## 4 Numerical Validation
### Example 1: Dynamic System with Cubic Nonlinearities  
To study the accuracy attained by applying the proposed approach to decouple nonlinear differential equations and to validate the proposed conjecture, first, the frequency-amplitude response curves of the original unforced, undamped system (4) will be compared to the backbone curves obtained from the Duffing equations described by Equation (29). To accomplish this goal, the nonlinear dynamic systems discussed in [20,21] are considered, in which the corresponding equations of motion are described by  
\[\left[ \begin{array}{cc}m&0\\ 0&m\end{array} \right]\left\{ \begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}c&-c\\ -c&2c\end{array} \right]\left\{ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}1+k&-k\\ -k&1+k\end{array} \right]\left\{ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right\}+\left\{ \begin{array}{c}\varepsilon x_{1}^{3}\\ 0\end{array} \right\}=\left\{ \begin{array}{c}f_{1}\cos\omega_{f}t\\ 0\end{array} \right\} \tag{30}\]where \(f_{1}\) is the magnitude of the driving force, and \(\omega_{f}\) represents the corresponding driving frequency. The system initial conditions are assumed to be given as \(x_{1}(0)=x_{10}\), \(\dot{x}_{1}(0)=0\), \(x_{2}(0)=x_{20}\), and \(\dot{x}_{2}(0)=0\). In order to study the dynamic response of the system (30), this is first transformed into the modal coordinates, \(\left\{ \begin{array}{cc}u_{1}&u_{2}\end{array} \right\}^{\mathrm{T}}\), via the following linear transformation  
\[\left\{ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right\}=\left[ \begin{array}{cc}\beta&\beta\\ \beta&-\beta\end{array} \right]\left\{ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right\} \tag{31}\]  
where \(\beta=1/\sqrt{2}\). Thus, the system (30) can be equivalently written as Equation (4) with  
\[ \begin{array}{c}\varphi_{1}\equiv\varepsilon/8,\qquad\varphi_{2}\equiv-3 \varepsilon/8,\qquad\varphi_{3}\equiv 3\varepsilon/8,\qquad\varphi_{4}\equiv- \varepsilon/8,\qquad\varphi_{5}\equiv-\varepsilon/8,\qquad\varphi_{6}\equiv 3 \varepsilon/8,\\ \qquad\varphi_{7}\equiv-3\varepsilon/8,\qquad\varphi_{8}\equiv\varepsilon/8, \qquad\
u_{1}=\
u_{2}=1/2c,\qquad\
u_{3}\equiv 5/2c,\qquad\omega_{n1}^{2} \equiv 1/m,\\ \qquad\omega_{n2}^{2}=(1+2k)/m,\qquad P_{1}(t)=P_{2}(t)=f_{1}/\sqrt{2}\cos \omega_{f}.\end{array}  \tag{32}\]  
In order to determine if the decoupled Equation (29) provides a good description of the dynamics of the original system (30), the frequency-amplitude backbone curves are plotted. In accordance with [PERSON] [47], the exact frequency-amplitude equations for the unforced, undamped Duffing Equation (4) are given as  
\[\text{mode 1}:\ \omega_{1}^{2}=a_{1}+a_{3}u_{10}^{2};\qquad\text{mode 2}:\ \omega_{2}^{2}=b_{1}+b_{3}u_{20}^{2} \tag{33}\]  
while the approximate frequency-amplitude relationships obtained from the modified Lindstedt-Poincare method are [48; 49]  
\[\text{mode 1}:\omega_{1}^{2}=\omega_{n1}^{2}+1/4\Big{(}3/8\varepsilon u_{10}^{ 2}+3/8\varepsilon u_{10}u_{20}+O\Big{(}\varepsilon^{2}\Big{)}\Big{)}, \tag{34}\]  
\[\text{mode 2}:\omega_{2}^{2}=\omega_{n2}^{2}+1/4\Big{(}3/8\varepsilon u_{20}^{ 2}+3/8\varepsilon u_{10}u_{20}+O\Big{(}\varepsilon^{2}\Big{)}\Big{)}. \tag{35}\]  
Figure 1 shows the simulation performed to verify the accuracy of the proposed methodology that decouples the nonlinear normal mode's differential equations. The system parameter values in dimensionless units are \(m=1\), \(k=1\), \(\varepsilon=0.5\) with \(c=0\) and \(f_{1}=0\); these are similar to those used in [20]. To plot the frequency-amplitude curve of the first mode, initial conditions of \(\big{(}u_{1},\ u_{2},\ \dot{u}_{1},\ \dot{u}_{2}\big{)}=(1,\ 0,\ 0,\ 0)\) were considered, and for the second mode, \(\big{(}u_{1},\ u_{2},\ \dot{u}_{1},\ \dot{u}_{2}\big{)}=(0,\ 1,\ 0,\ 0)\) was used. It is clear from the results exhibited in Figure 1 that the frequency-amplitude curves of the original system compare well with those of the uncoupled equations.