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Fractal Fract
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Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition
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Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 31 January 2025 | Viewed by 9230

Special Issue Editors

Prof. Dr. Song Zheng

E-Mail Website
Guest Editor
School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou 310018, China
Interests: fractional calculus; complex systems; nonlinear dynamics; synchronization; control; simulation
Special Issues, Collections and Topics in MDPI journals
Dr. Emad E. Mahmoud

E-Mail Website
Guest Editor
Faculty of Science, Taif University, Taif 21944, Saudi Arabia
Interests: fractional calculus; complex systems; nonlinear dynamics; synchronization; control; simulation
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yangquan Chen

grade E-Mail Website
Guest Editor
Department of Mechanical Engineering (ME), University of California, Merced, CA 95343, USA
Interests: mechatronics for sustainability; cognitive process control; small multi-UAV-based cooperative multi-spectral “personal remote sensing”; applied fractional calculus in controls, modeling, and complex signal processing; distributed measurements; control of distributed parameter systems with mobile actuators and sensor networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional-order calculus can be traced back to the work of Leibniz and Hospital in 1695. During the last twenty years, fractional calculus has attracted an increasing amount of attention. Fractional-order models have been proven to be an effective tool for the description of memory and hereditary properties of various materials and processes. The fractional-order nonlinear dynamic systems have displayed many complex dynamic behaviors such as chaos, bifurcation, attractor, and multistability. The fractional-order chaotic systems are extensively studied due to their potential applications in mathematics, biology, physics, finance, engineering, and other fields. As a collective behavior, the problem of synchronization and its control in fractional-order chaotic systems generally exists in many actual processes. Therefore, the analysis and synthesis of synchronization control problems play an important role in many practical systems.

The aim of this Special Issue is to explore recent trends and developments in the analysis and synthesis of synchronization control in fractional-order chaotic systems. Contributions can address all types of fractional-order chaotic systems and different synchronization control methods, as well as their practical applications. Review articles focused on a specific system behavior or specific synchronization methods are also welcome. Potential topics include, but are not limited to:

  • Synchronization;
  • Analysis and design of fractional-order controls;
  • Identification and optimization of fractional-order systems;
  • Event-based synchronization analysis of fractional-order systems;
  • Fractional-order chaos-based cryptography applications;
  • Fractional-order financial models and systems;
  • Fractional-order economic models and systems;
  • Intermittent control;
  • Impulsive control;
  • Complex networks;
  • Neural networks;
  • Consensus.

Prof. Dr. Song Zheng
Dr. Emad E. Mahmoud
Prof. Dr. Yangquan Chen
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional-order
  • complex systems
  • chaos
  • synchronization
  • control
  • stabilization
  • consensus
  • identification
  • modelling

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Related Special Issue

Published Papers (7 papers)

