(59), and then separating the real and imaginary parts of the resulting equations, we have  
\[\begin{array}{rl}\dot{a}_{1}&=f_{1}(a_{1},a_{2},a_{3},a_{4},\varphi_{1}, \varphi_{2},\varphi_{3},\varphi_{4})=-\frac{1}{2}\bigg{(}2\mu+\frac{\eta_{2 \overline{1}\overline{1}\overline{1}\overline{1}\overline{1}\overline{1}}}{ \rho_{1}^{2}+\omega^{2}}\bigg\[\dot{a}_{2} =f_{2}(a_{1},a_{2},a_{3},a_{4},\varphi_{1},\varphi_{2},\varphi_{3}, \varphi_{4})=-\frac{1}{2}\bigg{(}2\mu+\frac{\eta_{4}\eta_{8}}{\rho_{2}^{2}+\omega ^{2}}\bigg{)}a_{2}+\frac{1}{8}\bigg{(}a_{3}-\frac{2\gamma_{11}\eta_{8}}{\rho_{2 }^{2}+\omega^{2}} \tag{62}\] \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\] \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\] \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\] \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\] \[+\frac{1}{8\omega}\bigg{(}\gamma_{7}\omega-\frac{2\omega\gamma_{ 8}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{4}\cos(\varphi_{4})-\frac {1}{8\omega}\bigg{(}\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} \bigg{)}a_{2}a_{1}a_{3}\cos(\varphi_{3})\] \[+\frac{1}{8\omega}\bigg{(}\frac{\rho_{1}\gamma_{10}\eta_{7}}{\rho _{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}a_{3}\sin(\varphi_{3})+\frac{1}{8\omega }\gamma_{7}\eta_{1}^{2}a_{4}\sin(2\varphi_{2}-2\varphi_{1}+\varphi_{4})\] \[+\frac{1}{8\omega}\bigg{(}\gamma_{6}\omega-\frac{\omega\gamma_{ 10}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}a_{3}\cos(2\varphi_{2} -2\varphi_{1}-\varphi_{3})-\frac{1}{8\omega}(\omega+\sigma)^{2}f\cos(\varphi_ {2})\] \[+\frac{1}{8\omega}\bigg{(}\gamma_{4}+\frac{\rho_{11}\eta_{7}}{\rho _{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}a_{3}\sin(2\varphi_{2}-2\varphi_{1}- \varphi_{3})+\frac{1}{8\omega}\gamma_{5}a_{2}a_{3}^{2}\sin(2\varphi_{2}-2 \varphi_{1}-2\varphi_{3})\] \[\dot{a}_{3}=f_{3}(a_{1},a_{2},a_{3},a_{4},\varphi_{1},\varphi_{2},\varphi_{3},\varphi_{4})=-\mu_{1}a_{3}+\frac{\eta_{5}}{2(\omega+\sigma_{1})} a_{1}\sin(\varphi_{3}) \tag{63}\]  
\[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]0  
\[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]1 (65) \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]2 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]3 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]4 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]5 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]6 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]7 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]8 \[-\frac{2\rho_{2}\gamma_{13}\eta_{8}^{2}}{(\rho_{2}^{2}+\omega^{2} )}\bigg{)}a_{2}^{3}+\frac{1}{8}\bigg{(}2a_{4}-\frac{\gamma_{11}\eta_{7}}{\rho_ {1}^{2}+\omega^{2}}-\frac{a_{3}\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}} -\frac{2\gamma_{11}\eta_{8}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a_{1}^{2}\]9 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]0 (66) \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]1 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]2 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]3 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]4 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]5 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]6 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]7 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]8 \[+\frac{1}{8}\bigg{(}-a_{4}+\sigma_{7}-\frac{2\gamma_{11}\eta_{7} \rho_{7}^{2}}{\rho_{1}^{2}+\omega^{2}}-\frac{\gamma_{11}\eta_{7}}{\rho_{1}^{2 }+\omega^{2}}+\frac{\rho_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\bigg{)}a_{2}a _{1}^{2}\cos(2\varphi_{2}-2\varphi_{1})\]9 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]0 (67) \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]1 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]2 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]3 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]4 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]5 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]6 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]7 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]8 \[+\frac{1}{8}\big{(}\frac{\alpha_{2}}{\omega}-\alpha_{5}\omega+ \frac{(\rho_{2}^{2}-\omega^{2})\gamma_{12}\eta_{7}}{\omega(\rho_{1}^{2}+ \omega^{2})}\big{)}a_{2}a_{1}\frac{\rho_{11}\eta_{37}}{\omega(\rho_{1}^{2}+ \omega^{2})}+\frac{\omega\gamma_{11}\eta_{7}}{\rho_{1}^{2}+\omega^{2}}\big{)} a_{2}a_{1}^{2}\sin(2\varphi_{2}-2\varphi_{1})\]9 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]0 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]1 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]2 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]3 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]4 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]5 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]6 \[-\Big{(}\frac{1}{2\omega}\eta_{3}a_{4}+\frac{1}{8\omega}\gamma_{1 }a_{2}^{2}a_{4}+\frac{1}{8\omega}\gamma_{7}a_{1}^{2}a_{4}\Big{)}\sin(\varphi_ {4})-\frac{1}{8\omega}\gamma_{3}a_{2}a_{4}^{2}\sin(2\varphi_{4})\]7  
where \(\varphi_{1}=\sigma\tau-\theta_{1},\ \varphi_{2}=\sigma\tau-\theta_{2},\ \varphi_{3}=\theta_{1}-\theta_{3}-\sigma_{1}\tau\), and \(\varphi_{4}=\theta_{2}-\theta_{4}-\sigma_{2}\tau\).