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22 pages, 22865 KiB

Open AccessArticle

Fractional Discrete Computer Virus System: Chaos and Complexity Algorithms

by Ma’mon Abu Hammad, Imane Zouak, Adel Ouannas and Giuseppe Grassi

Algorithms 2025, 18(7), 444; https://doi.org/10.3390/a18070444 - 19 Jul 2025

Viewed by 187

The spread of computer viruses represents a major challenge to digital security, underscoring the need for a deeper understanding of their propagation mechanisms. This study examines the stability and chaotic dynamics of a fractional discrete Susceptible-Infected (SI) model for computer viruses, incorporating commensurate [...] Read more.

The spread of computer viruses represents a major challenge to digital security, underscoring the need for a deeper understanding of their propagation mechanisms. This study examines the stability and chaotic dynamics of a fractional discrete Susceptible-Infected (SI) model for computer viruses, incorporating commensurate and incommensurate types of fractional orders. Using the basic reproduction number

R_{0}

, the derivation of stability conditions is followed by an investigation of how varying fractional orders affect the system’s behavior. To explore the system’s nonlinear chaotic behavior, the research of this study employs a suite of analytical tools, including the analysis of bifurcation diagrams, phase portraits, and the evaluation of the maximum Lyapunov exponent (MLE) for the study of chaos. The model’s complexity is confirmed through advanced complexity algorithms, including spectral entropy, approximate entropy, and the

0 - 1

test. These measures offer a more profound insight into the complex behavior of the system and the role of fractional order. Numerical simulations provide visual evidence of the distinct dynamics associated with commensurate and incommensurate fractional orders. These results provide insights into how fractional derivatives influence behaviors in cyberspace, which can be leveraged to design enhanced cybersecurity measures. Full article

(This article belongs to the Section Algorithms for Multidisciplinary Applications)

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16 pages, 1576 KiB

Open AccessArticle

Comparison Principle Based Synchronization Analysis of Fractional-Order Chaotic Neural Networks with Multi-Order and Its Circuit Implementation

by Rongbo Zhang, Kun Qiu, Chuang Liu, Hongli Ma and Zhaobi Chu

Fractal Fract. 2025, 9(5), 273; https://doi.org/10.3390/fractalfract9050273 - 23 Apr 2025

Cited by 1 | Viewed by 350

Abstract

This article investigates non-fragile synchronization control and circuit implementation for incommensurate fractional-order (IFO) chaotic neural networks with parameter uncertainties. In this paper, we explore three aspects of the research challenges, i.e., theoretical limitations of uncertain IFO systems, the fragility of the synchronization controller, [...] Read more.

This article investigates non-fragile synchronization control and circuit implementation for incommensurate fractional-order (IFO) chaotic neural networks with parameter uncertainties. In this paper, we explore three aspects of the research challenges, i.e., theoretical limitations of uncertain IFO systems, the fragility of the synchronization controller, and the lack of circuit implementation. First, we establish an IFO chaotic neural network model incorporating parametric uncertainties, extending beyond conventional commensurate-order architectures. Second, a novel, non-fragile state-error feedback controller is designed. Through the formulation of FO Lyapunov functions and the application of inequality scaling techniques, sufficient conditions for asymptotic synchronization of master–slave systems are rigorously derived via the multi-order fractional comparison principle. Third, an analog circuit implementation scheme utilizing FO impedance units is developed to experimentally validate synchronization efficacy and accurately replicate the system’s dynamic behavior. Numerical simulations and circuit experiments substantiate the theoretical findings, demonstrating both robustness against parameter perturbations and the feasibility of circuit realization. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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16 pages, 589 KiB

Open AccessArticle

A New Orthogonal Least Squares Identification Method for a Class of Fractional Hammerstein Models

by Xijian Yin and Yanjun Liu

Algorithms 2025, 18(4), 201; https://doi.org/10.3390/a18040201 - 3 Apr 2025

Viewed by 329

Abstract

It is known that fractional-order models can effectively represent complex high-order systems with fewer parameters. This paper focuses on the identification of a class of multiple-input single-output fractional Hammerstein models. When the commensurate order is assumed to be known, a greedy orthogonal least [...] Read more.