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Research

23 pages, 5769 KiB  
Article
Observer-Based Prescribed Performance Adaptive Neural Network Tracking Control for Fractional-Order Nonlinear Multiple-Input Multiple-Output Systems Under Asymmetric Full-State Constraints
by Shuai Lu, Tao Yu and Changhui Wang
Fractal Fract. 2024, 8(11), 662; https://doi.org/10.3390/fractalfract8110662 - 13 Nov 2024
Viewed by 416
Abstract
In this work, the practical prescribed performance tracking issue for a class of fractional-order nonlinear multiple-input multiple-output (MIMO) systems with asymmetric full-state constraints and unmeasurable system states is investigated. A neural network (NN) nonlinear state observer is developed to estimate the unmeasurable states. [...] Read more.
In this work, the practical prescribed performance tracking issue for a class of fractional-order nonlinear multiple-input multiple-output (MIMO) systems with asymmetric full-state constraints and unmeasurable system states is investigated. A neural network (NN) nonlinear state observer is developed to estimate the unmeasurable states. Furthermore, the barrier Lyapunov functions with the settling time regulator are employed to deal with the asymmetric full-state constraint from the fractional-order MIMO system. On this ground, the prescribed performance adaptive tracking control approach is designed, assuring that all system states do not exceed the prescribed boundaries, and the tracking errors converge to the predetermined compact sets within a predefined time. Finally, two simulation examples are presented to show the effectiveness and practicability of the proposed control scheme. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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22 pages, 2746 KiB  
Article
Robust Design of Two-Level Non-Integer SMC Based on Deep Soft Actor-Critic for Synchronization of Chaotic Fractional Order Memristive Neural Networks
by Majid Roohi, Saeed Mirzajani, Ahmad Reza Haghighi and Andreas Basse-O’Connor
Fractal Fract. 2024, 8(9), 548; https://doi.org/10.3390/fractalfract8090548 - 20 Sep 2024
Viewed by 569
Abstract
In this study, a model-free  PIφ-sliding mode control ( PIφ-SMC) methodology is proposed to synchronize a specific class of chaotic fractional-order memristive neural network systems (FOMNNSs) with delays and input saturation. The fractional-order Lyapunov stability theory is [...] Read more.
In this study, a model-free  PIφ-sliding mode control ( PIφ-SMC) methodology is proposed to synchronize a specific class of chaotic fractional-order memristive neural network systems (FOMNNSs) with delays and input saturation. The fractional-order Lyapunov stability theory is used to design a two-level  PIφ-SMC which can effectively manage the inherent chaotic behavior of delayed FOMNNSs and achieve finite-time synchronization. At the outset, an initial sliding surface is introduced. Subsequently, a robust  PIφ-sliding surface is designed as a second sliding surface, based on proportional–integral (PI) rules. The finite-time asymptotic stability of both surfaces is demonstrated. The final step involves the design of a dynamic-free control law that is robust against system uncertainties, input saturations, and delays. The independence of control rules from the functions of the system is accomplished through the application of the norm-boundedness property inherent in chaotic system states. The soft actor-critic (SAC) algorithm based deep Q-Learning is utilized to optimally adjust the coefficients embedded in the two-level  PIφ-SMC controller’s structure. By maximizing a reward signal, the optimal policy is found by the deep neural network of the SAC agent. This approach ensures that the sliding motion meets the reachability condition within a finite time. The validity of the proposed protocol is subsequently demonstrated through extensive simulation results and two numerical examples. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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16 pages, 6073 KiB  
Article
Fractal Characteristics and Microstructure of Coal with Impact of Starch-Polymerized Aluminum Sulfate Fracturing Fluids
by Feng Cai, Qian Zhang and Lingling Yang
Fractal Fract. 2024, 8(4), 228; https://doi.org/10.3390/fractalfract8040228 - 15 Apr 2024
Viewed by 1331
Abstract
The degree of irregularity and complexity of the pore structure are comprehensively reflected in the fractal dimension. The porosity of coal was determined by its fractal dimension, where a larger dimension indicates a lower porosity. Fractal theory and the Frenkel–Halsey–Hill (FHH) model were [...] Read more.
The degree of irregularity and complexity of the pore structure are comprehensively reflected in the fractal dimension. The porosity of coal was determined by its fractal dimension, where a larger dimension indicates a lower porosity. Fractal theory and the Frenkel–Halsey–Hill (FHH) model were applied to explore the variation rules of concentration on functional groups and pore structure in this study. Combined with infrared spectroscopy (FTIR) and low-temperature nitrogen adsorption, a starch-polymerized aluminum sulfate composite fracturing fluid was prepared, which plays an important role in methane adsorption and permeability of coal samples. The test results showed that, compared with the original coal, the pore volume and specific surface area of each group of coal samples were reduced, the average pore diameter was initially enlarged and then declined, and fractal dimension D1 dropped by 5.4% to 15.4%, while fractal dimension D2 gained 1.2% to 7.9%. Moreover, the nitrogen adsorption of each group of coal samples was obviously lower than the original coal, and the concentration of starch-polymerized aluminum sulfate solution existed at a critical optimal concentration for the modification of the coal samples, and the nitrogen adsorption reached a minimum value of 0.6814 cm3/g at a concentration of 10%. The novel composite solution prepared by the combination of starch and flocculant in this paper enhanced the permeability of the coal seam, which is of great significance in improving the efficiency of coalbed methane mining. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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20 pages, 15205 KiB  
