Fractal and Fractional

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27 pages, 4505 KB

Open AccessArticle

A Variable-Order ABC Fractional Framework for Systemic Financial Stress Dynamics

by Saeed M. Ali

Fractal Fract. 2026, 10(5), 282; https://doi.org/10.3390/fractalfract10050282 - 23 Apr 2026

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This paper studies a novel nonlinear fractional-order financial stress model involving Atangana–Baleanu–Caputo (ABC) operators. It focuses on memory effects that are both constant and variable. The novelty of the proposed framework lies in combining multiple interconnected channels of systemic stress into one fractional [...] Read more.

This paper studies a novel nonlinear fractional-order financial stress model involving Atangana–Baleanu–Caputo (ABC) operators. It focuses on memory effects that are both constant and variable. The novelty of the proposed framework lies in combining multiple interconnected channels of systemic stress into one fractional dynamical model and looks at how they change over time and how they respond to sustained external perturbations. Theoretically, we prove well-posedness results and study the equilibrium structure and stability of the given model. On the computational side, we use numerical simulations of the individual stress components and an aggregate systemic stress index to look into short-term dynamics under different memory regimes. We also include a shock-response analysis to show how memory effects change the way stress builds up, relaxes, and spreads when forced. The sensitivity analysis shows that systemic stress is amplified by the forcing and interaction parameters and reduced by the damping parameters. These findings demonstrate that the model provides a new and effective tool for studying systemic financial instability in a fractional setting. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

17 pages, 2361 KB

Open AccessArticle

Fractional-Order Modelling of Pneumatic Transmission Dynamics in Soft Robotic Actuation

by Kutlo Popo, Andres San-Millan and Sumeet S. Aphale

Fractal Fract. 2026, 10(4), 254; https://doi.org/10.3390/fractalfract10040254 - 13 Apr 2026

Viewed by 306

Abstract

Pneumatic transmission lines play a critical role in the dynamic performance of soft robotic actuation systems, yet their behaviour is difficult to capture using conventional integer-order (IO) models. In long, slender pipelines, compressibility, viscothermal losses, and wave propagation give rise to distributed damping [...] Read more.

Pneumatic transmission lines play a critical role in the dynamic performance of soft robotic actuation systems, yet their behaviour is difficult to capture using conventional integer-order (IO) models. In long, slender pipelines, compressibility, viscothermal losses, and wave propagation give rise to distributed damping and non-exponential relaxation dynamics that are not well represented by finite-dimensional models. This paper presents a control-oriented, experimentally validated fractional-order (FO) modelling framework for pneumatic pipeline dynamics under closed-end boundary conditions. Models are calibrated using measured step-response data from a 13.2 m pipeline, with all parameters—including the fractional order—identified through a unified optimisation procedure. In addition to global fitting accuracy, model performance is evaluated using control-relevant metrics, including effective delay, initial slope and early transient behaviour, and early-time error. The results show that FO models provide a more compact and structurally consistent representation of long-memory dynamics while improving the accuracy of control-relevant features compared to their IO counterparts. These findings demonstrate that fractional dynamics offer a physically meaningful and practically useful framework for modelling pneumatic transmission lines, with direct implications for high-performance control design in soft robotic systems. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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20 pages, 555 KB

Open AccessArticle

Sampled-Data-Based Secondary Frequency Control for Fractional-Order Islanded Microgrid Subject to External Disturbance

by Yongjie Shi and Shuxian Fan

Fractal Fract. 2026, 10(4), 233; https://doi.org/10.3390/fractalfract10040233 - 31 Mar 2026

Viewed by 326

Abstract

The motivation for this paper is that most of the works on secondary frequency control are focused on conventional synchronous communication approaches. To extend this research, this paper investigates the sampled-data-based

H_{\infty}

load frequency control (LFC) problem for fractional-order islanded microgrids under [...] Read more.