It is known that fractional-order models can effectively represent complex high-order systems with fewer parameters. This paper focuses on the identification of a class of multiple-input single-output fractional Hammerstein models. When the commensurate order is assumed to be known, a greedy orthogonal least squares method is proposed to simultaneously identify the parameters and system orders, combined with a stopping rule based on the Bayesian information criterion. Subsequently, the commensurate order is determined by minimizing the normalized output error. The proposed method is validated by applying it to identify a CD-player arm system. Full article

(This article belongs to the Section Algorithms for Multidisciplinary Applications)

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24 pages, 5567 KiB

Open AccessArticle

The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization

by Louiza Diabi, Adel Ouannas, Amel Hioual, Giuseppe Grassi and Shaher Momani

Mathematics 2025, 13(2), 239; https://doi.org/10.3390/math13020239 - 12 Jan 2025

Cited by 4 | Viewed by 822

Abstract

The Ueda oscillator is one of the most popular and studied nonlinear oscillators. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on

Y

-th Caputo [...] Read more.

The Ueda oscillator is one of the most popular and studied nonlinear oscillators. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on

Y

-th Caputo fractional difference and thoroughly investigates its chaotic dynamics via commensurate and incommensurate orders. Applying numerical methods like maximum Lyapunov exponent spectra, bifurcation plots, and phase plane. We demonstrate the emergence of chaotic attractors influenced by fractional orders and system parameters. Advanced complexity measures, including approximation entropy (

A p E n

) and

C_{0}

complexity, are applied to validate and measure the nonlinear and chaotic nature of the system; the results indicate a high level of complexity. Furthermore, we propose a control scheme to stabilize and synchronize the introduced Ueda map, ensuring the convergence of trajectories to desired states. MATLAB R2024a simulations are employed to confirm the theoretical findings, highlighting the robustness of our results and paving the way for future works. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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21 pages, 4760 KiB

Open AccessArticle

Asymmetry and Symmetry in New Three-Dimensional Chaotic Map with Commensurate and Incommensurate Fractional Orders

by Hussein Al-Taani, Ma’mon Abu Hammad, Mohammad Abudayah, Louiza Diabi and Adel Ouannas

Symmetry 2024, 16(11), 1447; https://doi.org/10.3390/sym16111447 - 31 Oct 2024

Cited by 3 | Viewed by 1038

Abstract

According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and [...] Read more.

According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and incommensurate orders, revealing their impact on dynamics. Through the maximum Lyapunov exponent

(L E_{m a x})

, phase portraits, and bifurcation charts. In addition, we assess the complexity and confirm the chaotic features in the map using the approximation entropy

A p E n

and

C_{0}

complexity. Studies show that the commensurate and incommensurate derivative values influence the fractional chaotic map-based three functions, which exhibit a variety of dynamical behaviors, such as hidden attractors, asymmetry, and symmetry. Moreover, the new system’s stabilizing employing a 3D nonlinear controller is introduced. Finally, our study validates the research results using the simulation MATLAB R2024a. Full article

(This article belongs to the Special Issue Three-Dimensional Dynamical Systems and Symmetry)

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10 pages, 1329 KiB

Open AccessProceeding Paper

Comparative Analysis of Reduced Commensurate Fractional-Order Interval System Based on Artificial Bee Colony Method

by Kalyana Kiran Kumar, Gandi Ramarao, Gangu Suneetha and Budi Srinivasa Rao

Eng. Proc. 2024, 66(1), 45; https://doi.org/10.3390/engproc2024066045 - 16 Aug 2024

Viewed by 554

Abstract

Large dimensional systems are complicated and very tough to control. The effective solution is to reduce the large dimension of the systems to a lower dimension. This paper aims to reduce the dimension of a higher-order fractional commensurate interval system to a low-order [...] Read more.

