## 6 Conclusions  
A transformation approach has been proposed to find the equivalent representation form in the sense of Lyapunov of forced, damped, nonlinear, two degree-of-freedom dynamic systems. This proposed approach of finding equivalent expressions has the main advantage of providing simple algebraic relations whose coefficient values are determined by minimizing the error of replacing the original restoring forces by equivalent ones. By studying the dynamic responses of three nonlinear dynamic systems, the validity of the proposed conjecture has been numerically examined by comparing the LCEs of the original equations with those obtained from their equivalent uncoupled expressions. From the numerical predictions obtained, it is concluded that the LCEs and the LLE of the equivalent expression are similar to those obtained from the original equations of motion. In fact, numerical results have shown that the proposed procedure predicts the quantitative behavior of the nonlinear systems examined here well; however, there are some discrepancies which could be removed if additional terms in the power series expansion of the dynamical system's restoring forces are considered. Furthermore, when the system's restoring forces were replaced by a fifth-order truncated power series expansion in which decay rate terms were considered, numerical predictions computed from the nonlinear dynamic system of Example 1, indicated that the RMSE values were lower than those of the cubic truncated power series expansion, since, in this quintic approach, decay rate terms were considered. In this case, both the qualitative and the quantitative dynamic system behaviors in physical coordinates were predicted well in spite of having uncoupled modal equivalent oscillators, as shown in Figures 16 and 17.  
Finally, and contrary to other proposed approaches, the one introduced here is capable of describing the dynamic behavior of nonlinear systems when internal resonance exists between the two modes, and thus, it is possible to uncouple, in modal coordinates, the resonant modes, as illustrated in Figure 12. Therefore, it can be concluded, in accordance with the numerical evidence of the dynamic systems examined here, that it is possible to have two decoupled Duffing-type equivalent representation forms of the original equations of motion in the sense of Lyapunov that can predict the dynamics of the original system, and accuracy can be improved if additional terms in the truncated power series are considered, not only to describe elastic forces, but also to take into account decay rate effects.  
Of course, the system's restoring forces could be replaced by other forms rather than Duffing-type equivalent representation expressions.  
However, the applicability of the proposed approach to other nonlinear dynamic systems by using truncated power series expansions or alternative forms to replace the system restoring forces could be considered only for small deviations of the system's linear behavior, because for large amplitudes, coupling and nonlinear terms are important.  
This work was funded by Tecnologico de Monterrey-Campus Monterrey, through the Research Group in Nanomaterials and Devices Design. Additional support was provided from Consejo Nacional de Ciencia y Tecnologia (Conacyt), Mexico, Project Numbers 242269 and 255837.  
[PERSON] and [PERSON] conceived the decoupling approach and its mathematical modeling. [PERSON] and [PERSON] worked out the computation of the LCE and of the LLE, and helped in the computation of the numerical solutions of the three examined nonlinear dynamic systems. [PERSON] and [PERSON] directed the research and wrote the paper. All authors contributed to edition and revision of the manuscript.  
The authors declare no conflict of interest.