## 4 Numerical Validation
### Example 2: A Forced System with Cubic Nonlinearities  
Here, the nonlinear dynamic system introduced in [21] was examined, which has the form  
\[\left[ \begin{array}{cc}m_{1}&0\\ 0&m_{2}\end{array} \right]\left\{ \begin{array}{cc}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}c_{1}&0\\ 0&c_{2}\end{array} \right]\left\{ \begin{array}{cc}\dot{x}_{1}\\ \dot{x}_{2}\end{array} \right\}+\left[ \begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}+k_{3}\end{array} \right]\left\{ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right\}+\left\{ \begin{array}{c}\varepsilon_{1}x_{1}^{3}\\ \varepsilon_{2}x_{2}^{3}\end{array} \right\}=\left\{ \begin{array}{c}\mathcal{ Q}_{1}\cos\omega_{f}\\ 0\end{array} \right\}, \tag{36}\]  
Figure 4: Frequency-amplitude response curves: (**a**) \(x_{1}\) vs \(\omega_{t}\)**(b**) \(x_{2}\) vs \(\omega_{t}\). These were computed from the numerical integration solutions of Equations (29) and (30). The parameter values used to obtain these plots were \(m_{1}=m_{2}=1\), \(k=1\), \(\varepsilon=0.5\) with \(c=0.3\) and \(f_{1}=0.25\). Here, the black solid lines describe the numerical integration solutions of Equation (30), while the blue dashed lines represent the numerical solutions obtained from Equation (29). Figure 5: Largest Lyapunov Characteristic Exponent (LLE) curves computed from the numerical integration solutions of Equations (29) and (30). The parameter values used to obtain these plots were \(m_{1}=m_{2}=1\), \(k=1\) and \(\varepsilon=0.5\) with \(c=0.3\) and \(f_{1}=0.25\). Here, the black, solid line represents the LLE computed from the numerical integration solutions of Equation (30), while the blue, dashed line describes the LLE obtained from Equation (29). with initial conditions given by \(x_{1}(0)=x_{10}\), \(\dot{x}_{1}(0)=0\), \(x_{2}(0)=x_{20}\), and \(\dot{x}_{2}(0)=0\). To find the canonical representation form of Equation (36), the following transformation approach is applied [36]:  
\[\left\{ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right\}=\left[ \begin{array}{cc}R_{1}&R_{2}\\ R_{1}f_{1}&R_{2}f_{2}\end{array} \right]\left\{ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right\}, \tag{37}\]  
which yields the canonical form (4). In this case,  
\[\omega_{n1}^{2}=\frac{1}{2m_{1}m_{2}}[k_{1}m_{2}+k_{2}(m_{1}+m_{2})+k_{3}m_{1} -H_{1}], \tag{38}\]  
\[\omega_{n2}^{2}=\frac{1}{2m_{1}m_{2}}[k_{1}m_{2}+k_{2}(m_{1}+m_{2})+k_{3}m_{1} +H_{1}], \tag{39}\]  
where  
\[H_{1}=\sqrt{((k_{2}+k_{3})m_{1}+k_{1}+k_{2}m_{2})^{2}-4(k_{2}k_{3}+k_{1}(k_{2 }+k_{3}))m_{1}m_{2}} \tag{40}\]  
and  
\[ \begin{array}{c}\
u_{1}=\big{(}c_{1}+c_{2}f_{1}^{2}\big{)}r_{1}^{2},\qquad \
u_{2}=(c_{1}+c_{2}f_{1}f_{2})r_{1}r_{2},\qquad\
u_{3}=\big{(}c_{1}+c_{2}f_{2 }^{2}\big{)}r_{2}^{2},\\ \varphi_{1}\equiv\varepsilon_{1}R_{1}^{4}+f_{1}^{4}\varepsilon_{2}R_{1}^{4}, \qquad\varphi_{2}\equiv 3R_{1}^{3}R_{2}\varepsilon_{1}+3f_{1}^{3}f_{2} \varepsilon_{2}R_{1}^{3}R_{2},\qquad\varphi_{3}\equiv 3R_{1}^{2}R_{2}^{2} \varepsilon_{1}+3f_{1}^{2}f_{2}^{2}\varepsilon_{2}R_{1}^{2}R_{2}^{2},\\ \varphi_{4}\equiv R_{1}R_{2}^{3}\varepsilon_{1}+f_{1}^{3}\varepsilon_{2}R_{1} R_{2}^{3},\qquad\varphi_{5}\equiv R_{1}^{3}R_{2}\varepsilon_{1}+f_{1}^{3}f_{2} \varepsilon_{2}R_{1}^{3}R_{2},\qquad\varphi_{6}\equiv 3R_{1}^{2}R_{2}^{2} \varepsilon_{1}+3f_{1}^{2}f_{2}^{2}\varepsilon_{2}R_{1}^{2}R_{2}^{2},\\ \varphi_{7}\equiv 3R_{1}R_{2}^{3}\varepsilon_{1}+f_{1}f_{2}^{3}\varepsilon_{2}R_{1} R_{2}^{3},\qquad\varphi_{8}\equiv R_{2}^{4}\varepsilon_{1}+f_{2}^{4} \varepsilon_{2}R_{2}^{4},\\ f_{1}\equiv\frac{k_{1}+k_{2}-m_{1}\omega_{n1}^{2}}{k_{2}},\qquad f_{2}\equiv \frac{k_{1}+k_{2}-m_{1}\omega_{n2}^{2}}{k_{2}},\qquad P_{1}(t)\equiv R_{1}Q_ {1}\cos\omega_{f}t.\end{array}  \tag{41}\]  
Thus, the equivalent decoupled representation form of Equation (36), becomes  
\[\ddot{u}_{1}+2\mu_{1}\dot{u}_{1}+a_{1}u_{1}+a_{3}u_{1}^{3}=P_{1}(t),\qquad \ddot{u}_{2}+2\mu_{2}\dot{u}_{2}+b_{1}u_{2}+b_{3}u_{2}^{3}=P_{2}(t). \tag{42}\]  
where \(a_{i}\), \(b_{i}\), and \(\mu_{i}\), are defined by Equations (27) and (28).