The motivation for this paper is that most of the works on secondary frequency control are focused on conventional synchronous communication approaches. To extend this research, this paper investigates the sampled-data-based

H_{\infty}

load frequency control (LFC) problem for fractional-order islanded microgrids under a multi-region communication scheme. In contrast to conventional synchronous communication approaches, the proposed scheme allows each regional sensor to operate asynchronously based on its own sampling interval. To model this multi-region communication mechanism, a unified sampling sequence is constructed by collecting all sampling instants from regional sensors. Accordingly, a closed-loop system model is established through the introduction of virtual state variables. Furthermore, a novel class of looped functionals is developed to fully exploit the sampling interval characteristics of each regional sensor. By employing inequality techniques and stability analysis, sufficient conditions are derived to achieve multi-region sampled-data-based

H_{\infty}

LFC for fractional-order islanded microgrids. In addition, a co-design method is proposed to simultaneously determine the control gain and the maximum allowable sampling period. The simulations are conducted in MATLAB/Simulink (R2024a) and the LMI conditions are solved by using the LMI Toolbox and YALMIP. Finally, comprehensive simulations in MATLAB/Simulink validate the proposed scheme. For the two-region system, the method achieves a maximum sampling period of

ζ_{\max} = 0.106

s with an

H_{\infty}

performance ratio of

2.87

(below

γ = 5

) and settling times of

8.5

s and

9.2

s. Compared to synchronous sampling, it reduces the communication bandwidth by 50% for slower regions while maintaining comparable performance. For the single-region multi-rate case (

0.104

s and

0.140

s sampling periods), the

H_{\infty}

ratio is

3.12

, also satisfying

γ = 5

. The relationship between

γ

and

ζ_{\max}

is quantified:

ζ_{\max}

increases from

0.050

s to

0.106

s as

γ

increases from 3 to 5, confirming that relaxed disturbance attenuation allows larger sampling intervals. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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20 pages, 10684 KB

Open AccessArticle

Control and Synchronization of Julia Sets of the Discrete Three-Dimensional Fractional HCV Model

by Miao Ouyang, Yang Chen, Yuan Jiang, Junhua Li and Shutang Liu

Fractal Fract. 2026, 10(3), 207; https://doi.org/10.3390/fractalfract10030207 - 22 Mar 2026

Viewed by 278

Abstract

This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is [...] Read more.

This paper investigates the fractal dynamical behavior of a discrete Caputo fractional-order hepatitis C virus model. First, we analyze the stability of the system by using spectral radius and design the fractional-order controller based on coordinate transformation. Then, a nonlinear coupling controller is constructed to achieve synchronization between two fractional-order models with different parameters and different fractional orders, and the synchronization is supported by rigorous mathematical proof. Numerical simulations are used to verify the effectiveness of control and synchronization. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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20 pages, 907 KB

Open AccessArticle

Dynamics in a Fractional-Order Competitive–Competitive–Cooperative System with Beddington–DeAngelis Functional Responses and Delay

by Ting Zhou and Ahmadjan Muhammadhaji

Fractal Fract. 2026, 10(3), 176; https://doi.org/10.3390/fractalfract10030176 - 8 Mar 2026

Viewed by 338

Abstract

This study investigates the dynamics of a delayed fractional-order competition-competition-cooperative system with Beddington–DeAngelis functional responses. First, we prove the boundedness and uniqueness of the solutions. We analyze the existence conditions and local asymptotic stability of various equilibrium points using the stability theory. Second, [...] Read more.

This study investigates the dynamics of a delayed fractional-order competition-competition-cooperative system with Beddington–DeAngelis functional responses. First, we prove the boundedness and uniqueness of the solutions. We analyze the existence conditions and local asymptotic stability of various equilibrium points using the stability theory. Second, by taking the competition time delay

τ

as the bifurcation parameter, we derive explicit criteria for the stability of the system and the onset of aHopf bifurcation. Once the delay surpasses a critical threshold, the system loses its stability and displays periodic oscillatory behavior. Furthermore, the influence of the fractional order on the system dynamics is also examined. Finally, numerical simulations are performed to verify the theoretical results, providing significant insights into ecosystem complexity. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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21 pages, 3444 KB

Open AccessArticle

Toward Optimized Vibration Mitigation for Vehicle ISD Suspensions: A Fractional-Order Bridge Network Approach

by Yujie Shen, Lecun Wang, Jinpeng Yang, Hongbo Yang, Xiaofeng Yang and Miao Xu

Fractal Fract. 2026, 10(2), 115; https://doi.org/10.3390/fractalfract10020115 - 9 Feb 2026

Viewed by 386

Abstract

To further tap into the vibration isolation potential of vehicle mechatronic ISD (Inerter–Spring–Damper) suspension systems, a positive real synthesis design method for vehicle ISD suspensions is proposed based on the fractional-order bridge networks. Firstly, a quarter-car dynamic model of the mechatronic ISD suspension [...] Read more.

