Search Results (33)

This study presents the coupling-memory-sampled data control (CMSDC) design for the Takagi–Sugeno (T-S) fuzzy system that solves the stabilization issue of a surface-mounted permanent-magnet synchronous generator (PMSG)-based wind energy conversion system (WECS). A fuzzy CMSDC scheme that includes the sampled data control (SDC) and memory-sampled data control (MSDC) is designed by employing a Bernoulli distribution order. Meanwhile, the membership-function-dependent (MFD) $H_{\infty }$ performance index is presented, mitigating the continuous-time fuzzy system’s disturbances. Then, by using the Lyapunov–Krasovskii functional with the MFD $H_{\infty }$ performance index, the data of the sampling pattern, and a constant signal transmission delay, sufficient conditions are derived. These sufficient conditions are linear matrix inequalities (LMIs), ensuring the global asymptotic stability of a PMSG-based WECS under the designed control technique. The proposed method is demonstrated by a numerical simulation implemented on the PMSG-based WECS. Finally, Rossler’s system demonstrates the effectiveness and superiority of the proposed method. Full article

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. Full article

The applications of deep learning and artificial intelligence have permeated daily life, with time series prediction emerging as a focal area of research due to its significance in data analysis. The evolution of deep learning methods for time series prediction has progressed from the Convolutional Neural Network (CNN) and the Recurrent Neural Network (RNN) to the recently popularized Transformer network. However, each of these methods has encountered specific issues. Recent studies have questioned the effectiveness of the self-attention mechanism in Transformers for time series prediction, prompting a reevaluation of approaches to LTSF (Long Time Series Forecasting) problems. To circumvent the limitations present in current models, this paper introduces a novel hybrid network, Temporal Convolutional Network-Linear (TCN-Linear), which leverages the temporal prediction capabilities of the Temporal Convolutional Network (TCN) to enhance the capacity of LSTF-Linear. Time series from three classical chaotic systems (Lorenz, Mackey–Glass, and Rossler) and real-world stock data serve as experimental datasets. Numerical simulation results indicate that, compared to classical networks and novel hybrid models, our model achieves the lowest RMSE, MAE, and MSE with the fewest training parameters, and its R² value is the closest to 1. Full article

Mathematical models and numerical simulations are necessary to understand the functions of biological rhythms, to comprehend the transition from simple to complex behavior and to delineate the conditions under which they arise. The aim of this work is to investigate the R$\ddot{\mathrm{o}}$ssler-type system. This system could be proposed as a theoretical model for biological rhythms, generalizing this formula for chaotic behavior. It is assumed that the R$\ddot{\mathrm{o}}$ssler-type system has a Hamilton–Poisson realization. To semi-analytically solve this system, a Bratu-type equation was explored. The approximate closed-form solutions are obtained using the Optimal Parametric Iteration Method (OPIM) using only one iteration. The advantages of this analytical procedure are reflected through a comparison between the analytical and corresponding numerical results. The obtained results are in a good agreement with the numerical results, and they highlight that our procedure is effective, accurate and usefully for implementation in applicationssuch as an oscillator with cubic and harmonic restoring forces, the Thomas–Fermi equation and the Lotka–Voltera model with three species. Full article

This paper delves into precisely measuring liquid levels using a specific methodology with diverse real-world applications such as process optimization, quality control, fault detection and diagnosis, etc. It demonstrates the process of liquid level measurement by employing a chaotic observer, which senses multiple variables within a system. A three-dimensional computational fluid dynamics (CFD) model is meticulously created using ANSYS to explore the laminar flow characteristics of liquids comprehensively. The methodology integrates the system identification technique to formulate a third-order state–space model that characterizes the system. Based on this mathematical model, we develop estimators inspired by Lorenz and Rossler’s principles to gauge the liquid level under specified liquid temperature, density, inlet velocity, and sensor placement conditions. The estimated results are compared with those of an artificial neural network (ANN) model. These ANN models learn and adapt to the patterns and features in data and catch non-linear relationships between input and output variables. The accuracy and error minimization of the developed model are confirmed through a thorough validation process. Experimental setups are employed to ensure the reliability and precision of the estimation results, thereby underscoring the robustness of our liquid-level measurement methodology. In summary, this study helps to estimate unmeasured states using the available measurements, which is essential for understanding and controlling the behavior of a system. It helps improve the performance and robustness of control systems, enhance fault detection capabilities, and contribute to dynamic systems’ overall efficiency and reliability. Full article

