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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, 23418 KiB

Open AccessArticle

Criticality and Magnetic Phases of Ising Shastry–Sutherland Candidate Holmium Tetraboride

by Guga Khundzakishvili, Bishnu Prasad Belbase, Pravin Mahendran, Kevin Zhang, Hanjing Xu, Eliana Stoyanoff, Joseph George Checkelsky, Yaohua Liu, Linda Ye and Arnab Banerjee

Materials 2025, 18(11), 2504; https://doi.org/10.3390/ma18112504 - 26 May 2025

Cited by 1 | Viewed by 866

Abstract

Frustrated magnetic systems arising in geometrically constrained lattices represent rich platforms for exploring unconventional phases of matter, including fractional magnetization plateaus, incommensurate orders and complex domain dynamics. However, determining the microscopic spin configurations that stabilize such phases is a key challenge, especially when [...] Read more.

Frustrated magnetic systems arising in geometrically constrained lattices represent rich platforms for exploring unconventional phases of matter, including fractional magnetization plateaus, incommensurate orders and complex domain dynamics. However, determining the microscopic spin configurations that stabilize such phases is a key challenge, especially when in-plane and out-of-plane spin components coexist and compete. Here, we combine neutron scattering and magnetic susceptibility experiments with simulations to investigate the emergence of field-induced fractional plateaus and the related criticality in a frustrated magnet holmium tetraboride (HoB₄) that represents the family of rare earth tetraborides that crystalize in a Shastry–Sutherland lattice in the

a b

plane. We focus on the interplay between classical and quantum criticality near phase boundaries, as well as the role of material defects in the stabilization of the ordered phases. We find that simulations using classical annealing can explain certain observed features in the experimental Laue diffraction and the origin of multiple magnetization plateaus. Our results show that defects and out-of-plane interactions play an important role and can guide the route towards resolving microscopic spin textures in highly frustrated magnets. Full article

(This article belongs to the Special Issue Neutron Scattering in Materials)

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18 pages, 338 KiB

Open AccessFeature PaperArticle

Fundamental Matrix, Measure Resolvent Kernel and Stability Properties of Fractional Linear Delayed System with Discontinuous Initial Conditions

by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev

Mathematics 2025, 13(9), 1408; https://doi.org/10.3390/math13091408 - 25 Apr 2025

Viewed by 289

Abstract

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral [...] Read more.

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral system is introduced. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. As a consequence, an integral representation of the solutions of the studied system is obtained. Then, using the obtained results, relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions are established. Full article

(This article belongs to the Section C: Mathematical Analysis)

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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19 pages, 4102 KiB

Open AccessArticle

On Fractional Discrete Memristive Model with Incommensurate Orders: Symmetry, Asymmetry, Hidden Chaos and Control Approaches

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

Symmetry 2025, 17(1), 143; https://doi.org/10.3390/sym17010143 - 18 Jan 2025

Cited by 3 | Viewed by 1001

Abstract

Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using

ϑ_{i}

-th Caputo-like [...] Read more.

Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using

ϑ_{i}

-th Caputo-like operator. Bifurcation, phase portraits and the computation of the maximum Lyapunov Exponent

(L E_{m a x})

are used to demonstrate their impact on the system’s dynamics. Furthermore, we employ the sample entropy approach (SampEn),

C_{0}

complexity and the 0-1 test to quantify complexity and validate chaos in the incommensurate system. Studies indicate that the discrete memristive system with incommensurate fractional orders manifests diverse dynamical behaviors, including hidden chaos, symmetry, and asymmetry attractors, which are influenced by the incommensurate derivative values. Moreover, a 2D non-linear controller is presented to stabilize and synchronize the novel system. The work results are provided by numerical simulation obtained using MATLAB R2024a codes. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Chaos Theory and Application)

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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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29 pages, 7819 KiB

Open AccessArticle

Dynamic Behavior and Fixed-Time Synchronization Control of Incommensurate Fractional-Order Chaotic System

by Xianchen Wang, Zhen Wang and Shihong Dang

Fractal Fract. 2025, 9(1), 18; https://doi.org/10.3390/fractalfract9010018 - 30 Dec 2024

Viewed by 920

Abstract

In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial [...] Read more.

In this paper, an incommensurate fractional-order chaotic system is established based on Chua’s system. Combining fractional-order calculus theory and the Adomian algorithm, the dynamic phenomena of the incommensurate system caused by different fractional orders are studied. Meanwhile, the incommensurate system parameters and initial values are used as variables to study the dynamic characteristics of the incommensurate system. It is found that there are rich coexistence bifurcation diagrams and coexistence Lyapunov exponent spectra which are further verified with the phase diagrams. Moreover, a special dynamic phenomenon, such as chaotic degenerate dynamic behavior, is found in the incommensurate system. Secondly, for the feasibility of practical application, the equivalent analog circuit of incommensurate system is realized according to fractional-order time–frequency frequency domain algorithm. Finally, in order to overcome the limitation that the convergence time of the finite-time synchronization control scheme depends on the initial value, a fixed-time synchronization control scheme is proposed in the selection of synchronization control scheme. The rationality of this scheme is proved by theoretical analysis and numerical simulation. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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16 pages, 309 KiB

Open AccessFeature PaperArticle

Backward Continuation of the Solutions of the Cauchy Problem for Linear Fractional System with Deviating Argument

by Hristo Kiskinov, Mariyan Milev, Milena Petkova and Andrey Zahariev

Mathematics 2025, 13(1), 76; https://doi.org/10.3390/math13010076 - 28 Dec 2024

Viewed by 542

Abstract

Fractional calculus provides tools to model systems with memory effects; when coupled with delays, they model process histories inspired by two independent sources—the memory of the fractional derivative and the impact conditioned by the delays. This work considers a Cauchy (initial) problem for [...] Read more.

