Keywords: Riemann–Liouville derivative, Caputo fractional derivative, fractional differential equation, epidemiology

Keywords: control, grid resilience, nonlinear dynamics

Keywords: Mathematical methods, Special Functions
Abstract: We study fractional differential equations involving Erd'elyi--Kober derivatives with continuous variable coefficients. We solve these equations within a general framework and derive explicit solutions expressed as uniformly convergent series. The existence and uniqueness of solutions is proved using the Banach fixed point theorem in a weighted space of continuous functions. By considering several specific examples, we verify that our results are consistent with those already found in the literature.

Keywords: Mathematical methods
Abstract: This paper investigates the stability properties of fractional linear multistep methods (FLMMs) for solving fractional initial value problems (FIVPs). Lower order FLMMs are widely used to solve FIVP due to their simple stability properties. Higher order methods have not been used because their stability behaviors are highly restricted. We analyze the stability properties of implicit FLMMs and demonstrate that certain FLMMs fail to be unconditionally or A(π/2)- stable. We determine the maximum threshold values for the fractional order γ up to which the FLMM methods are A(0) or A(π/2)-stable. Theoretical stability conditions are derived, and numerical experiments are presented to validate the findings. We use these properties to modify the FIVP into a system of FIVP and use higher order FLMMs giving A(0) and A(π/2)- stabilities. The results provide crucial insights into the limitations of FLMMs and opens doors to improve the applicability in solving FIVPs with higher order FLMMs.

Keywords: Mathematical methods
Abstract: This paper investigates the chaotic dynamics and complexity of an incommensurate fractional-order discrete SI (Susceptible–Infected) model for computer virus propagation. Unlike commensurate systems, where all equations share the same fractional order, the proposed model assigns different fractional orders to the susceptible and infected compartments using the Caputo-like difference operator. By employing numerical techniques including bifurcation diagrams, Lyapunov exponents, Approximate Entropy, and the C_0 complexity measure, we demonstrate that the system exhibits rich and diverse dynamical behaviors, including chaos and high complexity. The results reveal the critical role of incommensurate fractional orders in shaping the system's nonlinear evolution.

Keywords: Mathematical methods
Abstract: A collocation-based numerical method with modified graded meshes is developed for a class of singular fractional differential equations (SFDEs) of order β, with β ∈ (0, 1). We first formulate a result about the unique solvability of the SFDE and then construct a numerical scheme with a modified graded mesh to ensure the solvability of the linear system arising from the discretization of the reformulated problem by a spline collocation method. Finally, numerical results are presented to illustrate the effectiveness of the proposed method.

Keywords: Mathematical methods, System identification & Modeling
Abstract: This paper studies discrete-time fractional systems described by the Grunwald- Letnikov-convolution type difference operator, with a particular focus on variable-order cases and systems incorporating infinite distributed delays. Since this type of operator is of convolution type, the Z-transform is used as an effective tool for analyzing stability. The work addresses the stability of solutions for linear systems, providing conditions for both stability and instability. Regions of stability are characterized in terms of the eigenvalue loci of an associated system matrix, and illustrative examples are provided.

Keywords: Mathematical methods, Special Functions, Others
Abstract: Fractional differential equations are powerful tools for modeling complex phenomena in science and engineering. This work addresses the numerical approximation of fourth-and fifth-order fractional Lane Emden Fowler equations with variable coefficients. We develop a collocation method using shifted fractional-order polynomials, where operational matrices for fractional product and integration transform the problem into a system of algebraic equations. A bound on the approximation error is established to confirm the method's reliability and convergence. Numerical experiments confirm the accuracy and efficiency of the proposed approach, demonstrating its capability to handle nonlinear terms and varying fractional orders.

