## 3 The Equations of ADFTV and the Mathematical Solutions of FOAM
The excitation frequencies \(\eta_{1},\eta_{3}\) of the GT and the GB of AD in the APPR must be defined by the expressions  
\[\eta_{3}=\frac{\mathrm{d}\Theta_{3}}{\mathrm{d}\mathrm{t}}=\frac{2\mathrm{d} \varphi_{3}}{\mathrm{d}\mathrm{t}}\simeq 2\overline{\Omega}_{3}\eta_{1}=\frac{\mathrm{d} \Theta_{1}}{\mathrm{d}\mathrm{t}}=\frac{2\mathrm{d}\varphi_{1}}{\mathrm{d} \mathrm{t}}\simeq 2\overline{\Omega}_{1}, \tag{11}\]  
and introducing the slowing time by imposing the condition that all the coefficients in Equations (9) and (10) are small yields  
\[\frac{\mathrm{d}\Theta_{1}}{\mathrm{d}\mathrm{t}}=\eta_{1}(\tau),\Theta_{1}=2 \varphi_{1},\xi=\varepsilon\xi,\mu_{1}=\varepsilon\mu_{1},\alpha_{1}= \varepsilon\alpha_{1},\alpha_{2}=\varepsilon\alpha_{2},\Gamma_{1}=\varepsilon \Gamma_{1}, \tag{12}\]  
\[\frac{\mathrm{d}\Theta_{3}}{\mathrm{d}\mathrm{t}}=\eta_{3}(\tau),\Theta_{3}=2 \varphi_{3},\xi=\varepsilon\xi,\mu_{3}=\varepsilon\mu_{3},parts are linear oscillators. For FOAM, there were meant to be analytical solutions for Equations (14) and (15), having the expressions  
\[\varphi_{1}=\Phi_{1}\cos\biggl{(}\frac{1}{2}\Theta_{1}+\Psi_{1}\biggr{)},\varphi_ {3}=\Phi_{3}\cos\biggl{(}\frac{1}{2}\Theta_{3}+\Psi_{3}\biggr{)}, \tag{16}\]  
where \(\Phi_{1},\Phi_{3}\) are the amplitudes and \(\Psi_{1},\Psi_{3}\) are the phase angles for global tulip and global bowl, given by the Equations  
\[ \begin{cases}\frac{\mathrm{d}\Phi_{1}}{\mathrm{d}t}=\varepsilon\mathrm{D}_{1 }(\tau,\Phi_{1},\Psi_{1})\\ \frac{\mathrm{d}\Psi_{1}}{\mathrm{d}t}=\overline{\Omega}_{1}-\frac{1}{2}\eta_ {1}(\tau)+\varepsilon\mathrm{E}_{1}(\tau,\Phi_{1},\Psi_{1})\\ \frac{\mathrm{d}\Psi_{1}}{\mathrm{d}t}=\varepsilon\mathrm{D}_{3}(\tau,\Phi_{ 3},\Psi_{3})\\ \frac{\mathrm{d}\Psi_{1}}{\mathrm{d}t}=\overline{\Omega}_{3}-\frac{1}{2}\eta_ {3}(\tau)+\varepsilon\mathrm{E}_{3}(\tau,\Phi_{3},\Psi_{3})\end{cases} . \tag{17}\]  
Differentiating concerning the time, the system of Equation (17) yields  
\[ \begin{cases}\frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}t^{2}}=\varepsilon\frac{ \partial\mathrm{D}_{1}}{\partial\Psi_{1}}\Bigl{(}\overline{\Omega}_{1}-\frac{ 1}{2}\eta_{1}\Bigr{)}+\varepsilon^{2}\dots\dots\dots\dots\\ \frac{\mathrm{d}^{2}\Psi_{1}}{\mathrm{d}t^{2}}=\varepsilon\Bigl{[}-\frac{1}{2} \frac{\partial\eta_{1}}{\partial\tau}+\frac{\partial\mathrm{E}_{1}}{\partial \Psi_{1}}\Bigl{(}\overline{\Omega}_{1}-\frac{1}{2}\eta_{1}\Bigr{)}\Bigr{]}+ \varepsilon^{2}\dots\\ \frac{\mathrm{d}^{2}\Phi_{1}}{\mathrm{d}t^{2}}=\varepsilon\frac{\partial\mathrm{ D}_{1}}{\partial\Psi_{3}}\Bigl{(}\overline{\Omega}_{3}-\frac{1}{2}\eta_{3} \Bigr{)}+\varepsilon^{2}\dots\dots\dots\dots\\ \frac{\mathrm{d}^{2}\Psi_{3}}{\mathrm{d}t^{2}}=\varepsilon\Bigl{[}-\frac{1}{2} \frac{\partial\eta_{3}}{\partial\tau}+\frac{\partial\mathrm{E}_{3}}{\partial \Psi_{3}}\Bigl{(}\overline{\Omega}_{3}-\frac{1}{2}\eta_{3}\Bigr{)}\Bigr{]}+ \varepsilon^{2}\dots\end{cases} . \tag{18}\]  
Using the first system of equations expressed in Equations (17) and (18) and the supposed solution (16) in Equation (14), balancing the coefficients of the mathematical expressions \(\varepsilon\cos\Bigl{(}\frac{1}{2}\Phi_{1}+\Psi_{1}\Bigr{)},\varepsilon\sin \Bigl{(}\frac{1}{2}\Phi_{1}+\Psi_{1}\Bigr{)}\) from the LHS with those from the RHS, and neglecting the terms containing \(\sin(\frac{\mathrm{k}}{2}\Phi_{1}+\Psi_{1}),\cos(\frac{\mathrm{k}}{2}\Phi_{1}+ \Psi_{1}),\mathrm{k}_{1}=2,3,4\) results in the differential system of equations (DSOE) for the GT:  
\[ \begin{cases}\frac{\partial\mathrm{D}_{1}}{\partial\overline{\Psi}_{1}}\Bigl{(} \overline{\Omega}_{1}-\frac{1}{2}\eta_{1}\Bigr{)}-2\Phi_{1}\overline{\Omega}_{ 1}\mathrm{E}_{1}=\mu_{1}\Phi_{1}\overline{\Omega}_{1}^{2}\cos(2\Psi_{1})- \frac{3}{4}\Gamma_{1}\Phi_{1}^{3}\\ \Phi_{1}\frac{\partial\mathrm{E}_{1}}{\partial\Psi_{1}}\Bigl{(}\overline{ \Omega}_{1}-\frac{1}{2}\eta_{1}\Bigr{)}+2\overline{\Omega}_{1}\mathrm{D}_{1}= -\mu_{1}\Phi_{1}\overline{\Omega}_{1}^{2}\sin(2\Psi_{1})-2\xi\overline{\Omega }_{1}^{2}\Phi_{1}\end{cases}