## 3 The Equations of ADFTV and the Mathematical Solutions of FOAM
, \tag{19}\]  
and proceeding with the same mathematical manipulations in Equation (15) by using the second system of equations expressed in Equations (17) and (18), and the supposed solution (16), balancing the coefficients of the mathematical expressions \(\varepsilon\cos\bigl{(}\frac{1}{2}\Phi_{3}+\Psi_{3}\bigr{)},\varepsilon\sin \bigl{(}\frac{1}{2}\Phi_{3}+\Psi_{3}\bigr{)}\) from the LHS with those from the RHS, and neglecting the terms containing \(\sin(\frac{\mathrm{k}}{2}\Phi_{3}+\Psi_{3}),\cos(\frac{\mathrm{k}}{2}\Phi_{3}+ \Psi_{3})\),\(\mathrm{k}_{2}=2,3,4,5\) results in the differential system of equations (DSOE) for the GB:  
\[ \begin{cases}\frac{\partial\mathrm{D}_{3}}{\partial\Psi_{3}}\Bigl{(} \overline{\Omega}_{3}-\frac{1}{2}\eta_{3}\Bigr{)}-2\Phi_{3}\overline{\Omega}_{ 3}\mathrm{E}_{3}=\mu_{3}\Phi_{3}\overline{\Omega}_{3}^{2}\cos(2\Psi_{3})- \frac{3}{4}\Gamma_{3}\Phi_{1}^{3}-\frac{5}{6}\Gamma_{5}\Phi_{3}^{5}\\ \Phi_{1}\frac{\partial\mathrm{E}_{1}}{\partial\Psi_{3}}\Bigl{(}\overline{ \Omega}_{3}-\frac{1}{2}\eta_{3}\Bigr{)}+2\overline{\Omega}_{3}\mathrm{D}_{3}= -\mu_{3}\Phi_{3}\overline{\Omega}_{3}^{2}\sin(2\Psi_{3})-2\xi\overline{\Omega }_{3}^{2}\Phi_{3}\end{cases} , \tag{20}\]  
which yields the solutions of DSOE (19) for the GT and (20) for the GB:  
\[ \begin{cases}\mathrm{D}_{1}=-\frac{\mu_{1}\overline{\Omega}_{1}^{2}\Phi_{1}}{ \eta_{1}}\sin 2\Psi_{1}-\varepsilon\overline{\Omega}_{1}\Phi_{1}\\ \mathrm{E}_{1}=-\frac{\mu_{1}\overline{\Omega}_{1}^{2}}{\eta_{1}}\cos 2\Psi_{1}+ \frac{3}{8}\frac{\Gamma_{1}\Phi_{1}^{2}}{\Omega_{1}}\end{cases} , \begin{cases} \mathrm{D}_{3}=-\frac{\mu_{3}\overline{\Omega}_{3}^{2}\Phi_{3}}{\eta_{3}}\sin 2\Psi_{3}- \varepsilon\overline{\Omega}_{3}\Phi_{3}\\ \mathrm{E}_{3}=-\frac{\mu_{3}\overline{\Omega}_{3}^{2}\overline{\Omega}_{3}^{2}}{ \eta_{3}}\cos 2\Psi_{3}+\frac{3}{8}\frac{\Gamma_{3}\Phi_{3}^{2}}{\Omega_{3}}+\frac{5}{ 16}\frac{\Gamma_{5}\Phi_{3}^{4}}{\Omega_{3}}\end{cases} . \tag{21}\]Using the system of Equation (21) in the DSOE (17) and considering all the characteristics of the systems in real-time values yields the DSOEs of the analytical amplitudes and the phases angles of FTV for GT and GB in the APPR for each element:  
\[\left\{\begin{array}{c}\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}t}=-\frac{\mathrm{ u}_{1}\overline{\mathrm{r}}_{3}^{2}\Phi_{1}}{\mathrm{n}_{1}}\sin 2\Psi_{1}-\xi \overline{\Omega}_{1}\Phi_{1}\\ \frac{\mathrm{d}\overline{\
u}_{1}}{\mathrm{d}t}=\overline{\Omega}_{1}-\frac{ 1}{2}\mathrm{n}_{1}-\frac{\mathrm{u}_{1}\overline{\Omega}_{1}^{2}}{\mathrm{n} _{1}}\cos 2\Psi_{1}+\frac{3}{2}\frac{\Gamma_{1}\Phi_{2}^{2}}{\Omega_{1}}^{ \prime}\\ \left\{ \begin{array}{c}\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}t}=-\frac{\mathrm{ u}_{1}\overline{\Omega}_{1}^{2}}{\mathrm{n}_{3}}\sin 2\Psi_{3}-\xi \overline{\Omega}_{3}\Phi_{3}\\ \frac{\mathrm{d}\overline{\
u}_{3}}{\mathrm{d}t}=\overline{\Omega}_{3}-\frac{ 1}{2}\mathrm{n}_{3}-\frac{\mathrm{u}_{3}\overline{\Omega}_{1}^{2}}{\mathrm{n }_{3}}\cos 2\Psi_{3}+\frac{3}{8}\frac{\Gamma_{1}\Phi_{2}^{2}}{\Omega_{3}}+ \frac{5}{16}\frac{\Gamma_{3}\Phi_{3}^{4}}{\Omega_{3}}.\end{array} \right. \tag{22}\]  
The investigation of dynamic instability for the stationary FTV of AD consists of computing the dynamic instability boundary frontiers (DIBF) of GT and GB in the APPR. The stationary dynamic behavior of FTV for the GT and the GB is characterized by zeroing \(\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}t}\), \(\frac{\mathrm{d}\Psi_{1}}{\mathrm{d}t}\), \(\frac{\mathrm{d}\Phi_{2}}{\mathrm{d}t}\), \(\frac{\mathrm{d}\Psi_{3}}{\mathrm{d}t}\) in DSOE (22) and expressing the trigonometric identity \(\sin^{2}\Psi_{1}+\cos^{2}\Psi_{1}=1,i=1,3\).