## 5 Fifth-Order Power Series Expansion
By following the procedure described in Section 3, the coefficients, \(a_{i}\), \(b_{i}\), and \(\mu_{i}\), were found to be  
\[a_{1}=\omega_{n1}^{2}+\frac{1206645\varphi_{1}\eta_{1}\eta_{2}}{10446976\eta_ {1}}+\frac{394725}{614528\eta_{1}}\Big{(}\varphi_{4}\eta_{2}^{3}+2\
u_{2}V_{2} \Big{)}+\frac{\varphi_{3}\eta_{2}^{2}}{3}, \tag{55}\]  
\[a_{2}=\varphi_{1}+\frac{105\big{[}57921\varphi_{1}\eta_{1}^{2}\eta_{2}-109055 \big{(}\varphi_{4}\eta_{2}^{3}+2\
u_{2}V_{2}\big{)}\big{]}}{10446976\eta_{1}^{ 3}}, \tag{56}\]  
\[a_{3}=\frac{231\big{[}-9199\varphi_{1}\eta_{1}^{2}\eta_{2}+31659\big{(} \varphi_{4}\eta_{2}^{3}+2\
u_{2}V_{2}\big{)}\big{]}}{10446976\eta_{1}^{5}}, \tag{57}\]  
\[\mu_{1}=\frac{17\big{(}4875\varphi_{4}\eta_{2}^{3}+19204\
u_{1}V_{1}+9750\
u_ {2}V_{2}\big{)}-2727\varphi_{2}\eta_{1}^{2}\eta_{2}}{652936V_{1}}, \tag{58}\]  
\[\mu_{2}=\frac{135\big{[}1013\varphi_{2}\eta_{1}^{2}\eta_{2}-35275\big{(} \varphi_{4}\eta_{2}^{3}+2\
u_{2}V_{2}\big{)}\big{]}}{10446976\eta_{1}V_{1}^{ 2}}, \tag{59}\]  
\[\mu_{3}=\frac{3\big{[}-52\varphi_{2}\eta_{1}^{2}\eta_{2}+3825\big{(}\varphi_{ 4}\eta_{2}^{3}+2\
u_{2}V_{2}\big{)}\big{]}}{81617\eta_{1}^{2}V_{1}}, \tag{60}\]  
\[b_{1}=\omega_{n2}^{2}+\frac{15\big{(}447355\varphi_{5}\eta_{11}^{3}+804432 \varphi_{7}\eta_{11}\eta_{22}^{2}+894710\
u_{2}V_{11}\big{)}}{10446976\eta_{2 }^{5}}+\frac{\varphi_{6}\eta_{11}^{2}}{3}, \tag{61}\]  
\[b_{2}=\frac{34\big{(}307264\varphi_{8}\eta_{22}^{3}-673575\
u_{2}V_{11}\big{)} +6081705\varphi_{7}\eta_{11}\eta_{22}^{3}-11450775\varphi_{5}\eta_{11}^{3}}{10 446976\eta_{11}^{3}}, \tag{62}\]  
\[b_{3}=\frac{231\big{(}31659\varphi_{5}\eta_{11}^{3}-9199\varphi_{7}\eta_{11} \eta_{22}^{2}+63138\
u_{2}V_{11}\big{)}}{10446976\eta_{2}^{5}}, \tag{63}\]  
\[\mu_{4}=\frac{\
u_{3}}{2}+\frac{3\big{(}2762  
For comparison purposes, only the dynamic system examined in Example 1 was considered, since the oscillators studied in Examples 2 and 3 had small errors when the equivalent expressions in the sense of Lyapunov were compared to the original equations of motion. First, we focused on studying the unforced dynamic system (30) and plotted the amplitude-time curves by using the equivalent cubic, and quintic expressions: (29), (53) and (54). The parameter values selected were \(m=1\), \(k=1\), \(c=0.3\), \(\varepsilon=0.5\), with \(\big{(}x_{1},\ x_{2},\ \dot{x}_{1},\ \dot{x}_{2}\big{)}=(0,\ 0,\ 2,\ 0)\). Projection of the initial condition onto the linear transformation (3), yielded \(\big{(}u_{10},\ u_{20},\ \dot{u}_{10},\ \dot{u}_{20}\big{)}=(1.9118,\ -0.9165,\ 0,\ 0)\). Figure 16 shows the time-amplitude curves for the two modes of the system. Notice that as a consequence of the fifth-order and nonlinear decay terms of the quintic approach, the equivalent representation forms (53) and (54) provide an improvement in the RMSE values. Next, the driving force magnitude of \(f_{1}=0.25\) was considered and then, the frequency-amplitude response, as well as the LLE curves were plotted. As can be seen in Figure 17, the quintic equivalent representation form in the sense of Lyapunov provided a better approximation to the exact numerical values. An improvement was achieved in the frequency-amplitude response curves obtained from the quintic approach since the estimated curves match the numerical results well; however, one can notice from Figure 18, that when the quintic approach is used to compute the LLE values, the RMSE value is slightly higher than that computed by using the cubic model, which is mainly due to the numerical procedure used to compute these values rather than to the decoupling proposed approach. Figure 16: Time-amplitude response curves computed from the numerical integration solutions of Equations (53) and (54). Cubic approximation: (**a**) \(x_{1}\) vs \(t\), (**b**) \(x_{2}\) vs \(t\). Quintic approximation: (**c**) \(x_{1}\) vs \(t\), (**d**) \(x_{2}\) vs \(t\).