Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
3.6
5.4
Journals
Fractal Fract
Special Issues
Fractional-Order Chaotic System: Control and Synchronization
share announcement

Fractional-Order Chaotic System: Control and Synchronization

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

Deadline for manuscript submissions: closed (30 January 2023) | Viewed by 27053

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: drone-based remote sensing; 3D photogrammetry; optimal sensing in cyber-physical systems; distributed collaborative sensing in distributed parameter systems; mobile sensor networks; virtual metrology
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 increasing attention. Fractional-order models have been proven to be an effective tool for the description of memory and the hereditary properties of various materials and processes. Fractional-order nonlinear dynamic systems have displayed many complex dynamic behaviors such as chaos, bifurcation, attractor, and multistability. Fractional-order chaotic systems have been 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 the synchronization control problem plays 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
Prof. 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

Related Special Issue

Published Papers (18 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

16 pages, 1523 KiB  
Article
Control and Synchronization of a Novel Realizable Nonlinear Chaotic System
by Mohammed Almuzaini and Abdullah Alzahrani
Fractal Fract. 2023, 7(3), 253; https://doi.org/10.3390/fractalfract7030253 - 10 Mar 2023
Cited by 4 | Viewed by 1115
Abstract
The study proposes a novel chaotic system with a cubic non-linear term. Different system characteristics are investigated including equilibria, stability, invariance, dissipation, Lyapunov dimension, and Lyapunov exponents. Also, the electronic circuit and Signal flow graph of the system are carried out to show [...] Read more.
The study proposes a novel chaotic system with a cubic non-linear term. Different system characteristics are investigated including equilibria, stability, invariance, dissipation, Lyapunov dimension, and Lyapunov exponents. Also, the electronic circuit and Signal flow graph of the system are carried out to show the applicability of the chaotic system. Lyapunov stability theorem converts the system’s chaotic behavior to unstable trivial fixed point. The study also focuses on demonstrating complete synchronization between two similar novel chaotic systems. According to Lyapunov stability theorem, simple application in secure communication was developed by employing the chaos synchronization results. Numerical simulations for the systems are performed for establishing the synchronization strategy effectiveness and proposed control. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

16 pages, 2864 KiB  
Article
Fixed-Time Sliding Mode Synchronization of Uncertain Fractional-Order Hyperchaotic Systems by Using a Novel Non-Singleton-Interval Type-2 Probabilistic Fuzzy Neural Network
by Ke-Yong Shao, Ao Feng and Ting-Ting Wang
Fractal Fract. 2023, 7(3), 247; https://doi.org/10.3390/fractalfract7030247 - 09 Mar 2023
Cited by 4 | Viewed by 1015
Abstract
In this study, we proposed a sliding mode control method based on fixed-time sliding mode surface for the synchronization of uncertain fractional-order hyperchaotic systems. In addition, we proposed a novel self-evolving non-singleton-interval type-2 probabilistic fuzzy neural network (SENSIT2PFNN) to estimate the uncertain dynamics [...] Read more.
In this study, we proposed a sliding mode control method based on fixed-time sliding mode surface for the synchronization of uncertain fractional-order hyperchaotic systems. In addition, we proposed a novel self-evolving non-singleton-interval type-2 probabilistic fuzzy neural network (SENSIT2PFNN) to estimate the uncertain dynamics of the system. Moreover, an adaptive compensator was designed to eliminate the influences of random uncertainty and fuzzy uncertainty, thereby yielding an asymptotically stable controlled system. Furthermore, an adaptive law was introduced to optimize the consequence parameters of SENSIT2PFNN. The membership layer and rule base of SENSIT2PFNN were optimized using the self-evolving algorithm and whale optimization algorithm, respectively. The simulation results verified the effectiveness of the proposed methods for the synchronization of uncertain fractional-order hyperchaotic systems. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