Keywords: System Analysis & Dynamics, Automatic Control & Stability
Abstract: This research presents a novel implementation of Fractional Mixed-Integer Model Predictive Control (FMIMPC) for the stabilization of the chaotic Rössler system, incorporating fractional-order dynamics via the Grünwald–Letnikov characterization. The system is represented using a discrete-time linear approximation around its equilibrium point, and the control strategy is based on a finite-horizon optimization that includes binary variables to allow switching between different cost levels adaptively. The Grünwald–Letnikov (GL) characterization is utilized to capture the system’s inherent memory effects, thereby enhancing the fidelity of the fractional-order modeling. The optimization problem is formulated in YALMIP and solved using the Gurobi solver, facilitating efficient handling of mixed-integer constraints. To assess the closed-loop stability under the proposed FMIMPC strategy, the largest Lyapunov exponent is estimated by monitoring the divergence of two initially adjacent trajectories governed by the same control policy. Numerical simulations demonstrate successful trajectory tracking and chaos suppression, as evidenced by convergence to the reference state and negative Lyapunov exponent trends. The results validate the potential of FMIMPC as a robust control strategy for fractional-order chaotic systems, where discrete switching logic, enabled by binary decision variables, allows adaptive transitions between multiple control modes based on system performance.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: This paper deals with longitudinal vehicle control provided by a cruise control system using the Crone approach. A comparison with a conventional proportional-integral (PI) controller, widely used in the automotive industry, is presented. Time-domain simulations with both controllers are performed considering a nonlinear and uncertain vehicle model including wheel dynamics, vehicle mass uncertainties, aerodynamic and rolling resistances. In the case of the Crone control, the results highlight the robustness of the stability degree with respect to uncertainties.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: This paper presents a fractional-order adaptive control design for a class of fractional-order nonlinear systems with uncertainties. The proposed control scheme guarantees the stability of the system and output convergence to a reference signal. Although the system is subject to nonlinear uncertainty, the proposed adaptive control design proved to be efficient while easy set up. A numerical simulation example illustrates the satisfying performance obtained with this proposed control scheme.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: This paper presents a novel approach to trajectory tracking for quadrotor unmanned aerial vehicles (UAVs) by integrating Particle Swarm Optimization (PSO) with Fractional Order Sliding Mode Control (FOSMC). Unlike classical sliding mode schemes, the proposed FOSMC leverages fractional-order derivatives to enrich the controller’s memory and improve chattering attenuation. A PSO-only implementation is employed to tune fifteen key control and fractional parameters—spanning position gains, attitude gains, and fractional orders—subject to multi-objective fitness criteria that explicitly balance trajectory error, settling time, control effort, and robustness against disturbances. To validate performance, two representative flight scenarios (sinusoidal and high frequency) are tested under both ideal and disturbed conditions, including moderate wind gusts (5.0 m/s), translational perturbations (0.5 m/s², 5.1% of gravity), rotational disturbances (0.05 rad/s²), and measurement noise (0.01 m). Extensive simulation results and Lyapunov stability analysis with constructive gain selection procedure demonstrate that the PSO-optimized FOSMC achieves up to 84.8% reduction in RMS tracking error in disturbed conditions and maintains excellent robustness with average 1.45% performance degradation. Additionally, comprehensive 3D trajectory visualizations, time-series plots, and Lyapunov analysis illustrate enhanced disturbance rejection, control smoothness, and stability.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: 时间稳定性 （FTS） 对于现代至关重要 具有严格实时要求的工程系统。 本文研究了随机非线性的FTS 由点阵布朗运动 （fBm） 驱动的系统， 过程表现出长程依赖性。这 fBm 的非半鞅性质呈现分析性 挑战。使用分数 Wick-Itô-Skorohod 积分和分数 Itô 公式，有限时间 动态经过严格分析。事实证明 FTS 在 Lipschitz 条件下无法实现。 建立了 FTS 的充分条件，并且 提供了建立时间的明确估计。 数值模拟验证了理论结果。

Keywords: Artificial Intelligence, Mathematical methods, Others

Keywords: System identification & Modeling, Automatic Control & Stability
Abstract: Fractional (or non-integer) differentiation has played an important role in various fields notably in signal and image processing and control theory. In these last fields, important considerations such as modeling, system identification and observability are now linked to long-range dependence phenomena. It is expected that such an open invited track attracts new researchers and developers that use fractional calculus in the areas of mathematics, physics, engineering and particularly in automatic control. The latest developments for continuous-time modeling and system identification with fractional order models are proposed in the newest CRONE toolbox (version 2.0). Fully compatible with the latest Matlab® versions (since 2020a), it includes time-domain identification algorithms for estimating continuous-time models. Thanks to this new programming, the options arguments of the proposed functions have been simplified and updated. In order to help a new user, a tutorial has been completely revised as the CRONE demos command which allows handling the new options. A Guided User Interface (GUI) is now available, as the CroneIdentification application so that a user, familiar with the Matlab SystemIdentification GUI, can easily handle the system identification methods for preprocessing data, defining a model structure and estimating as well the coefficients as the differentiation orders.

