## 3 Results
Figure 3: Flow diagram of the proposed convolutional neural network (CNN) model for the Lorenz system. where \(\mathbf{q}=\left[ \begin{array}{cc}q&p&v\end{array} \right]^{\mathsf{T}}\) is the state vector and \(\sigma\), \(\rho\), and \(\beta\) are parameters. Figure 4 shows the Lorenz system with chaotic attractors. In the Lorenz system, infinitesimal differences in the parameters or initial conditions alter the state of the system. As shown in Figure 4a, the trajectory shows a butterfly shape in phase space and moves in an infinite trajectory at a bounded steady state. Figure 4b illustrates the time series result of \(q\). Figure 5a presents the Lyapunov exponent of the Lorenz equation with the change in the control parameter. Figure 5b shows the unthresholded recurrence plot of visualized chaos and non-chaos. [1 0] is a chaos image and [0 1] is a non-chaos image. The explanation of critical A for the Lyapunov exponent of the Lorenz system has been investigated [6]. To summarize this system briefly, if \(215.8>\rho>24.86\), it represents a strange attractor for the control parameter \(\rho\) and the initial condition. On the other hand, strange attractors are not displayed under some conditions. On the other hand, if \(\rho>215.8\), the system was represented as a stable attractor. In 3D phase space, the Lyapunov exponent has four types of attractors, such as stable fixed points, stable limit cycles, stable two-torus, and strange attractors (Chaos), but only chaos and non-chaos were distinguished in this system. In other words, the unthresholded recurrence matrix can be obtained from the flow of the proposed method in the control parameter representing the chaotic attractor (\(\max(\lambda_{i})>0\)) of the dynamic system, which is classified using the CNN. Figure 4: Chaotic attractor for the Lorenz system when \(\sigma=10\), \(\beta=8/3\), and \(\rho=28\); the initial conditions were \(q(0)=v(0)=0\) and \(p(0)=1\), (**a**) Phase portrait, (**b**) Time series plot of \(q(t)\). Figure 5: Dataset labeling with respect to the chaotic signal, (**a**) Lyapunov exponent in the range \(0<\rho<350\), (**b**) recurrence matrix. The Lorenz system consisted of 3800 datasets and 200 \(\times\) 200 pixels images. The time series analysis and Lyapunov exponents were calculated with a time step of 0.01 and an orthonormalization of 0.1, respectively. The dataset was divided into three main parts. The total dataset was divided into 70% of the training dataset and 30% of the testing dataset. From the divided training dataset, it was divided into 20% of the validation dataset. The dataset samples used for training are described in Table 2. Because the images are generated sequentially as the Lyapunov exponent increases, errors due to the sequential dataset were removed by shuffling the dataset. Figure 6 presents the results of the proposed CNN model. The chaos characteristics were trained for 2128 training data samples and simultaneously verified by 532 data samples during each epoch. The tests were then performed on 1140 data samples using the trained CNN model. The batch size was set to 10, and the learning rate of the optimization function was 0.0001. The accuracies and losses were collected as the training and verification data for each epoch, and are plotted in Figure 6a,b. As shown in the learning results, the accuracies were illustrated as a logarithmic function and reached almost 100%. The losses were indicated as a negative exponential function and converge close to 0%. This suggests that the proposed model detects the recurrence properties of the Lorenz system very well. Figure 7 shows the feature map in the first layer. Figure 7a shows the filter map for weight and basis using the He Gaussian method, and Figure 7b presents the feature map corresponding to each filter. In Figure 7b, the white image means that the feature cannot be found by the corresponding filter. Overall, the filter finds chaos characteristics well. Over 100 epochs, the proposed CNN model classified the chaotic characteristics with 96% accuracy.