## 1 Introduction  
[PERSON] [1], [PERSON] [2], [PERSON] [3, 4], [PERSON] and [PERSON] [5], [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7, 8], [PERSON] [9], [PERSON] and [PERSON] [10], and many others have reported approaches that replace linear and nonlinear dynamic systems by equivalent ones with known solutions that are closed to the original system, which produce the same oscillations that appear in the original equations of motion.. There, the authors focused on using linearization, weighted mean-square, and least squares techniques to determine the equivalent expressions that provide solutions with the same quantitative and qualitative original system dynamics response behavior. On the other hand, a nonlinear transformation approach that is based on a cubication method, was recently introduced to obtain the equivalent representation form of conservative two degree-of-freedom nonlinear oscillators [11]. There, the authors developed an approach to replace a two degree-of-freedom homogeneous, undamped system by another equivalent system with known solutions that were closed to the original one. In that approach, they first replaced the system restoring forces by polynomial expressions and then used a transformation technique to replace the resulting equations by two uncoupled nonlinear differential equations of the Duffing type. Here, some steps of that approach are used to convert forced, damped, nonlinear systems of the form [12; 13; 14; 15]  
\[\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]\left\{ \begin{array}{cc}\ddot{x}_{1}\\ \ddot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}c_{11}&c_{12}\\ c_{21}&c_{22}\end{array} \right]\left\{ \begin{array}{cc}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array} \right]\left\{ \begin{array}{c}x_{1}\\ z_{2}\end{array} \right\}+\left\{ \begin{array}{c}f_{1}(x_{1},x_{2})\\ f_{2}(x_{1},x_{2})\end{array} \right\}=\left\{ \begin{array}{c}Q_{1}^{*}(t)\\ Q_{2}^{*}(t)\end{array} \right\} \tag{1}\]  
into equivalent decoupled, nonlinear equations of the Duffing type [16; 17] in the Lyapunov sense, with similar quantitative and qualitative dynamic behaviors. In Equation (1), \(m_{1}\) and \(m_{2}\) are the masses of the system, \(c_{ij}\) and \(k_{ij}\) are the damping and stiffness system coefficients, respectively, \(f_{1}(x_{1},x_{2})\) and \(f_{2}(x_{1},x_{2})\) are even, nonlinear restoring forces, and \(Q_{1}^{*}(t)\) and \(Q_{2}^{*}(t)\) are driving periodic forces. The system initial conditions are assumed to be given by \(x_{1}(0)=x_{10}\), \(x_{2}(0)=x_{20}\), \(\dot{x}_{1}(0)=\dot{x}_{10}\), and \(\dot{x}_{2}(0)=\dot{x}_{20}\). To achieve such equivalence in the sense of Lyapunov, first, the system (1) is written in its normal canonical form in an attempt to predict the system's dynamic behavior. Then, from the normal form representation of the system (1) it is assumed that the modal system's generalized coordinates can be equivalently expressed as a power series expansion [18; 19; 20; 21] to decouple each normal mode equation into a forced, damped, nonlinear differential equation of the Duffing type, whose approximate solution has been widely discussed in the literature [22; 23; 24]. It is conjectured that the equivalent form representation of the decoupled equations will have the same Lyapunov characteristic exponents value (LCEs) as those for the normal canonical form of the original equations of motion. In other words, it can be conjectured that the resulting decoupled equations are equivalent in the sense of Lyapunov if the solutions of the decoupled expressions are solutions of the normal canonical form of the original equations of motion, and vice versa [25; 26].