19 pages, 4484 KiB  
Article
On the Development of a Data-Driven-Based Fractional-Order Controller for Unmanned Aerial Vehicles
by Fawaz W. Alsaade, Hadi Jahanshahi, Qijia Yao, Mohammed S. Al-zahrani and Ali S. Alzahrani
Fractal Fract. 2023, 7(3), 236; https://doi.org/10.3390/fractalfract7030236 - 07 Mar 2023
Cited by 3 | Viewed by 1363
Abstract
Proper control is necessary for ensuring that UAVs successfully navigate their surroundings and accomplish their intended tasks. Undoubtedly, a perfect control technique can significantly improve the performance and reliability of UAVs in a wide range of applications. Motivated by this, in the current [...] Read more.
Proper control is necessary for ensuring that UAVs successfully navigate their surroundings and accomplish their intended tasks. Undoubtedly, a perfect control technique can significantly improve the performance and reliability of UAVs in a wide range of applications. Motivated by this, in the current paper, a new data-driven-based fractional-order control technique is proposed to address this issue and enable UAVs to track desired trajectories despite the presence of external disturbances and uncertainties. The control approach combines a deep neural network with a robust fractional-order controller to estimate uncertainties and minimize the impact of unknown disturbances. The design procedure for the controller is outlined in the paper. To evaluate the proposed technique, numerical simulations are performed for two different desired paths. The results show that the control method performs well in the presence of dynamic uncertainties and control input constraints, making it a promising approach for enabling UAVs to track desired trajectories in challenging environments. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

16 pages, 3821 KiB  
Article
Fractal Dimension and Nuclear Magnetic Resonance Characteristics of Surfactants for Coal Gas Desorption
by Lingling Yang, Feng Cai and Yuan Yuan
Fractal Fract. 2023, 7(3), 217; https://doi.org/10.3390/fractalfract7030217 - 26 Feb 2023
Cited by 2 | Viewed by 1243
Abstract
In this paper, the fractal dimension of coal surfaces was calculated using the fractal theory, and the influence of different surfactants on the fractal dimension of coal surface was studied. Six kinds of sustainable and traditional surfactants used in coal gas desorption processes [...] Read more.
In this paper, the fractal dimension of coal surfaces was calculated using the fractal theory, and the influence of different surfactants on the fractal dimension of coal surface was studied. Six kinds of sustainable and traditional surfactants used in coal gas desorption processes were compared and analyzed. We used mineral gas fertilizer coal from Huainan Liuzhuang, China, as the research object and studied sodium dodecyl benzene sulfonate (SDBS), cohol diethanolamide (CDEA), nonylphenol polyoxyethylene ether (NP-10), tea saponin, sucrose ester and rhamnolipid surfactants used to modify coal. The methane adsorption capacity of coal before and after surfactant modification was measured by low field nuclear magnetic resonance methane adsorption, and Langmuir volume and Langmuir pressure were obtained according to the Langmuir fitting equation. The results showed that from the perspective of fractal dimensions, the best surfactant in the context of sustainability (in order) is rhamnolipid, CDEA, tea saponin, sucrose ester. According to the two important parameters VL and PL in the Langmuir equation, the coal sample modified by sucrose ester had the strongest gas adsorption capacity. According to the numerical analysis of the surface fractal dimension DS of the coal modified by surfactants, the soluble organic matter in the raw coal samples dissolved, and the micropore morphology evolved to become mesoporous or macroporous, which is more favorable for desorption. The coal samples modified by rhamnolipid and SDBS had the strongest gas desorption ability. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

14 pages, 1660 KiB  
Article
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 24 | Viewed by 1443
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)
►▼ Show Figures

Figure 1

15 pages, 1363 KiB  
Article
Sliding Mode Control for a Class of Nonlinear Fractional Order Systems with a Fractional Fixed-Time Reaching Law
by Yuquan Chen, Bing Wang, Yangquan Chen and Yong Wang
Fractal Fract. 2022, 6(11), 678; https://doi.org/10.3390/fractalfract6110678 - 16 Nov 2022
Cited by 5 | Viewed by 1249
Abstract
In this paper, the sliding-mode control method was used to control a class of general nonlinear fractional-order systems which covers a wide class of chaotic systems. A novel sliding manifold with an additional nonlinear part which achieved better control performance was designed. Furthermore, [...] Read more.
In this paper, the sliding-mode control method was used to control a class of general nonlinear fractional-order systems which covers a wide class of chaotic systems. A novel sliding manifold with an additional nonlinear part which achieved better control performance was designed. Furthermore, a novel fixed-time reaching law with a fractional adaptive gain is proposed, where the reaching time to the sliding manifold is determined by the first positive zero of a Mittag–Leffler function and is independent of initial conditions. We have provided some instructions on tuning the parameters of the proposed reaching law to avoid exacerbating the chattering phenomenon. Finally, simulation examples are presented to validate all results. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