To further tap into the vibration isolation potential of vehicle mechatronic ISD (Inerter–Spring–Damper) suspension systems, a positive real synthesis design method for vehicle ISD suspensions is proposed based on the fractional-order bridge networks. Firstly, a quarter-car dynamic model of the mechatronic ISD suspension system is established. Subsequently, based on the impedance solution method for bridge networks, a basic bridge network structure and its corresponding series-parallel network configuration are constructed for comparative analysis. Then, with the aim of improving the overall performance of the vehicle suspension system, the particle swarm optimization algorithm is employed to determine the optimal parameters for the mechatronic ISD suspension. Finally, simulation results under random road excitation at a vehicle speed of 20 m/s indicate that, compared to the traditional passive suspension system, the fractional-order bridge network-based ISD suspension system achieves reductions of 3.9%, 27.5%, and 11.8% in the root mean square (RMS) values of vehicle body acceleration, suspension working space, and dynamic tire load, respectively. In contrast, the integer-order bridge network-based ISD suspension system only achieves reductions of 3.0%, 24.2%, and 4.5% in these respective performance metrics. The results confirm that, with an equivalent number of components, the fractional-order bridge network exhibits superior vibration isolation performance relative to series-parallel networks, thereby providing novel methodological guidance for the design of vehicle mechatronic ISD suspension. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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25 pages, 2056 KB

Open AccessArticle

Analysis of Stability and Quasi-Synchronization in Fractional-Order Neural Networks with Mixed Delays, Uncertainties, and External Disturbances

by Tian-Zeng Li, Xiao-Wen Tan, Yu Wang and Qian-Kun Wang

Fractal Fract. 2026, 10(1), 73; https://doi.org/10.3390/fractalfract10010073 - 22 Jan 2026

Viewed by 280

Abstract

This study addresses the stability and quasi-synchronization of fractional-order neural networks that incorporate mixed delays, system uncertainties, and external disturbances. Accordingly, a more realistic neural network model is constructed. For fractional-order neural networks incorporating mixed delays and uncertainties (FONNMDU), this study establishes a [...] Read more.

This study addresses the stability and quasi-synchronization of fractional-order neural networks that incorporate mixed delays, system uncertainties, and external disturbances. Accordingly, a more realistic neural network model is constructed. For fractional-order neural networks incorporating mixed delays and uncertainties (FONNMDU), this study establishes a criterion for uniform asymptotic stability and proves the existence and uniqueness of the equilibrium solution. Furthermore, it investigates the global uniform stability and stability regions of fractional-order neural networks with mixed delays, uncertainties, and external disturbances (FONNMDUED). Then, to address the quasi-synchronization problem, a controller is designed and some novel sufficient conditions for achieving quasi-synchronization are established. The results show that tuning the control parameters can adjust the error bound. These findings not only enrich the theoretical foundation of fractional-order neural networks but also offer practical insights for applications in complex systems. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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27 pages, 2899 KB

Open AccessArticle

Adaptive Fuzzy Finite-Time Synchronization Control of Fractional-Order Chaotic Systems with Uncertain Dynamics, Unknown Parameters and Input Nonlinearities

by Xiyu Zhang, Chun Feng, Youjun Zhou and Xiongfeng Deng

Fractal Fract. 2025, 9(12), 805; https://doi.org/10.3390/fractalfract9120805 - 9 Dec 2025

Cited by 1 | Viewed by 513

Abstract

This work focuses on the finite-time synchronization control (FTSC) for fractional-order chaotic systems (FOCSs) subject to uncertain dynamics, unknown parameters and input nonlinearities. In the control law design, the uncertain dynamics of the FOCSs are addressed by using fuzzy logic systems (FLSs), while [...] Read more.

This work focuses on the finite-time synchronization control (FTSC) for fractional-order chaotic systems (FOCSs) subject to uncertain dynamics, unknown parameters and input nonlinearities. In the control law design, the uncertain dynamics of the FOCSs are addressed by using fuzzy logic systems (FLSs), while the unknown control direction caused by unknown input nonlinearity is handled through applying the Nussbaum gain function (NGF) method. Parameter adaptive laws are derived to estimate the unknown parameters of the given FOCSs, the parameter vectors of the FLSs, and unknown bounded constants, respectively. By integrating these parameter-adaptive laws with the FT backstepping control framework and FO Lyapunov direct method, an adaptive fuzzy FTSC strategy is developed. This strategy ensures that the synchronization error (SE) can converge to a small neighborhood of zero (SNoZ) within a FT and all signals of the closed-loop system (CLS) remain ultimately bounded. In the end, three simulation cases are utilized to demonstrate the efficiency of the proposed control method. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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13 pages, 280 KB

