## 7 Largest Lyapunov Exponent Parameter  
In the design, the largest Lyapunov exponent parameter is one of the indicators to detect the non-periodic motion and chaos of the bending deflection of the stiffened composite laminated plate. The higher the value of the Lyapunov exponent parameter, the more non-periodic the motion and chaos. The value of the Lyapunov exponent of the bending deflection being positive means that the motion of the bending deflection is non-periodic. The negative largest Lyapunov exponent value of the bending deflection indicates periodic motion. The Wolff algorithm code based on MATLAB software is used to extract the values of the largest Lyapunov exponent by monitoring the orbital divergence for the contact force [17]. Equations (19) and (20) were used to build the Wolff algorithm code of the dynamic tool [18].  
\[\mathrm{d(t)=De^{At}} \tag{19}\]  
\[\mathrm{y(i)=\frac{1}{\Delta t}\big{[}Ind_{J}(i)\big{]}} \tag{20}\]  
Figure 15 shows the numerical value of the largest Lyapunov exponent against the number of samples at a fiber volume fraction (80%) and aspect ratio (2.5) for an un-stiffened composite laminated plate. The value of the Lyapunov exponent parameter is taken at its respective equilibrium point. One column of the bending deflection with the values of time delay and embedding dimension are used in the Wolff algorithm to extract the numerical value of the largest Lyapunov exponent parameter. One column of the bending deflection has been taken from the ANSYS program, in which it used the dynamic tool of the Wolff algorithm code.  
Figure 14: Convergence test of bending deflection against numbers of elements for plate with and without stiffeners.  
Time delay and embedding dimensions are numeric values in which it is needed in the Wolff algorithm code in the estimation of the local Lyapunov exponent parameter. The algorithm of the dynamic code of the global false nearest neighbors (GFNN) of the bending deflection is used to obtain the value of the embedding dimensions, in which it is estimated when the trend of global false nearest neighbors approaches zero [19]. In this paper, a local and optimal time delay was estimated in which it is recommended that the local time delay should be chosen to be dependent on the embedding dimensions [20]. The algorithm code of the average mutual information (AMI) is used to obtain the value of the time delay, and the first minimum time in the average mutual information trend represents the  
Figure 19: Embedding dimensions of experiment test.  
Figure 17: Time delay of experiment test.  
Figure 18: Embedding dimensions of numerical simulation.  
There is another way for estimating the largest Lyapunov exponent of the bending deflection that is called the average logarithmic divergence. The straight line represents the slope of the average logarithmic divergence, which reflects the value of the Lyapunov exponent parameter. The curve represents the logarithm function against time of the bending deflection. As known, the logarithm function treats any set of data by a straight line, which gives the value of the largest Lyapunov exponent directly. Figure 20 shows the comparison of the average logarithmic divergence of bending deflection against time at a fiber volume fraction (80%) and aspect ratio (2.5) for a plate with two stiffeners. The analytic largest Lyapunov exponent of the bending deflection is calculated after applying Equation (9) on one column of the bending deflection using the average logarithmic divergence approach, while the numerical simulation of the largest Lyapunov exponent of the bending deflection is determined using the ANSYS program. The experiment value of the largest Lyapunov exponent has been obtained after tracking the bending deflection in the direction of the plate thickness using the strain gauge through the strain meter.