This tutorial prepares the audience with: 1. System modeling with the different classes (frac_poly_exp, frac_tf, frac_lti…) 2. Simulation of fractional order models 3. System Identification with fractional order model: coefficient and order estimations 4. CroneIdentification application

Keywords: Automatic Control & Stability, Mathematical methods, System identification & Modeling
Abstract: This paper presents a Weighted Fractional Order PID (WFO-PID) controller that enhances the conventional Fractional Order PID (FOPID) structure by introducing explicit weighting factors for the tuning terms. These weights, combined with fractional orders and gain parameters, provide greater flexibility in tuning, enabling for more precise control over system dynamics. The proposed controller is applied to a mixing tank process with variable time delay, a representative case of industrial processes with challenging dynamics. Its performance is benchmarked against both PID and FOPID controllers, demonstrating improved reference tracking and disturbance rejection, particularly under varying delay conditions and noise in the transmitter. The findings highlight the WFO-PID method’s efficiency in managing intricate control situations.

Keywords: Automatic Control & Stability, Mathematical methods, Electrical Engineering & Electromagnetism
Abstract: This paper presents two fractional-order state-feedback controllers: MOIF and MOPIF. Their parameters, including proportional, integral, and fractional gains, are optimized using the MOPSO algorithm to enhance system performance. Stability is verified via Matignon's and Mikhailov's criteria. Applied to DC motor speed control, both controllers ensure minimal overshoot and satisfactory transient response.

Keywords: Automatic Control & Stability, Mathematical methods, System Analysis & Dynamics
Abstract: In this paper we extend the classical high gain theorem to linear fractional systems of commensurate order. The main purpose of the present work is to stabilize this special class of fractional order systems using a high gain output feedback. The basis for extending this theorem lies in the stability results derived by Matignon. The derived theorem ensures the stabilization of linear fractional systems of arbitrary commensurate order under certain assumptions. A numerical example illustrates the effectiveness of the approach. Furthermore, We employ the Particle Swarm Optimization (PSO) algorithm to tune the high gain feedback so that the output of the closed-loop system reaches a desired value. The study provides valuable insights for linear fractional order systems of commensurate order.

Keywords: Automatic Control & Stability, Mathematical methods, System Analysis & Dynamics
Abstract: Internet has become the primary medium for cybercrimes committed by developing malicious codes or programs which invade private and public computers, gathering information and posing a security threat. The dynamics of worm propagation in cryptovirology in blockchain system based on SEI nonlinear epidemic model is represented by three-dimensional fractional-order chaotic systems. This paper studies the robust stabilization of fractional-order cryptovirology in blockchain system in presence of disturbance, by means of an adaptive sliding mode control based on Lyapunov stability theory. It focuses on an important issue in the research of fractional order cryptovirology in blockchain systems that is chaos stabilization. Based on this adaptive sliding mode control design, the states of the fractional-order system have been stabilized. Numerical simulation results are also provided to illustrate the effectiveness of proposed schemes.

Keywords: Automatic Control & Stability, Thermal Engineering, System Analysis & Dynamics
Abstract: Electric furnaces play a vital role in industrial thermal processes, offering high energy efficiency and precise temperature control. However, challenges such as nonlinear system dynamics, inherent time delays, and thermal inertia limit the effectiveness of conventional control methods. This paper presents a novel adaptive Fractional order proportional-integral-derivative accelerated (FO-PIDA) controller optimized via a Modified Flower Pollination Algorithm (MFPA) to enhance temperature regulation in electric furnaces. The optimization strategy explicitly incorporates performance criteria including transient response measures and robustness analysis to ensure reliable operation under varying conditions. The proposed controller is evaluated against established benchmarks, namely classical MFPA-based PIDA and MFPA-based PID controllers. Simulation results demonstrate that the MFPA-optimized FO-PIDA controller achieves superior tracking accuracy, reduced rise time, and improved overall dynamic performance. These findings underscore the potential of integrating fractional-order control with advanced metaheuristic optimization to significantly improve thermal system control in industrial applications.