Open AccessArticle

A Note on Controllability of Time Invariant Linear Fractional h-Difference Equations

by Ferhan M. Atıcı, Jagan Mohan Jonnalagadda and Amber Wu

Fractal Fract. 2025, 9(12), 784; https://doi.org/10.3390/fractalfract9120784 - 1 Dec 2025

Cited by 1 | Viewed by 452

Abstract

In this paper, we establish and prove two main results: (i) a Kalman-like controllability criterion, and (ii) a rank condition on the controllability matrix, defined via the discrete Mittag–Leffler function, for time-invariant linear fractional-order h-discrete systems. Using some properties of the Mittag–Leffler-type [...] Read more.

In this paper, we establish and prove two main results: (i) a Kalman-like controllability criterion, and (ii) a rank condition on the controllability matrix, defined via the discrete Mittag–Leffler function, for time-invariant linear fractional-order h-discrete systems. Using some properties of the Mittag–Leffler-type function within the framework of fractional h-discrete calculus, we state and prove the variation of constants formula for an initial value problem. Then we use this formula to prove the equivalence between two notions of controllability: complete controllability and controllability to the origin. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

24 pages, 6140 KB

Open AccessArticle

Stabilization of DC Microgrids Using Frequency-Decomposed Fractional-Order Control and Hybrid Energy Storage

by Sherif A. Zaid, Hani Albalawi, Hazem M. El-Hageen, Abdul Wadood and Abualkasim Bakeer

Fractal Fract. 2025, 9(10), 670; https://doi.org/10.3390/fractalfract9100670 - 17 Oct 2025

Cited by 1 | Viewed by 1300

Abstract

In DC microgrids, the combination of pulsed loads and renewable energy sources significantly impairs system stability, especially in highly dynamic operating environments. The resilience and reaction time of conventional proportional–integral (PI) controllers are often inadequate when managing the nonlinear dynamics of hybrid energy [...] Read more.

In DC microgrids, the combination of pulsed loads and renewable energy sources significantly impairs system stability, especially in highly dynamic operating environments. The resilience and reaction time of conventional proportional–integral (PI) controllers are often inadequate when managing the nonlinear dynamics of hybrid energy storage systems. This research suggests a frequency-decomposed fractional-order control strategy for stabilizing DC microgrids with solar, batteries, and supercapacitors. The control architecture divides system disturbances into low- and high-frequency components, assigning high-frequency compensation to the ultracapacitor (UC) and low-frequency regulation to the battery, while a fractional-order controller (FOC) enhances dynamic responsiveness and stability margins. The proposed approach is implemented and assessed in MATLAB/Simulink (version R2023a) using comparison simulations against a conventional PI-based control scheme under scenarios like pulsed load disturbances and fluctuations in renewable generation. Grey Wolf Optimizer (GWO), a metaheuristic optimization procedure, has been used to tune the parameters of the FOPI controller. The obtained results using the same conditions were compared using an optimal fractional-order PI controller (FOPI) and a conventional PI controller. The microgrid with the best FOPI controller was found to perform better than the one with the PI controller. Consequently, the objective function is reduced by 80% with the proposed optimal FOPI controller. The findings demonstrate that the proposed method significantly enhances DC bus voltage management, reduces overshoot and settling time, and lessens battery stress by effectively coordinating power sharing with the supercapacitor. Also, the robustness of the proposed controller against parameters variations has been proven. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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24 pages, 1878 KB

Open AccessArticle

Advancements in Sustainable Mobility: Fractional-Order FOC of IM in an Electric Vehicle Powered by an Autonomous PV Battery System

by Fatma Ben Salem, Jaouhar Mouine and Nabil Derbel

Fractal Fract. 2025, 9(10), 661; https://doi.org/10.3390/fractalfract9100661 - 14 Oct 2025

Viewed by 749

Abstract

This paper presents a novel fractional-order field-oriented control (FO-FOC) strategy for induction motor drives in electric vehicles (EVs) powered by an autonomous photovoltaic (PV) battery energy system. The proposed control approach integrates a fractional-order sliding mode controller (FO-SMC) into the conventional FOC framework [...] Read more.

