## 3 Determination of \(a_{i}\), \(b_{i}\), and \(\mu_{i}\)
\begin{array}{ll}\frac{\partial S_{2}}{\partial V_{11}}=\frac{\eta_{11}\eta_{2 }V_{22}}{20699280}&(1890375\varphi_{5}^{2}\eta_{11}^{6}+1838900\varphi_{5} \varphi_{6}\eta_{11}^{5}\eta_{22}+613312\varphi_{6}^{2}\eta_{11}^{4}\eta_{22} ^{2}\\ &+1103340\varphi_{5}\varphi_{7}\eta_{11}^{4}\eta_{22}^{2}+862470\varphi_{6} \varphi_{7}\eta_{11}^{3}\eta_{22}^{3}+344988\varphi_{7}^{2}\eta_{11}^{2}\eta_{ 22}^{4}\\ &+4957575\varphi_{5}\
u_{2}\eta_{11}^{3}V_{11}+2755760\varphi_{6}\
u_{2}\eta_{ 11}^{2}\eta_{22}V_{11}+2066820\varphi_{7}\
u_{2}\eta_{11}\eta_{22}^{2}V_{11} \\ &+7429380\
u_{2}^{2}V_{11}^{2}),\end{array}  \tag{25}\]  
\[\begin{array}{ll}\frac{\partial S_{2}}{\partial V_{2}}=\frac{\eta_{11}\eta_{2 }V_{11}}{41398560}&(3780750\varphi_{5}^{2}\eta_{11}^{6}+3677800\varphi_{5} \varphi_{6}\eta_{11}^{5}\eta_{22}+1226624\varphi_{6}^{2}\eta_{11}^{4}\eta_{22} ^{2}  
which are set equal to zero to find the critical points at which Equations (7) and (8) are minimized. Notice from Equation (9) through (16) that the denominator terms depend, respectively, on \(\eta_{1}\), \(V_{1}\), \(\eta_{22}\), and \(V_{22}\). Therefore, these coefficients cannot be zero. Thus, \(\partial S_{1}/\partial\eta_{1}=0\), \(\partial S_{1}/\partial\eta_{2}=0\), \(\partial S_{1}/\partial V_{1}=0\), \(\partial S_{1}/\partial V_{2}=0\), \(\partial S_{2}/\partial\eta_{11}=0\), \(\partial S_{2}/\partial\eta_{22}=0\), \(\partial S_{2}/\partial V_{11}=0\), \(\partial S_{2}/\partial V_{22}=0\) are identically satisfied, if, and only if, \(\eta_{2}=0\), \(V_{2}=0\), \(\eta_{11}=0\), and \(V_{11}=0\). Then, the expressions for \(a_{i}\) and \(b_{i}\) that minimize the weighted mean square error of using Expressions (7) and (8) to determine the coefficients, \(a_{i}\), \(b_{i}\), and \(\mu_{i}\) that describes the equivalent representation form of the original equation of motion, into the approximate forms (5) and (6), simplify to  
\[a_{1}=\omega_{n1}^{2},\qquad a_{2}=0,\qquad a_{3}=\varphi_{1},\qquad\mu_{1}= \frac{\
u_{1}}{2}, \tag{27}\]  
\[b_{1}=\omega_{n2}^{2},\qquad b_{2}=0,\qquad b_{3}=\varphi_{8},\qquad\mu_{2}= \frac{\
u_{3}}{2}. \tag{28}\]  
Thus, the above approach and the usage of power series expansion up to the third-order to replace the modal restoring forces of Equations (5) and (6) provide the following uncoupled Duffing-type equivalent equations:  
\[\ddot{u}_{1}+2\mu_{1}\dot{u}_{1}+a_{1}u_{1}+a_{3}u_{1}^{3}=P_{1}(t),\qquad \ddot{u}_{2}+2\mu_{2}\dot{u}_{2}+b_{1}u_{2}+b_{3}u_{2}^{3}=P_{2}(t). \tag{29}\]  
It is important to point out that such equivalence in this two degree-of-freedom (DOF), non-integrable system (4) is only equivalent, in an approximate form, to two single degree-of-freedom integrable equations. Furthermore, in the conservative case, almost periodic solutions can be observed in two-DOF nonlinear system of coupled equations, but they are absent in the uncoupled, single-DOF, nonlinear Equation (29). However, the linear combination of their approximate solutions, in accordance with Equation (3), could exhibit, as in the case of the original system of differential Equation (1), almost periodic solutions, as will be demonstrated in Example 2. In accordance with [PERSON] and [PERSON] [20], the advantage of having decoupled modal equations is related to the possibility of obtaining information about the system modal amplitudes and thus, the numerical integration solution of these equivalent expressions, which could provide valuable information about the qualitative and quantitative behaviors of the system dynamics. On the other hand, if the restoring forces, \(f_{1}(x_{1},x_{2})\) and \(f_{2}(x_{1},x_{2})\), of the dynamic system (1) contain even, nonlinear effects, then, their equivalent representation forms in the sense of Lyapunov could also be expanded to power series in which decay terms must be considered. Section 5 focuses on studying the effects of these new terms in the decoupled form of a nonlinear, dynamic system.