A recurrent wavelet first-order neural network (RWFONN) is proposed to select a desired attractor in a multistable erbium-doped fiber laser (EDFL). A filtered error algorithm is used to classify coexisting EDFL states and train RWFONN. The design of the intracavity laser power controller is developed according to the RWFONN states with the block control linearization technique and the super-twisting control algorithm. Closed-loop stability analysis is performed using the boundedness of synaptic weights. The efficiency of the control method is demonstrated through numerical simulations.

Considering that chaotic systems are immersed in multiple areas of science and nature and that their dynamics are governed by a great sensitivity to the initial conditions and variations in their parameters, it is of great interest for the scientific community to have tools to characterize and reproduce these trajectories. Two dynamic chaotic systems whose equations are based on the jerky system are used as benchmarks, i.e., the Memristive Shaking Chaotic System (MSCS) and the Unstable Dissipative System of type I (UDSI). One characteristic common to them is their simple mathematical structure and the complexity of their solutions. Therefore, this paper presents a strategy for identifying chaotic trajectories using a recurrent wavelet first-order neural network (RWFONN) that is trained online with an error filtering algorithm and considering the Morlet-wavelet as an activation function. The parameters of the network are adjusted considering the Euclidean distance between the solutions. Finally, the results depict proper identification of the chaotic systems studied through analysis and numerical simulation to validate the behavior and functionality of the proposed network.

This paper presents a quantitative study of the effects of using arbitrary-order operators in Neural Networks. It is based on a Recurrent Wavelet First-Order Neural Network (RWFONN), which can accurately identify several chaotic systems (measured by the mean square error and the coefficient of determination, also known as R-Squared, $r^2$) under a fixed parameter scheme in the neural algorithm. Using fractional operators, we analyze whether the identification capabilities of the RWFONN are improved, and whether it can identify signals from fractional-order chaotic systems. The results presented in this paper show that using a fractional-order Neural Network does not bring significant advantages in the identification process, compared to an integer-order RWFONN. Nevertheless, the neural algorithm (modeled with an integer-order derivative) proved capable of identifying fractional-order dynamical systems, whose behavior ranges from periodic and multi-stable to chaotic oscillations. That is, the performances of the Neural Network model with an integer-order derivative and the fractional-order network are practically identical, making the use of fractional-order RWFONN-type networks meaningless. The results deepen the work previously published by the authors, and contribute to developing structures based on robust and generic neural algorithms to identify more than one chaotic oscillator without retraining the Neural Network.

This paper presents a First-Order Recurrent Neural Network activated by a wavelet function, in particular a Morlet wavelet, with a fixed set of parameters and capable of identifying multiple chaotic systems. By maintaining a fixed structure for the neural network and using the same activation function, the network can successfully identify the three state variables of several different chaotic systems, including the Chua, PWL-Rössler, Anishchenko–Astakhov, Álvarez-Curiel, Aizawa, and Rucklidge models. The performance of this approach was validated by numerical simulations in which the accuracy of the state estimation was evaluated using the Mean Square Error (MSE) and the coefficient of determination ($r^2$), which indicates how well the neural network identifies the behavior of the individual oscillators. In contrast to the methods found in the literature, where a neural network is optimized to identify a single system and its application to another model requires recalibration of the neural algorithm parameters, the proposed model uses a fixed set of parameters to efficiently identify seven chaotic systems. These results build on previously published work by the authors and advance the development of robust and generic neural network structures for the identification of multiple chaotic oscillators.

In the study of dynamic systems, bifurcation diagrams are a very popular graphical tool for studying stability and nonlinear changes in behavior. They are instrumental in analyzing the system’s response to parameter changes. These diagrams show the system’s various dynamic patterns and phase transitions by plotting the relationship between the system response and the parameters. This paper presents a computational algorithm for studying bifurcations in dynamic systems, especially for systems with chaotic behavior that depends on parameter changes. Depending on the type of system to be analyzed, the following two strategies for computing bifurcation diagrams are described: (i) detecting crossing points through the Poincaré plane and (ii) the identification of local maxima (or minima) in one of the system solutions. In addition, this paper presents a method for implementing parallel computation in MATLAB using the Parallel Computing Toolbox from MathWorks, which significantly reduces the computational time required to generate bifurcation diagrams. This work contributes to the study of dynamic systems by providing the reader with accessible tools for studying any dynamic system under established standards and reducing the computational time required for these types of studies by implementing these algorithms utilizing the multi-core processors found in modern computers.

In this work, a neural super-twisting algorithm is applied to the design of a controller for a flywheel energy storage system (FESS) emulator. Emulation of the FESS is achieved through driving a Permanent Magnet Synchronous Machine (PMSM) coupled to a shaft to shaft DC-motor. The emulation of the FESS is carried out by controlling the velocity of the PMSM in the energy storage stag and then by controlling the DC-motor velocity in the energy feedback stage, where the plant’s states of both electrical machines are identified via a neural network. For the neural identification, a Recurrent Wavelet First-Order Neural Network (RWFONN) is proposed. For the design of the velocity controller, a super-twisting algorithm is applied by using a sliding surface as the argument; the latter is designed based on the states of the RWFONN, in combination with the block control linearization technique to the control of the angular velocity from both machines in their respective operation stage. The RWFONN is trained online using the filtered error algorithm. Closed-loop stability analysis is included when assuming boundedness of the synaptic weights. The results obtained from Matlab/Simulink validate the performance of the proposal in the control of an FESS.