This paper presents a novel fractional-order field-oriented control (FO-FOC) strategy for induction motor drives in electric vehicles (EVs) powered by an autonomous photovoltaic (PV) battery energy system. The proposed control approach integrates a fractional-order sliding mode controller (FO-SMC) into the conventional FOC framework to enhance dynamic performance, improve robustness, and reduce sensitivity to parameter variations. The originality of this work lies in the combined use of fractional-order control and real-time adaptive parameter updating, applied within a PV battery-powered EV platform. This dual-layer control structure allows the system to effectively reject disturbances, maintain torque and flux tracking, and mitigate the effects of component aging or thermal drift. Furthermore, to address the chattering phenomenon typically associated with sliding mode control, a continuous saturation function was employed, resulting in smoother voltage and current responses more suitable for real-time implementation. Extensive simulation studies were conducted under ideal conditions, with parameter mismatch, and with the proposed adaptive update laws. Results confirmed the superiority of the FO-based approach over classical integer-order designs in terms of speed tracking, flux regulation, torque ripple reduction, and system robustness. The proposed methodology offers a promising solution for next-generation sustainable mobility systems requiring high-performance, energy-efficient, and fault-tolerant electric drives. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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24 pages, 9636 KB

Open AccessArticle

Finite-Time Modified Function Projective Synchronization Between Different Fractional-Order Chaotic Systems Based on RBF Neural Network and Its Application to Image Encryption

by Ruihong Li, Huan Wang and Dongmei Huang

Fractal Fract. 2025, 9(10), 659; https://doi.org/10.3390/fractalfract9100659 - 13 Oct 2025

Cited by 1 | Viewed by 771

Abstract

This paper innovatively achieves finite-time modified function projection synchronization (MFPS) for different fractional-order chaotic systems. By leveraging the advantages of radial basis function (RBF) neural networks in nonlinear approximation, this paper proposes a novel fractional-order sliding-mode controller. It is designed to address the [...] Read more.

This paper innovatively achieves finite-time modified function projection synchronization (MFPS) for different fractional-order chaotic systems. By leveraging the advantages of radial basis function (RBF) neural networks in nonlinear approximation, this paper proposes a novel fractional-order sliding-mode controller. It is designed to address the issues of system model uncertainty and external disturbances. Based on Lyapunov stability theory, it has been demonstrated that the error trajectory can converge to the equilibrium point along the sliding surface within a finite time. Subsequently, the finite-time MFPS of the fractional-order hyperchaotic Chen system and fractional-order chaotic entanglement system are realized under conditions of periodic and noise disturbances, respectively. The effects of the neural network parameters on the performance of the MFPS are then analyzed in depth. Finally, a color image encryption scheme is presented integrating the above MFPS method and exclusive-or operation, and its effectiveness and security are illustrated through numerical simulation and statistical analysis. In the future, we will further explore the application of fractional-order chaotic system MFPS in other fields, providing new theoretical support for interdisciplinary research. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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18 pages, 1082 KB

Open AccessArticle

Dynamics in a Fractional-Order Four-Species Food Web System with Top Predator and Delays

by Xiao Tang and Ahmadjan Muhammadhaji

Fractal Fract. 2025, 9(10), 650; https://doi.org/10.3390/fractalfract9100650 - 8 Oct 2025

Cited by 1 | Viewed by 881

Abstract

The predator–prey model is a fundamental mathematical tool in ecology used to understand the dynamic relationship between predator and prey populations. This study develops a fractional-order delayed dynamical model for a four-species food web, which includes an intermediate predator feeding on two prey [...] Read more.

The predator–prey model is a fundamental mathematical tool in ecology used to understand the dynamic relationship between predator and prey populations. This study develops a fractional-order delayed dynamical model for a four-species food web, which includes an intermediate predator feeding on two prey species and a top predator preying on all three species. The boundedness of the system’s solutions is first rigorously established using the Laplace transform method. Next, a nonlinear dynamical analysis is performed to determine the existence conditions and local stability of both the trivial and positive equilibrium points. In particular, by treating the time delay as a bifurcation control parameter, explicit criteria for the onset of Hopf bifurcation are derived. Theoretically, when the delay magnitude exceeds a critical threshold, the system loses stability and exhibits sustained oscillatory behavior. Finally, systematic numerical simulations are performed under specific parameter settings. The effects of varying fractional orders and delay magnitudes on the system’s dynamics are quantitatively explored, and the results show strong agreement with the theoretical predictions. Full article

(This article belongs to the Special Issue Advances in Dynamics and Control of Fractional-Order Systems)

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