Search Results (40)

This article investigates the chaotic features of a novel variable-order fractional Rössler system built with Liouville–Caputo derivatives of variable order. Variable-order fractional (VOF) operators incorporated in the system render its dynamics more flexible and richer with memory and hereditary effects. We run numerical simulations to see how different fractional-order functions alter the qualitative behavior of the system. We demonstrate this via phase portraits and time-series responses. The research analyzes bifurcation development, chaotic oscillations, and stability transition and demonstrates dynamic patterns impossible to describe with integer-order models. Lyapunov exponent analysis also demonstrates system sensitivity to initial conditions and small disturbances. The outcomes confirm that the variable-order procedure provides a faithful representation of nonlinear and intricate processes of engineering and physical sciences, pointing out the dominant role of memory effects on the transitions among periodic, quasi-periodic, and chaotic regimes. Full article

In this paper, we first used a Modified Numerical Approximation Method (NAM) and then a fractional Laplace Decomposition Method (LDM) to find the solution to the symmetric Rössler attractor. The newly proposed NAM is obtained through a nuanced discretization of the Caputo derivative, rendering it exceptionally effective in emulating the inherent sensitivity and memory-dependent characteristics of fractional-order systems. Second, a comprehensive analysis is conducted to examine how variations in the fractional parameters ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$ influence the dynamic response of the system. Third, the simulation results, which include time series, bifurcation diagrams, and Lyapunov exponent spectra, show that the proposed method works well to find changes in system behavior that integer-order or lower-accuracy schemes cannot find. The fractional Laplace Decomposition Method (LDM) is straightforward to implement, computationally efficient, and exhibits outstanding accuracy. Other widely used approximation approaches achieve comparable results. The comparisons between NAM and LDM reveal that these two methodologies are not only highly consistent but also mutually reinforcing. Their straightforward application and robust consistency of numerical solutions indicate that these methods can be effectively utilized in the majority of fractional-order systems, resulting in more accurate and practical answers. Full article

This study examines the contrast in the nonlinear dynamics of Thrinax radiata Lodd. ex Schult. & Schult. f. Seed germplasm explored by optical and electrical signals. By integrating chaotic attractors for the modulation of the optical and electrical measurements, the research ensures high sensitivity monitoring of seed germplasm dynamics. Reflectance measurements and electrical responses were analyzed across different laser pulse energies using Newton–Leipnik and Rössler chaotic attractors for signal characterization. The optical attractor captured laser-induced changes in reflectance, highlighting nonlinear thermal effects, while the electrical attractor, through a custom-designed circuit, revealed electromagnetic interactions within the seed. Results showed that increasing laser energy amplified voltage magnitudes in both systems, demonstrating their sensitivity to energy inputs and distinct energy-dependent chaotic patterns. Fractional calculus, specifically the Caputo fractional derivative, was applied for modeling temperature distribution within the seeds during irradiation. Simulations revealed heat transfer about 1 °C in central regions, closely correlating with observed changes in chaotic attractor morphology. This interdisciplinary approach emphasizes the unique strengths of each method: optical attractors effectively analyze photoinduced thermal effects, while electrical attractors offer complementary insights into bioelectrical properties. Together, these techniques provide a realistic framework for studying seed germplasm dynamics, advancing knowledge of their responses to external perturbations. The findings pave the way for future applications and highlight the potential of chaos theory for early detection of structural and bioelectrical changes induced by external energy inputs, thereby contributing to sample protection. Our results provide quantitative dynamical descriptors of laser-evoked seed responses that establish a tractable framework for future studies linking these metrics to physiological outcomes. Full article

This paper proposes a numerical technique to study dynamical systems and uncover new behaviors in chaotic fractional-order models, a field that continues to attract significant research interest due to its broad applicability and the ongoing development of innovative methods. Through various types of simulations, this approach is able to uncover novel dynamic behaviors that were previously undiscovered. The results guarantee that initial conditions and fractional-order derivatives have a significant contribution to system dynamics, thus distinguishing fractional systems from traditional integer-order models. The approach demonstrated has excellent consistency with traditional approaches for integer-order systems while offering higher accuracy for fractional orders. Consequently, this approach serves as a powerful and efficient tool for studying complex chaotic models. Fractional-order dynamical systems (FDSs) are particularly noteworthy for their ability to model memory and hereditary characteristics. The method identifies new complex phenomena, including new chaos, unusual attractors, and complex time-series patterns, not documented in the existing literature. We use Lyapunov exponents, bifurcation analysis, and Poincaré sections to thoroughly investigate the system dynamics, with particular emphasis on the effect of fractional-order and initial conditions. Compared to traditional integer-order approaches, our approach is more accurate and gives a more efficient device for facilitating research on fractional-order chaos. Full article

Chaotic systems, governed by deterministic nonlinear equations yet exhibiting highly complex and unpredictable behaviors, have emerged as valuable tools at the intersection of mathematics, engineering, and information security. This paper presents a comparative study of the Lorenz and Rössler systems, focusing on their dynamic complexity and statistical independence—two critical properties for applications in chaos-based cryptography. By integrating techniques from nonlinear dynamics (e.g., Lyapunov exponents, KS entropy, Kaplan–Yorke dimension) and statistical testing (e.g., chi-square and Gaussian transformation-based independence tests), we provide a quantitative framework to evaluate the pseudo-randomness potential of chaotic trajectories. Our results show that the Lorenz system offers faster convergence to chaos and superior statistical independence over time, making it more suitable for rapid encryption schemes. In contrast, the Rössler system provides complementary insights due to its simpler attractor and longer memory. These findings contribute to a multidisciplinary methodology for selecting and optimizing chaotic systems in secure communication and signal processing contexts. Full article

In this study, the numerical scheme for the Caputo fractional derivative (NCFD) method and the He–Laplace method (H-LM) are two powerful methods used for analyzing fractional-order systems. These two approaches are used in the study of the complex dynamics of the fractional-order inverted Rössler system, particularly for the detection of chaotic behavior. The enhanced NCFD method is used for reliable and accurate numerical simulations by capturing the intricate dynamics of chaotic systems. Further, analytical solutions are obtained using the H-LM for the fractional-order inverted Rössler system. This method is popular due to its simplicity, numerical stability, and ability to handle most initial values, yielding very accurate results. Combining analytical insights from the H-LM with the robust numerical accuracy of the NCFD approach yields a comprehensive understanding of this system’s dynamics. The advantages of the NCFD method include its high numerical accuracy and ability to capture complex chaotic dynamics. The H-LM offers simplicity and stability. The proposed methods prove to be capable of detecting chaotic attractors, estimating their behavior correctly, and finding accurate solutions. These findings confirm that NCFD- and H-LM-based approaches are promising methods for the modeling and solution of complex systems. Since these results provide improved numerical simulations and solutions for a broad class of fractional-order models, they will thus be of greatest use in forthcoming applications in engineering and science. Full article

We propose using the ordinal pattern transition (OPT) entropy measured at sentinel central nodes as a potential predictor of explosive transitions to synchronization in networks of various dynamical systems with increasing complexity. Our results demonstrate that the OPT entropic measure surpasses traditional early warning signal (EWS) measures and could be valuable to the tools available for predicting critical transitions. In particular, we investigate networks of diffusively coupled phase oscillators and chaotic Rössler systems. As maps, we consider a neural network of Chialvo maps coupled in star and scale-free configurations. Furthermore, we apply this measure to time series data obtained from a network of electronic circuits operating in the chaotic regime. Full article

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