Keywords: Mathematical methods, System identification & Modeling, System Analysis & Dynamics
Abstract: By giving system dynamics more degrees of freedom, fractional discrete systems provide flexible modeling and control frameworks. The main dynamic characteristics of the 3D-fractional Layla and Majnun model (3D-LMM) are addressed in this study as we suggest and examine variations of the model. The presence of constant and asymptotically steady zero solutions is guaranteed by the necessary and sufficient conditions we define for the stability and asymptotic stability of the 3D-LMM. As a specific case, we analyze the system in terms of its equilibrium and fixed points, looking at the situation where Layla and Majnun have equal feelings. We demonstrate that, in the presence of marginal stability conditions, the system hovers on a precarious trajectory influenced by its fractional dynamics, neither convergent nor divergent, but on the edge between stability and instability.

Keywords: Mathematical methods, Biology & Biomedicine, Artificial Intelligence
Abstract: Abstract:

Deoxyribonucleic acid (DNA) cryptography is a notion which merges the principles of biotechnology and cryptography, situated at the intersection of mathematics, molecular biology and computer science. Correspondingly, DNA coding and DNA-based cryptography mark a significant paradigm toward secure information management by addressing information density, biological stability of DNA as well as inherent parallelism while considering the complexity and unpredictability of chaotic systems, paving the way for more effective processes of data encryption, steganography and secure storage. Biomedical data is regarded as one of the most sensitive types of data since patients’ medical data may contain their medical history, diagnoses, prescriptions, and so on. Thus, the security of biomedical data and information should be ensured to prevent unauthorized users from obtaining private data. Addressing these aspects and concerns, an innovative hybrid encryption algorithm integrating fractional-order hyper-chaotic systems with DNA-based cryptography is proposed in the present paper, considering fractional-order chaotic systems’ exhibiting more complex and richer dynamic behaviors compared to classical systems. The algorithm developed based on mathematical model established was applied to the heart disease dataset and evaluated through comprehensive experimental analyses. The results demonstrate that the encryption and decryption processes exhibit low latency, high entropy values and maintain data integrity with 100% accuracy. These findings indicate that the proposed hybrid encryption approach offers a robust and practical solution for real-time and high-security applications, particularly in the protection of sensitive biomedical data.

Keywords: Hyper-Chaos, Fractional-Order Chaotic Systems, DNA-based Cryptography, DNA Coding, Complexity, Biomedical Data Security, Data Privacy in Healthcare.

Keywords: Automatic Control & Stability, Artificial Intelligence, Electrical Engineering & Electromagnetism
Abstract: Photovoltaic (PV) systems are among the most widely used sources of renewable energy, offering a promising alternative to conventional energy. However, their efficiency is significantly affected by weather variations, leading to fluctuations in output voltage and power generation. To maintain a stable output voltage, DC converters require robust control strategies that take into account their non-linear behavior. This work presents a maximum power point tracking (MPPT) control strategy based on the incremental conductance method, enhanced by a fractional-order proportional-integral (FOPI) controller. Additionally, a novel non-linear controller topology (FONPI) is introduced by integrating a non-linear gain to the conventional FOPI controller, the modification helps cope with the non-linear nature of PV systems and improve the dynamic response to abrupt environmental changes. The controller parameters are optimized using two metaheuristic algorithms: Grey Wolf Optimizer (GWO) and Ant Lion Optimizer (ALO). System performance is evaluated through metrics including overshoot, settling time, and efficiency, demonstrating the effectiveness of the proposed approach.

Keywords: History of Fractional-Order Calculus, Automatic Control & Stability
Abstract: Due to the finite lifespan of particles and constraints imposed by physical space, tempered fractional calculus has attracted increasing attention. This paper focuses on discretetime tempered fractional neural networks (DTFNNs) with time delays and investigates their quasi-projective synchronization. By employing discrete convolution techniques and the discrete Laplace transform of tempered fractional operators, a fractional difference inequality is derived. Combining inequality methods with the Lyapunov function approach, easily verifiable sufficient conditions are established to guarantee synchronization of the proposed neural networks under a designed adaptive controller. Finally, numerical simulations are presented to validate the effectiveness and applicability of the theoretical results.

Keywords: Artificial Intelligence, Automatic Control & Stability, Electrical Engineering & Electromagnetism
Abstract: This study introduces an adaptive fractional-order radial basis function neural network (RBFNN) controller with disturbance rejection for doubly fed induction generator (DFIG)-based wind turbines. A fractional-order disturbance observer estimates time-varying bounded disturbances, enhancing robustness and dynamic performance through fractional calculus. The control scheme ensures practical finite-time convergence of tracking and estimation errors. Simulations demonstrate the method's effectiveness.