In this article, a considerably efficient predictor-corrector method (PCM) for solving Atangana–Baleanu Caputo (ABC) fractional differential equations (FDEs) is introduced. First, we propose a conventional PCM whose computational speed scales with quadratic time complexity $\mathcal{O}\left(N^2\right)$ as the number of time steps N grows. A fast algorithm to reduce the computational complexity of the memory term is investigated utilizing a sum-of-exponentials (SOEs) approximation. The conventional PCM is equipped with a fast algorithm, and it only requires linear time complexity $\mathcal{O}\left(N\right)$. Truncation and global error analyses are provided, achieving a uniform accuracy order $\mathcal{O}\left(h^2\right)$ regardless of the fractional order for both the conventional and fast PCMs. We demonstrate numerical examples for nonlinear initial value problems and linear and nonlinear reaction-diffusion fractional-order partial differential equations (FPDEs) to numerically verify the efficiency and error estimates. Finally, the fast PCM is applied to the fractional-order Rössler dynamical system, and the numerical results prove that the computational cost consumed to obtain the bifurcation diagram is significantly reduced using the proposed fast algorithm. Full article

Ordinal measures provide a valuable collection of tools for analyzing correlated data series. However, using these methods to understand information interchange in the networks of dynamical systems, and uncover the interplay between dynamics and structure during the synchronization process, remains relatively unexplored. Here, we compare the ordinal permutation entropy, a standard complexity measure in the literature, and the permutation entropy of the ordinal transition probability matrix that describes the transitions between the ordinal patterns derived from a time series. We find that the permutation entropy based on the ordinal transition matrix outperforms the rest of the tested measures in discriminating the topological role of networked chaotic Rössler systems. Since the method is based on permutation entropy measures, it can be applied to arbitrary real-world time series exhibiting correlations originating from an existing underlying unknown network structure. In particular, we show the effectiveness of our method using experimental datasets of networks of nonlinear oscillators. Full article

This study focuses on the solution of the rotationally symmetric Rossler attractor by using the adaptive predictor–corrector algorithm (Apc-ABM-method) and the fractional Laplace decomposition method ($\rho$-Laplace DM). Furthermore, a comparison between the proposed methods and Runge–Kutta Fourth Order (RK4) is made. It is discovered that the proposed methods are effective and yield solutions that are identical to the approximate solutions produced by the other methods. Therefore, we can generalize the approach to other systems and obtain more accurate results. In addition to this, it has been shown to be useful for correctly discovering examples via the demonstration of attractor chaos. In the future, the two methods can be used to find the numerical solution to a variety of models that can be used in science and engineering applications. Full article

This paper discusses the analysis and computations of chaos–hyperchaos (or vice versa) transition in Rössler–Nikolov–Clodong O (RNC-O) hyperchaotic system. Our work is motivated by our previous analysis of hyperchaotic transitional regimes of RNC-O system and the results recently obtained from another researchers. The analysis and numerical simulations show that chaos–hyperchaos transition in RNC-O system is coupled to change in the equilibria type as one large hyperchaotic attractor occurs. Moreover, we show that for this system, a zero-Hopf bifurcation is not possible. We also consider the cases when the divergence of the system is a constant and detected two families of exact solutions. Full article

The dynamical interplay of coupled non-identical chaotic oscillators gives rise to diverse scenarios. The incoherent dynamics of these oscillators lead to the structural impairment of attractors in phase space. This paper investigates the couplings of Lorenz–Rössler, Lorenz–HR, and Rössler–HR to identify the dominant attractor. By dominant attractor, we mean the attractor that is less changed by coupling. For comparison and similarity detection, a cost function based on the return map of the coupled systems is used. The possible effects of frequency and amplitude differences between the systems on the results are also examined. Finally, the inherent chaotic characteristic of systems is compared by computing the largest Lyapunov exponent. The results suggest that in each coupling case, the attractor with the greater largest Lyapunov exponent is dominant. Full article

Interest in chaotic time series prediction has grown in recent years due to its multiple applications in fields such as climate and health. In this work, we summarize the contribution of multiple works that use different machine learning (ML) methods to predict chaotic time series. It is highlighted that the challenge is predicting the larger horizon with low error, and for this task, the majority of authors use datasets generated by chaotic systems such as Lorenz, Rössler and Mackey–Glass. Among the classification and description of different machine learning methods, this work takes as a case study the Echo State Network (ESN) to show that its optimization can lead to enhance the prediction horizon of chaotic time series. Different optimization methods applied to different machine learning ones are given to appreciate that metaheuristics are a good option to optimize an ESN. In this manner, an ESN in closed-loop mode is optimized herein by applying Particle Swarm Optimization. The prediction results of the optimized ESN show an increase of about twice the number of steps ahead, thus highlighting the usefulness of performing an optimization to the hyperparameters of an ML method to increase the prediction horizon. Full article

