## 3 Results  
This study aimed to show that it is possible to classify chaos signals and non-chaos signals using a CNN and verify it using the Lyapunov exponent. Figure 2 shows the flow chart of the proposed methodology. First, the nonlinear time series data onto the parametric dynamic system were obtained by numerical analysis using the Runge-Kutta method. Second, the obtained time series data were determined using the FNN algorithm to determine the embedding dimension, and the phase space was reconstructed. Third, the reconstructed data were presented as an unthresholded recurrence plot to visualize the dynamic characteristics. The Lyapunov exponent corresponding to the change in the control parameter was calculated, and it was trained and verified using the proposed CNN model. Table 1 shows the layer type, filter size, and shape of each layer of the proposed CNN model. For image classification using a CNN, many sophisticated models have been developed that transcend human cognitive ability, but the proposed model is simply composed of a two-stage structure to distinguish only chaos and non-chaos, as shown in Figure 3. In each stage, the convolution layer, activation function, and pooling layer were included. The proposed model had two convolution layers with a \(32-3\times 3\) filter and a \(64-3\times 3\) filter, respectively. Figure 2: Flow diagram of the proposed methodology. As mentioned above, the unthresholded recurrence plot is a tool to visualize the recurrence characteristics of a dynamic system. In the case of chaotic signals, the repetition characteristics can occur in a very short interval. Therefore, the filter size was set as small as possible. In addition, two max pooling layers of 2 \(\times\) 2 were used. At the end of the last stage, through two convolution layers, the feature maps were flattened into a column vector, and the features of the image were classified into a fully connected layer for the two types of signals. Softmax was used as the activation function of the output. One of the gradient-based optimization methods was used to estimate the minimized cost function of the proposed model. The Adam optimizer is an optimization function based on the gradient decent algorithm and achieved faster convergence [25]. The weight initialization is a very important problem. Many problems can occur if the weight setup is wrong, such as convergence problems and local minimum problems. LeCun initialization follows a Gaussian distribution and uniform distribution of weight initialization for effective backpropagation [26]. Xavier initialization sets the weight depending on the number of previous and next nodes [27]. This is the most generalized method, but the output value showed inefficient results when used in the ReLU function. He initialization was developed to compensate for this [28]. In the proposed model, the weight initialization was performed using the He initialization method following a Gaussian distribution. The Lorenz system was used as the typical chaotic system for preliminary analysis and CNN verification. \(\rho\) was selected as the control parameter; the other parameters were \(\sigma=10\) and \(\beta=8/3\). The initial conditions were \(q(0)=v(0)=0\) and \(p(0)=1\). The Lorenz equation is described in the state space such that:  
\[ \begin{array}{l}\dot{q}=-\sigma(q+p)\\ \dot{p}=q(\rho-v)-p\\ \dot{v}=qp-\beta v\end{array}  \tag{30}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Layer** & **Output Shape** & **Param \#** \\ \hline Conv2d(Conv2d) & (None, 200, 200, 32) & 896 \\ Batch normalization & (None, 200, 200, 32) & 128 \\ Max pooling2d & (None, 100, 100, 32) & 0 \\ Conv2d 1 & (None, 100, 100, 64) & 18,496 \\ Batch normalization 1 & (None, 100, 100, 64) & 256 \\ Max pooling2d 1 & (None, 50, 50, 64) & 0 \\ Flatten & (None, 160,000) & 0 \\ Dense & (None, 256) & 40,960,256 \\ Batch normalization 2 & (None, 256) & 1024 \\ Dense 1 & (None, 2) & 514 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: CNN model for the Lorenz system.