Large dimensional systems are complicated and very tough to control. The effective solution is to reduce the large dimension of the systems to a lower dimension. This paper aims to reduce the dimension of a higher-order fractional commensurate interval system to a low-order fractional commensurate interval system by using evolutionary techniques. Kharitonov’s theorem and artificial bee colony optimization technique are used to determine the interval numerator and denominator polynomials for the simplified models. The algorithm is very modest and generates a stable reduced dimensional commensurate fractional interval model preserving the properties of the original system. The efficiency of the suggested strategy is illustrated with a numerical example. Full article

(This article belongs to the Proceedings of The 5th International Conference on Innovative Product Design and Intelligent Manufacturing Systems (IPDIMS 2023))

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20 pages, 5790 KiB

Open AccessArticle

FOMCON Toolbox-Based Direct Approximation of Fractional Order Systems Using Gaze Cues Learning-Based Grey Wolf Optimizer

by Bala Bhaskar Duddeti, Asim Kumar Naskar, Veerpratap Meena, Jitendra Bahadur, Pavan Kumar Meena and Ibrahim A. Hameed

Fractal Fract. 2024, 8(8), 477; https://doi.org/10.3390/fractalfract8080477 - 15 Aug 2024

Cited by 10 | Viewed by 1773

Abstract

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer [...] Read more.

This study discusses a new method for the fractional-order system reduction. It offers an adaptable framework for approximating various fractional-order systems (FOSs), including commensurate and non-commensurate. The fractional-order modeling and control (FOMCON) toolbox in MATLAB and the gaze cues learning-based grey wolf optimizer (GGWO) technique form the basis of the recommended method. The fundamental advantage of the offered method is that it does not need intermediate steps, a mathematical substitution, or an operator-based approximation for the order reduction of a commensurate and non-commensurate FOS. The cost function is set up so that the sum of the integral squared differences in step responses and the root mean squared differences in Bode magnitude plots between the original FOS and the reduced models is as tiny as possible. Two case studies support the suggested method. The simulation results show that the reduced approximations constructed using the methodology under consideration have step and Bode responses more in line with the actual FOS. The effectiveness of the advocated strategy is further shown by contrasting several performance metrics with some of the contemporary approaches disseminated in academic journals. Full article

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19 pages, 4631 KiB

Open AccessArticle

On New Symmetric Fractional Discrete-Time Systems: Chaos, Complexity, and Control

by Ma’mon Abu Hammad, Louiza Diabi, Amer Dababneh, Amjed Zraiqat, Shaher Momani, Adel Ouannas and Amel Hioual

Symmetry 2024, 16(7), 840; https://doi.org/10.3390/sym16070840 - 3 Jul 2024

Cited by 2 | Viewed by 1735

Abstract

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, [...] Read more.

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the

C_{0}

algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map’s states in commensurate and incommensurate cases. Full article

(This article belongs to the Special Issue Nonlinear Symmetric Systems and Chaotic Systems in Engineering)

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37 pages, 460 KiB

Open AccessArticle

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

by Sabir Umarov

Fractal Fract. 2024, 8(5), 254; https://doi.org/10.3390/fractalfract8050254 - 25 Apr 2024

Cited by 2 | Viewed by 1212

Abstract

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational [...] Read more.

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties. Full article

(This article belongs to the Section General Mathematics, Analysis)

19 pages, 2526 KiB

Open AccessArticle

Bifurcation, Hidden Chaos, Entropy and Control in Hénon-Based Fractional Memristor Map with Commensurate and Incommensurate Orders

by Mayada Abualhomos, Abderrahmane Abbes, Gharib Mousa Gharib, Abdallah Shihadeh, Maha S. Al Soudi, Ahmed Atallah Alsaraireh and Adel Ouannas

Mathematics 2023, 11(19), 4166; https://doi.org/10.3390/math11194166 - 5 Oct 2023

Cited by 4 | Viewed by 1750

Abstract

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting [...] Read more.

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting bifurcation diagrams, and applying the sample entropy test to assess the complexity and validate the chaotic characteristics. However, since the proposed fractional map has no fixed points, the outcomes reveal that the map can exhibit a wide range of hidden dynamical behaviors. This phenomenon significantly augments the complexity of the fractal structure inherent to the chaotic attractors. Moreover, we introduce nonlinear controllers designed for stabilizing and synchronizing the proposed fractional Hénon-based memristor map. The research emphasizes the system’s sensitivity to fractional-order parameters, resulting in the emergence of distinct dynamic patterns. The memristor-based chaotic map exhibits rich and intricate behavior, making it a captivating and significant area of investigation. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

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21 pages, 2720 KiB

Open AccessArticle

On Ikeda-Based Memristor Map with Commensurate and Incommensurate Fractional Orders: Bifurcation, Chaos, and Entropy

by Omar Alsayyed, Abderrahmane Abbes, Gharib Mousa Gharib, Mayada Abualhomos, Hassan Al-Tarawneh, Maha S. Al Soudi, Nabeela Abu-Alkishik, Abdallah Al-Husban and Adel Ouannas

Fractal Fract. 2023, 7(10), 728; https://doi.org/10.3390/fractalfract7100728 - 1 Oct 2023

Cited by 2 | Viewed by 1848

Abstract

This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the [...] Read more.

