## 2 Transformation Technique  
Before the decoupling procedure to find the equivalent representation form of Equation (1) in the sense of Lyapunov is introduced, Equation (1) is first written into its canonical, normal mode by applying the following linear transformation [36]  
\[\left\{ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right\}=\kappa\left\{ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right\} \tag{3}\]  
which yields:  
\[ \begin{array}{c}\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right]\left\{\begin{array}{c}\ddot{u}_{1}\\ \ddot{u}_{2}\end{array}\right\}+\left[\begin{array}{cc}v_{1}&v_{2}\\ v_{2}&v_{3}\end{array}\right]\left\{\begin{array}{c}\dot{u}_{1}\\ \dot{u}_{2}\end{array}\right\}+\left[\begin{array}{cc}\omega_{n1}^{2}&0\\ 0&\omega_{n2}^{2}\end{array}\right]\left\{\begin{array}{c}u_{1}\\ u_{2}\end{array}\right\}\\ +\left\{\begin{array}{c}\varphi_{1}u_{1}^{3}+\varphi_{2}u_{1}^{2}u_{2}+ \varphi_{3}u_{1}u_{2}^{2}+\varphi_{4}u_{2}^{3}\\ \varphi_{5}u_{1}^{3}+\varphi_{6}u_{1}^{2}u_{2}+\varphi_{7}u_{1}u_{2}^{2}+ \varphi_{8}u_{2}^{3}\end{array}\right\}=\left\{\begin{array}{c}P_{1}(t)\\ P_{2}(t)\end{array}\right\}\end{array}  \tag{4}\]  
where \(u_{1}\) and \(u_{2}\) are the normal coordinates of the linear, undamped, free vibration system, \(\kappa\) is the modal matrix which consists of the characteristic vectors that represent the natural modes of the linear system obtained from (1), and these are given by \(\kappa=\left[R_{1}\{x\}_{1},R_{2}\{x\}_{2}\right]\), where \(R_{i}\) are scale factors that can be determined from \(\sqrt{1/M_{ii}}\), in which \(M_{ii}=\kappa_{i}^{T}\mathbf{M}\kappa_{i}\) is the generalized mass and \(\{\kappa\}_{1}\) and \(\{\kappa\}_{2}\) denote column eigenvectors. If the scale factors are chosen so that \(M_{ii}=\kappa_{i}^{T}\mathbf{M}\kappa_{i}=1\), \(i=1,2\), then the normal modes are mass-orthonormal, and thus, \(\omega_{ni}^{2}=\kappa_{i}^{T}\mathbf{M}\kappa_{i}\), \(i=1,2\). Furthermore, \(\varphi_{1}\) through \(\varphi_{8}\), and \(P_{1}(t)\) and \(P_{2}(t)\) are parameters that are defined in accordance with the physics of the system. Here, the initial conditions are assumed to be given as \(u_{1}(0)=u_{10}\), \(u_{2}(0)=u_{20}\), \(\dot{u}_{1}(0)=\dot{u}_{10}\) and \(\dot{u}_{2}(0)=\dot{u}_{20}\). It is further assumed that the linear transformation (3) preserves the Lyapunov exponents of the original system (1); this implies that Lyapunov exponents of the original system (1) are equal to those computed from (4) and hold for any solution of the original system and the corresponding solution of the transformed one, as stated by [PERSON] [37]. On the other hand, one can notice that the nonlinear Equation (4) does not have known exact solutions and is as difficult to solve as the original equations of motion (1). In order to avoid these difficulties, it is now assumed that the modal system (4) can be expressed, for the system generalized coordinates (displacements or rotations), in a third-order power series expansion [18; 19; 20; 21]. Thus, Equation (4) becomes  
\[ \begin{array}{c}\ddot{u}_{1}+\omega_{n1}^{2}u_{1}+\
u_{1}\dot{u}_{1}+\
u_{2 }\dot{u}_{2}+\varphi_{1}u_{1}^{3}+\varphi_{2}u_{1}^{2}u_{2}+\varphi_{3}u_{1} u_{2}^{2}+\varphi_{4}u_{2}^{3}\\ \equiv\ddot{u}_{1}+\dot{u}_{1}u_{1}+a_{2}u_{1}^{2}+a_{3}u_{1}^{3}+2\dot{u}_{1} \dot{u}_{1}+\cdots=P_{1}(t)\\ \\ \ddot{u}_{2}+\omega_{n2}^{2}u_{2}+\
u_{2}\dot{u}_{1}+\
u_{3}\dot{u}_{2}+\varphi _{5}u_{1}^{3}+\varphi_{6}u_{1}^{2}u_{2}+\varphi_{7}u_{1}u_{2}^{2}+\varphi_{8}u _{2}^{3}\\ \equiv\ddot{u}_{2}+\dot{p}_{1}u_{2}+b_{2}u_{2}^{2}+b_{3}u_{2}^{3}+2\dot{p}_{2} \dot{u}_{2}+\cdots=P_{2}(t)\end{array}  \tag{6}\]  
Expressions (5) and (6) provide approximate decoupled representation forms of the original equations of motion whose accuracy depends on the unknown coefficients: \(a_{i}\), \(b_{i}\), and \(\mu_{i}\). Here, the external forces and the driving frequencies are considered to be the same in both systems [16; 38].