Haris Calgan

The numerical investigation of fractional-order chaotic systems is computationally demanding due to the long-memory effect of fractional derivatives and the need for small integration step sizes. In particular, computing bifurcation diagrams and Lyapunov exponent spectra over wide parameter ranges becomes time-consuming when conventional serial for-loop implementations are used. In this study, a parallel computational approach is proposed for the simultaneous extraction of bifurcation diagrams and Lyapunov exponents of fractional-order chaotic systems. For each parameter value, the system equations and the corresponding variational equations are integrated simultaneously, while periodic Gram–Schmidt reorthonormalization is applied for Lyapunov exponent estimation. MATLAB’s parfor structure is employed to distribute independent parameter realizations across multiple workers. The proposed approach is validated using the commensurate fractional-order Lorenz system with q1 = q2 = q3 = 0.99, where the bifurcation parameter β is varied over [1, 10]. The obtained bifurcation structures are consistent with the sign variation of the largest Lyapunov exponent, confirming the transition between chaotic and regular regimes. The serial and parallel implementations produce numerically identical Lyapunov spectra within the reported precision, indicating that the proposed parallelization strategy preserves the dynamical outputs of the corresponding serial computation. In terms of computational performance, the serial implementation required 6617.57 s, whereas the parallel implementation completed the same analysis in 1473.20 s, corresponding to a speed-up of approximately 4.49× with a parallel efficiency of 74.87%. In addition, an incommensurate fractional-order case with q1 = 0.99, q2 = 0.98, and q3 = 0.97 was tested, where the parallel implementation produced the same largest Lyapunov exponent curve as the serial implementation while reducing the computation time from 6843.45 s to 1974.79 s. These results show that the proposed parfor-based approach is a numerically consistent and computationally efficient tool for high-resolution bifurcation and Lyapunov exponent analyses of fractional-order chaotic systems.

This study comparatively and innovatively examines linear state feedback control (LSFC) of the Conformable Sliding Mode Control (CSMC) methods on the conformable fractional-order Lorenz system. The extreme sensitivity of chaotic systems to initial conditions and their nonlinear nature present significant challenges for classical control methods. In this context, the conformable fractional derivative approach integrates the memory effect and fractional order character of the system into the model by adding a tq−1 multiplier to the system dynamics, while preserving the classical differential form. Thus, both analytical simplification is achieved and the physical interpretation of fractional dynamics is preserved. One of the original contributions of the study is the investigation of the conformable fractional-order form of the Lorenz system under control and the systematic comparison of two different control strategies under the same performance criteria. Innovatively, the effect of the conformable fractional-order on control performance is parametrically analysed, and the decisive role of the fractional order (q) on system stability and convergence speed is revealed. By activating the control signals after a specific delay, the natural evolution of the system in a chaotic regime was observed, and a realistic suppression scenario was created. This approach offers a more practical engineering perspective compared to instantaneous control applications. The results showed that the LSFC method provides a simple control structure and smooth control action. In contrast, the CSMC method achieves shorter settling time, faster convergence, improved disturbance rejection capability, and higher robustness against external disturbances and parameter uncertainties.

This paper investigates the conformable fractional-order King Cobra chaotic system and proposes a robust synchronization scheme based on multi-state Proportional–Integral Active Disturbance Rejection Control (PI-ADRC). The conformable fractional derivative is introduced to generalize the classical King Cobra model which offers a simple and numerically efficient way to capture fractional-order dynamics. A detailed parametric study is carried out to explore how the fractional order and system parameters affect the onset of chaos, supported by phase portraits, Lyapunov exponents, and bifurcation analyses. To synchronize two identical chaotic systems with different initial conditions, a multi-state PI-ADRC controller equipped with an Extended State Observer (ESO) is developed. The ESO estimates and compensates for internal uncertainties and external disturbances and, allows the controller to maintain stability and accuracy even under stochastic noise. Numerical simulations show that an efficient synchronization is achieved in about one second, with smooth control signals and minimal error. The results demonstrate that the proposed approach provides a practical and robust framework for the control and synchronization of chaotic systems with conformable fractional-order behavior.

Recent developments in fractional calculus reveal that fractional operators enable the emergence of new chaotic behaviors that cannot be observed in integer-order systems. When the characteristics of chaotic systems are combined with fractional calculus, the complexity and unpredictability of the system are further enhanced. This study examines two simple yet topologically similar six-term chaotic systems, namely Sprott H and Sprott K, to explore their application in deep learning within the framework of fractional calculus. Through the analysis of time series data, phase portraits, Lyapunov exponents, and bifurcation diagrams, it is demonstrated that the fractional-order Sprott H (FOS-H) and fractional-order Sprott K (FOS-K) chaotic systems exhibit chaotic behavior under specific system parameters and fractional orders. A dataset of 28,800 time series samples is generated and classified using pre-trained deep learning models, including GoogleNet, MobileNet-v2, DarkNet-19, DarkNet-53, and EfficientNet-b0, with transfer learning techniques applied. Using the SGDM optimizer with a learning rate of 0.0003, all models, except DarkNet-19, achieve high classification accuracy. To assess generalization, a 3160-sample test dataset from lower order fractional chaotic systems (0.8 for FOS-H, 0.82 for FOS-K) is introduced, where DarkNet-53 achieves 100% accuracy and GoogleNet reaches 99.77%, outperforming other models. Additionally, a comparison with classical machine learning methods (Support Vector Machines (SVM), Decision Tree, Random Forest) reveals that while SVM achieves 99.77% validation accuracy, its performance declines for lower fractional orders. Test results confirm that GoogleNet offers the best balance between accuracy and resource efficiency, making it suitable for real-time applications. These findings demonstrate that deep learning models, particularly DarkNet-53 and GoogleNet, can effectively classify chaotic time series with varying fractional orders, even when encountering previously unseen system dynamics.