Article
Fractional-Order Modeling of Piezoelectric Actuators with Coupled Hysteresis and Creep Effects
by Yifan Xu, Ying Luo, Xin Luo, Yangquan Chen and Wei Liu
Fractal Fract. 2024, 8(1), 3; https://doi.org/10.3390/fractalfract8010003 - 19 Dec 2023
Cited by 3 | Viewed by 1670
Abstract
A novel fractional-order model, incorporating coupled hysteresis and creep effects, is proposed for typical piezoelectric actuators in this study. Throughout the actuation process, various nonlinear behaviors such as piezoelectric hysteresis, non-local memory, peak transition, and creep nonlinearity are accurately characterized by the model. [...] Read more.
A novel fractional-order model, incorporating coupled hysteresis and creep effects, is proposed for typical piezoelectric actuators in this study. Throughout the actuation process, various nonlinear behaviors such as piezoelectric hysteresis, non-local memory, peak transition, and creep nonlinearity are accurately characterized by the model. Offering a simpler structure and superior tracking performance compared to conventional models, the proposed fractional-order model parameters are identified using a method that integrates actuator dynamics and employs the particle swarm optimization algorithm. Experimental validation on a piezoelectric actuation platform confirms the model’s superior accuracy and simplified structure, contributing to a deeper understanding of piezoelectric actuation mechanisms and providing an efficient modeling tool for enhanced system performance. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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13 pages, 427 KiB  
Article
General Methods to Synchronize Fractional Discrete Reaction–Diffusion Systems Applied to the Glycolysis Model
by Tareq Hamadneh, Amel Hioual, Rania Saadeh, Mohamed A. Abdoon, Dalal Khalid Almutairi, Thwiba A. Khalid and Adel Ouannas
Fractal Fract. 2023, 7(11), 828; https://doi.org/10.3390/fractalfract7110828 - 20 Nov 2023
Cited by 13 | Viewed by 1563
Abstract
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this [...] Read more.
Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study’s deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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14 pages, 407 KiB  
Article
An Enhanced Numerical Iterative Method for Expanding the Attraction Basins When Computing Matrix Signs of Invertible Matrices
by Lei Shi, Malik Zaka Ullah, Hemant Kumar Nashine, Monairah Alansari and Stanford Shateyi
Fractal Fract. 2023, 7(9), 684; https://doi.org/10.3390/fractalfract7090684 - 14 Sep 2023
Cited by 1 | Viewed by 1119
Abstract
The computation of the sign function of a matrix plays a crucial role in various mathematical applications. It provides a matrix-valued mapping that determines the sign of each eigenvalue of a nonsingular matrix. In this article, we present a novel iterative algorithm designed [...] Read more.
The computation of the sign function of a matrix plays a crucial role in various mathematical applications. It provides a matrix-valued mapping that determines the sign of each eigenvalue of a nonsingular matrix. In this article, we present a novel iterative algorithm designed to efficiently calculate the sign of an invertible matrix, emphasizing the enlargement of attraction basins. The proposed solver exhibits convergence of order four, making it highly efficient for a wide range of matrices. Furthermore, the method demonstrates global convergence properties. We validate the theoretical outcomes through numerical experiments, which confirm the effectiveness and efficiency of our proposed algorithm. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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19 pages, 3774 KiB  
Article
Synchronization of Fractional-Order Delayed Neural Networks Using Dynamic-Free Adaptive Sliding Mode Control
by Majid Roohi, Chongqi Zhang, Mostafa Taheri and Andreas Basse-O’Connor
Fractal Fract. 2023, 7(9), 682; https://doi.org/10.3390/fractalfract7090682 - 13 Sep 2023
Cited by 17 | Viewed by 1651
Abstract
In this work, a dynamic-free adaptive sliding mode control (adaptive-SMC) methodology for the synchronization of a specific class of chaotic delayed fractional-order neural network systems in the presence of input saturation is proposed. By incorporating the frequency distributed model (FDM) and the fractional [...] Read more.
In this work, a dynamic-free adaptive sliding mode control (adaptive-SMC) methodology for the synchronization of a specific class of chaotic delayed fractional-order neural network systems in the presence of input saturation is proposed. By incorporating the frequency distributed model (FDM) and the fractional version of the Lyapunov stability theory, a dynamic-free adaptive SMC methodology is designed to effectively overcome the inherent chaotic behavior exhibited by the delayed FONNSs to achieve synchronization. Notably, the decoupling of the control laws from the nonlinear/linear dynamical components of the system is ensured, taking advantage of the norm-boundedness property of the states in chaotic systems. The effectiveness of the suggested adaptive-SMC method for chaos synchronization in delayed fractional-order Hopfield neural network systems is validated through numerical simulations, demonstrating its robustness and efficiency. The proposed dynamic-free adaptive-SMC approach, incorporating the FDM and fractional Lyapunov stability theorem, offers a promising solution for synchronizing chaotic delayed FONNSs with input saturation, with potential applications in various domains requiring synchronization of such systems. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization, 2nd Edition)
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