15 pages, 8151 KiB  
Article
Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model
by Min Huang, Shichang Lu, Stanford Shateyi and Hassan Saberi-Nik
Fractal Fract. 2022, 6(11), 630; https://doi.org/10.3390/fractalfract6110630 - 28 Oct 2022
Cited by 4 | Viewed by 1042
Abstract
In this paper, the global attractive set (GAS) and positive invariant set (PIS) of the five-dimensional Lorenz model with the fractional order derivative are studied. Using the Mittag-Leffler function and Lyapunov function method, the ultimate boundedness of the proposed system are estimated. An [...] Read more.
In this paper, the global attractive set (GAS) and positive invariant set (PIS) of the five-dimensional Lorenz model with the fractional order derivative are studied. Using the Mittag-Leffler function and Lyapunov function method, the ultimate boundedness of the proposed system are estimated. An effective control strategy is also designed to achieve the finite time stability of this fractional chaotic system. The corresponding boundedness and control scheme are numerically verified to show the effectiveness of the theoretical analysis. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

17 pages, 616 KiB  
Article
A Novel Nonlinear Pseudorandom Sequence Generator for the Fractal Function
by Yelai Feng, Huaixi Wang, Chao Chang, Hongyi Lu, Fang Yang and Chenyang Wang
Fractal Fract. 2022, 6(10), 589; https://doi.org/10.3390/fractalfract6100589 - 13 Oct 2022
Cited by 3 | Viewed by 1215
Abstract
A pseudorandom sequence is a repeatable sequence with random statistical properties that is widely used in communication encryption, authentication and channel coding. The pseudorandom sequence generator based on the linear feedback shift register has the problem of a fixed sequence, which is easily [...] Read more.
A pseudorandom sequence is a repeatable sequence with random statistical properties that is widely used in communication encryption, authentication and channel coding. The pseudorandom sequence generator based on the linear feedback shift register has the problem of a fixed sequence, which is easily tracked. Existing methods use the secret linear feedback shift register (LFSR) and built-in multiple LFSRs and is difficult to prevent cracking based on the hardware analysis. Since the plaintext depends on a specific language to be generated, using pseudo-random sequence encryption, it faces the problem that the encryptor cannot hide the characteristics of the plaintext data. Fractal functions have the following properties: chaotic, unpredictable and random. We propose a novel pseudorandom sequence generator based on the nonlinear chaotic systems, which is constructed by the fractal function. Furthermore, we design a data processing matrix to hide the data characteristics of the sequence and enhance the randomness. In the experiment, the pseudo-random sequences generator passed 16 rigorous test items from the National Institute of Standards and Technology (NIST), which means that the nonlinear pseudorandom sequence generator for the fractal function is effective and efficient. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

16 pages, 753 KiB  
Article
Synchronizability of Multilayer Directed Dutch Windmill Networks
by Yongqing Wu and Xiao Zhang
Fractal Fract. 2022, 6(10), 537; https://doi.org/10.3390/fractalfract6100537 - 23 Sep 2022
Cited by 2 | Viewed by 1023
Abstract
This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and [...] Read more.
This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

21 pages, 5332 KiB  
Article
Fractional-Order Financial System and Fixed-Time Synchronization
by Yingjin He, Jun Peng and Song Zheng
Fractal Fract. 2022, 6(9), 507; https://doi.org/10.3390/fractalfract6090507 - 10 Sep 2022
Cited by 10 | Viewed by 1672
Abstract
This study is concerned with the dynamic investigation and fixed-time synchronization of a fractional-order financial system with the Caputo derivative. The rich dynamic behaviors of the fractional-order financial system with variations of fractional orders and parameters are discussed analytically and numerically. Through using [...] Read more.
This study is concerned with the dynamic investigation and fixed-time synchronization of a fractional-order financial system with the Caputo derivative. The rich dynamic behaviors of the fractional-order financial system with variations of fractional orders and parameters are discussed analytically and numerically. Through using phase portraits, bifurcation diagrams, maximum Lyapunov exponent diagrams, 0–1 testing and time series, it is found that chaos exists in the proposed fractional-order financial system. Additionally, a complexity analysis is carried out utilizing approximation entropy SE and C0 complexity to detect whether chaos exists. Furthermore, a synchronization controller and an adaptive parameter update law are designed to synchronize two fractional-order chaotic financial systems and identify the unknown parameters in fixed time simultaneously. The estimate of the setting time of synchronization depends on the parameters of the designed controller and adaptive parameter update law, rather than on the initial conditions. Numerical simulations show the effectiveness of the theoretical results obtained. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