The Historical Consistent Neural Networks (HCNN) are an extension of the standard Recurrent Neural Networks (RNN): they allow the modeling of highly-interacting dynamical systems across multiple time scales. HCNN do not draw any distinction between inputs and outputs, but model observables embedded in the dynamics of a large state space. In this paper, we propose to improve the predictive power of the (Vanilla) HCNN using three methods: (1) HCNN with Partial Teacher Forcing, (2) HCNN with Sparse State Transition Matrix, and (3) a Long Short Term Memory Formulation of HCNN. We investigated the effect of those long memory improvement methods on three chaotic time-series mathematically generated from the Rabinovich–Fabrikant, the Rossler System and the Lorenz system. To complement our study, we compared the accuracy of the different HCNN variants with well-known recurrent neural networks methods such as Vanilla RNN and LSTM for the same prediction tasks. Overall, our results show that the Vanilla HCNN is superior to RNN and LSTM. This is even more the case if you include the above long memory extensions (1), (2) and (3). We demonstrate that (1) and (3) are superior for the modeling of our chaotic dynamical systems. We show that for these deterministic systems, the ensembles are narrowed. Full article

Uncovering causal interdependencies from observational data is one of the great challenges of a nonlinear time series analysis. In this paper, we discuss this topic with the help of an information-theoretic concept known as Rényi’s information measure. In particular, we tackle the directional information flow between bivariate time series in terms of Rényi’s transfer entropy. We show that by choosing Rényi’s parameter $\alpha$, we can appropriately control information that is transferred only between selected parts of the underlying distributions. This, in turn, is a particularly potent tool for quantifying causal interdependencies in time series, where the knowledge of “black swan” events, such as spikes or sudden jumps, are of key importance. In this connection, we first prove that for Gaussian variables, Granger causality and Rényi transfer entropy are entirely equivalent. Moreover, we also partially extend these results to heavy-tailed $\alpha$-Gaussian variables. These results allow establishing a connection between autoregressive and Rényi entropy-based information-theoretic approaches to data-driven causal inference. To aid our intuition, we employed the Leonenko et al. entropy estimator and analyzed Rényi’s information flow between bivariate time series generated from two unidirectionally coupled Rössler systems. Notably, we find that Rényi’s transfer entropy not only allows us to detect a threshold of synchronization but it also provides non-trivial insight into the structure of a transient regime that exists between the region of chaotic correlations and synchronization threshold. In addition, from Rényi’s transfer entropy, we could reliably infer the direction of coupling and, hence, causality, only for coupling strengths smaller than the onset value of the transient regime, i.e., when two Rössler systems are coupled but have not yet entered synchronization. Full article

In this paper, by adopting sliding mode control, an adaptive memoryless control scheme has been developed for uncertain Rössler chaotic systems with unknown time delays. Firstly, the proposed adaptive control can force the trajectories of controlled Rössler time-delayed chaotic systems into the specified sliding manifold. Then, the Riemann sum is introduced to analyze the stability of the equivalent dynamics in the sliding manifold. The control performance can be predicted even if the controlled systems have unmatched uncertainties and unknown time delays, which have not been well addressed in the literature. Numerical simulations are included to demonstrate the feasibility of the proposed scheme. Full article

We discuss chaos and its quality as measured through the 0-1 test for chaos. When the 0-1 test indicates deteriorating quality of chaos, because of the finite precision representations of real numbers in digital implementations, then the process may eventually lead to a periodic sequence. A simple method for improving the quality of a chaotic signal is to mix the signal with another signal by using the XOR operation. In this paper, such mixing of weak chaotic signals is considered, yielding new signals with improved quality (with K values from the 0-1 test close to 1). In some sense, such a mixing of signals could be considered as a two-layer prevention strategy to maintain chaos. That fact may be important in those applications when the hardware resources are limited. The 0-1 test is used to show the improved chaotic behavior in the case when a continuous signal (for example, from the Chua, Rössler or Lorenz system) intermingles with a discrete signal (for example, from the logistic, Tinkerbell or Henon map). The analysis is presented for chaotic bit sequences. Our approach can further lead to hardware applications, and possibly, to improvements in the design of chaotic bit generators. Several illustrative examples are included. Full article