## 3 Results
### Test Case 1: Willamoski-Rossler Reaction  
This benchmark is widely known because it has been the first case that showed that deterministic chaos can be generated by a chemical reaction. The process consists of a multi-step catalytic reaction in the open system, which involves five species, between initiators and products and three intermediates. The following equations describe the five steps of the process:  
\[ \begin{array}{c}K_{1}\\ A_{1}+X\rightleftharpoons 2X\\ K_{-1}\\ K_{2}\\ X+Y\rightleftharpoons 2Y\\ K_{-2}\\ K_{3}\\ A_{5}+Y\rightleftharpoons A_{2}\\ K_{-3}\\ K_{4}\\ X+Z\rightleftharpoons A_{3}\\ K_{-4}\\ K_{5}\\ A_{4}+Z\rightleftharpoons 2Z\\ K_{-5}\end{array}  \tag{25}\]  
where \(A_{1},~{}A_{4}\), and \(A_{5}\) are the initiators, \(A_{2}\) and \(A_{3}\) are the products, and \(X,~{}Y\), and \(Z\) are the intermediates. By assuming the five-step equations, the following overall reaction is obtained:  
\[A_{1}+A_{4}+A_{5}\Leftrightarrow A_{2}+A_{3} \tag{26}\]  
In an open system, the concentration of both initiators and products is constant in nominal conditions, while the concentrations of the three intermediate species assume chaotic behavior. By assuming \(X,~{}Y\), and \(Z\) as state variables, the phase space can be represented graphically. The evolution of the state variables can be studied by means of the following set of differential equations:  
\[\left\{ \begin{array}{l}\dot{X}=K_{1}X-K_{-1}X^{2}-K_{2}XY+K_{-2}Y^{2}-K_{4} XY+K_{-4}\\ \dot{Y}=K_{2}XY-K_{-2}Y^{2}-K_{3}Y+K_{-3}\\ \dot{Z}=-K_{4}XZ+K_{5}Z-K_{5}Z^{2}+K_{-4}\end{array} \right. \tag{27}\]  
In Figure 4, the phase space trajectory of the Willamowski-Rossler model, depicted in a three-dimensional coordinate system, is shown corresponding to the following reaction rates: \(K_{1}=30\), \(K_{-1}=0.25\), \(K_{2}=1\), \(K_{-2}=10^{-4}\), \(K_{3}=10\), \(K_{-3}=10^{-3}\), \(K_{4}=1\), \(K_{-4}=10^{-3}\), \(K_{5}=16.5\), \(K_{-5}=0.5\), and initial conditions are: \(X_{0}=0.21\), \(Y_{0}=0. In the chemical process, chaotic behavior is necessary to obtain an efficient mixing of the reacting species, avoiding spending a great quantity of energy. Therefore, it is important to forecast any variation in the parameters before the chaotic behavior of the system is lost. Figure 5 illustrates the evolution of the system's state resulting from variations in the state variables. The trends depicted in Figure 5 have been obtained by setting \(K_{1}=29\). The results clearly demonstrate that the chaotic behavior characteristic of the system is entirely suppressed across all state variables after only a few seconds. This loss of chaotic dynamics highlights the sensitivity of the system's behavior to changes in its parameters, as well as the stabilizing effect introduced by the specific value of \(K_{1}\). Three LRNNs have been trained, one for each state variable, to predict the value of the respective variable one time step ahead. The optimal network structure has been determined through a trial-and-error approach to maximize forecasting accuracy. The neural network used to perform the test has a 1-25-1 structure (\(K=25\)), with only one delay (\(r=1\)) for each hidden neuron and no delay in the output neuron. As an activation function, the hyperbolic tangent has been assigned to the hidden neurons, while the output neuron is linear. A sequence of 20 s of each state variable of the system in  
Figure 4: Phase space trajectory of the Willamowski–Rössler model. Figure 5: State evolution in presence of a 3% deviation of the reaction rate \(K_{1}\). nominal conditions has been acquired with a sample time of 5 ms, obtaining sequences of 4000 samples, and these sequences have been used to train the neural networks. Figure 6 presents the evolution of the Mean Squared Error (MSE) observed during the training phase for variable X. The plot demonstrates that the MSE decreases steadily during training, reaching a value of approximately 0.003 after around 350 iterations.