Fractional calculus provides tools to model systems with memory effects; when coupled with delays, they model process histories inspired by two independent sources—the memory of the fractional derivative and the impact conditioned by the delays. This work considers a Cauchy (initial) problem for a linear delayed system with derivatives in Caputo’s sense of incommensurate order, distributed delays, and piecewise initial functions. For this initial problem, we study the important problem of the backward continuation of its solutions. We consider the backward continuation of the solutions as a problem of the renewal of a process with aftereffect under given final observation. Sufficient conditions for backward continuation of the solutions of these systems have been obtained. As application, a formal (Lagrange) adjoint system for the studied homogeneous system is introduced, and using the backward continuation, it is proved that for this system there exists a unique matrix solution called by us as the formal adjoint fundamental matrix, which can play the same role as the fundamental matrix in the forward case. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

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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17 pages, 3103 KiB

Open AccessArticle

Distributed Consensus Tracking of Incommensurate Heterogeneous Fractional-Order Multi-Agent Systems Based on Vector Lyapunov Function Method

by Conggui Huang and Fei Wang

Fractal Fract. 2024, 8(10), 575; https://doi.org/10.3390/fractalfract8100575 - 30 Sep 2024

Viewed by 987

Abstract

This paper investigates the tracking problem of fractional-order multi-agent systems. Both the order and parameters of the leader are unknown. Firstly, based on the positive system approach, the asymptotically stable criteria for incommensurate linear fractional-order systems are derived. Secondly, the models of incommensurate [...] Read more.

This paper investigates the tracking problem of fractional-order multi-agent systems. Both the order and parameters of the leader are unknown. Firstly, based on the positive system approach, the asymptotically stable criteria for incommensurate linear fractional-order systems are derived. Secondly, the models of incommensurate heterogeneous multi-agent systems are introduced. To cope with incommensurate and heterogeneous situations among followers and the leader, radial basis function neural networks (RBFNNs) and a discontinuous control method are used. Thirdly, the consensus criteria are derived by using the Vector Lyapunov Function method. Finally, a numerical example is presented to illustrate the effectiveness of the proposed theoretical method. Full article

(This article belongs to the Special Issue Analysis and Modeling of Fractional-Order Dynamical Networks)

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17 pages, 338 KiB

Open AccessArticle

The Well-Posedness of Incommensurate FDEs in the Space of Continuous Functions

by Babak Shiri, Yong-Guo Shi and Dumitru Baleanu

Symmetry 2024, 16(8), 1058; https://doi.org/10.3390/sym16081058 - 16 Aug 2024

Cited by 4 | Viewed by 1168

Abstract

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. [...] Read more.

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality. Full article

(This article belongs to the Section Mathematics)

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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)

14 pages, 1152 KiB

Open AccessArticle

A Novel Approach to Modeling Incommensurate Fractional Order Systems Using Fractional Neural Networks

by Meshach Kumar, Utkal Mehta and Giansalvo Cirrincione

Mathematics 2024, 12(1), 83; https://doi.org/10.3390/math12010083 - 26 Dec 2023

Cited by 5 | Viewed by 1880

Abstract

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly

^{R L} F σ

, activation function in modeling the chaotic dynamics of Chua’s circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address [...] Read more.

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly

^{R L} F σ

, activation function in modeling the chaotic dynamics of Chua’s circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address the limitations of conventional activation functions in capturing complex relationships within datasets. Employing a structured approach, the methods involve training MLP models with various activation functions, including

^{R L} F σ

, sigmoid, swish, and proportional Caputo derivative

^{P C} σ

, and subjecting them to rigorous comparative analyses. The main findings reveal that the proposed

^{R L} F σ

consistently outperforms traditional counterparts, exhibiting superior accuracy, reduced Mean Squared Error, and faster convergence. Notably, the study extends its investigation to scenarios with reduced dataset sizes and network parameter reductions, demonstrating the robustness and adaptability of

^{R L} F σ

. The results, supported by convergence curves and CPU training times, underscore the efficiency and practical applicability of the proposed activation function. This research contributes a new perspective on enhancing neural network architectures for system modeling, showcasing the potential of

^{R L} F σ

in real-world applications. Full article

(This article belongs to the Special Issue New Trends on Identification of Dynamic Systems)

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11 pages, 454 KiB

Open AccessCommunication

Positivity and Stability of Fractional-Order Coupled Neural Network with Time-Varying Delays

by Jiyun Gong, Hongling Qiu and Jun Shen

Electronics 2023, 12(23), 4782; https://doi.org/10.3390/electronics12234782 - 26 Nov 2023

Viewed by 1197

Abstract

This brief paper analyzes the positivity and asymptotic stability of incommensurate fractional-order coupled neural networks (FOCNNs) with time-varying delays. Under a reasonable assumption about the activation functions of neurons, a sufficient and necessary condition is proposed to guarantee that FOCNNs are positive systems. [...] Read more.

This brief paper analyzes the positivity and asymptotic stability of incommensurate fractional-order coupled neural networks (FOCNNs) with time-varying delays. Under a reasonable assumption about the activation functions of neurons, a sufficient and necessary condition is proposed to guarantee that FOCNNs are positive systems. Furthermore, the sufficient and necessary condition ensuring the asymptotic stability of FOCNNs is also given via introducing a linear auxiliary system. Finally, a simulation experiment was carried out to justify the effectiveness of the derived results. Full article

(This article belongs to the Special Issue Resilient Estimation, Control, Optimization, and Their Applications in Positive Networked Systems)

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