This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the interplay of different fractional orders in the proposed map. Furthermore, the study employs the sample entropy test to quantify complexity and validate the presence of chaos. Non-linear controllers are also presented to stabilize and synchronize the model. The research emphasizes the system’s sensitivity to the fractional order parameters, leading to distinct dynamic patterns and stability regimes. The memristor-based chaotic map exhibits rich and intricate behavior, making it an interesting and important area of research. Full article

(This article belongs to the Special Issue Bifurcation, Chaos, and Fractals in Fractional-Order Electrical and Electronic Systems)

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19 pages, 459 KiB

Open AccessArticle

Handling a Commensurate, Incommensurate, and Singular Fractional-Order Linear Time-Invariant System

by Iqbal M. Batiha, Omar Talafha, Osama Y. Ababneh, Shameseddin Alshorm and Shaher Momani

Axioms 2023, 12(8), 771; https://doi.org/10.3390/axioms12080771 - 9 Aug 2023

Cited by 11 | Viewed by 1525

Abstract

From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems [...] Read more.

From the perspective of the importance of the fractional-order linear time-invariant (FoLTI) system in plenty of applied science fields, such as control theory, signal processing, and communications, this work aims to provide certain generic solutions for commensurate and incommensurate cases of these systems in light of the Adomian decomposition method. Accordingly, we also generate another general solution of the singular FoLTI system with the use of the same methodology. Several more numerical examples are given to illustrate the core points of the perturbations of the considered singular FoLTI systems that can ultimately generate a variety of corresponding solutions. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications II)

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22 pages, 1697 KiB

Open AccessArticle

Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems

by Dalibor Biolek, Viera Biolková, Zdeněk Kolka and Zdeněk Biolek

Algorithms 2023, 16(7), 335; https://doi.org/10.3390/a16070335 - 13 Jul 2023

Cited by 3 | Viewed by 1931

Abstract

In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of s^α-domain poles, the corresponding basis functions are found, the kernel [...] Read more.

In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of s^α-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag–Leffler function E_α_,β, β = α. It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for α = 1. This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny‘s function of type ε_k (t, c; α, α), where c and k are the corresponding pole and its multiplicity, respectively. Full article

(This article belongs to the Collection Feature Papers in Algorithms for Multidisciplinary Applications)

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16 pages, 2881 KiB

Open AccessArticle

Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator–Prey Leslie–Gower Model with Fractional Orders

by Tareq Hamadneh, Abderrahmane Abbes, Ibraheem Abu Falahah, Yazan Alaya AL-Khassawneh, Ahmed Salem Heilat, Abdallah Al-Husban and Adel Ouannas

Axioms 2023, 12(6), 561; https://doi.org/10.3390/axioms12060561 - 6 Jun 2023

Cited by 14 | Viewed by 1837

Abstract

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model’s nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate [...] Read more.

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model’s nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator–prey Leslie–Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0–1 method, the approximate entropy test and the

C_{0}

algorithm to measure complexity and confirm chaos in the proposed system. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Fractional Chaotic Systems and Their Applications)

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14 pages, 1660 KiB

Open AccessArticle

The Fractional Discrete Predator–Prey Model: Chaos, Control and Synchronization

by Rania Saadeh, Abderrahmane Abbes, Abdallah Al-Husban, Adel Ouannas and Giuseppe Grassi

Fractal Fract. 2023, 7(2), 120; https://doi.org/10.3390/fractalfract7020120 - 27 Jan 2023

Cited by 42 | Viewed by 2435

Abstract

This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and [...] Read more.

This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and bifurcation diagrams. These behaviours imply that the fractional predator–prey discrete system of Leslie type has rich and complex dynamical properties that are influenced by commensurate and incommensurate orders. Moreover, the sample entropy test is carried out to measure the complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model. Full article

(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)

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