19 pages, 6706 KiB  
Article
A Variational Level Set Image Segmentation Method via Fractional Differentiation
by Xiangguo Liu, Guojun Liu, Yazhen Wang, Gengsheng Li, Rui Zhang and Weicai Peng
Fractal Fract. 2022, 6(9), 462; https://doi.org/10.3390/fractalfract6090462 - 23 Aug 2022
Cited by 2 | Viewed by 1354
Abstract
To solve the issues with conventional level set segmentation algorithms, which are sensitive to the initial contours and less noise-resistant, a segmentation model based on the coupling of texture information and structural information is developed. In this model, a rotation invariant mask produced [...] Read more.
To solve the issues with conventional level set segmentation algorithms, which are sensitive to the initial contours and less noise-resistant, a segmentation model based on the coupling of texture information and structural information is developed. In this model, a rotation invariant mask produced by fractional-order differentiation is used to first describe the image’s global information. Then, the power function of the energy generalization function is solved by applying factorization theory, and for each pixel of the image, not only its information but also its surrounding pixel information is taken into account and integrated into the energy generalization function via weight scaling. At the same time, the L2 norm of the fractional-order image and the difference from the fitted image are used to generate the energy generalization function of the model. The final results of this study demonstrate that the proposed model achieved a better segmentation performance than the current active contour models in terms of robustness to Gaussian noise and pretzel noise, as well as the segmentation accuracy and algorithm running time. These results were obtained in synthetic images, real images, and natural images. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

16 pages, 363 KiB  
Article
Stability Analysis on Nabla Discrete Distributed-Order Dynamical System
by Xiang Wu, Xujun Yang, Qiankun Song and Xiaofeng Chen
Fractal Fract. 2022, 6(8), 429; https://doi.org/10.3390/fractalfract6080429 - 07 Aug 2022
Cited by 3 | Viewed by 1164
Abstract
This paper addresses the problems of the stability of a nabla discrete distributed-order dynamical system (NDDS). Firstly, based on a proposed generalized definition of discrete integral, some related definitions of nabla discrete distributed-order calculus are given. Then, several useful inequalities in sense of [...] Read more.
This paper addresses the problems of the stability of a nabla discrete distributed-order dynamical system (NDDS). Firstly, based on a proposed generalized definition of discrete integral, some related definitions of nabla discrete distributed-order calculus are given. Then, several useful inequalities in sense of nabla discrete fractional-order difference are extended to distributed-order cases. Meanwhile, on basis of the proposed inequalities and Lyapunov direct method, some sufficient conditions guaranteeing the asymptotic stability of the origin of NDDS are established under both the Caputo and Riemann–Liouville sense. Finally, some designed simulation examples are given to validate the correctness and practicability of the obtained results. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

12 pages, 1231 KiB  
Article
Nonlinear Control and Circuit Implementation in Coupled Nonidentical Fractional-Order Chaotic Systems
by Yanbin Zhang, Ping Lin and Weigang Sun
Fractal Fract. 2022, 6(8), 428; https://doi.org/10.3390/fractalfract6080428 - 05 Aug 2022
Cited by 5 | Viewed by 1311
Abstract
In this paper, circuit implementation and anti-synchronization are studied in coupled nonidentical fractional-order chaotic systems where a fractance module is introduced to approximate the fractional derivative. Based on the open-plus-closed-loop control, a nonlinear coupling strategy is designed to realize the anti-synchronization in the [...] Read more.
In this paper, circuit implementation and anti-synchronization are studied in coupled nonidentical fractional-order chaotic systems where a fractance module is introduced to approximate the fractional derivative. Based on the open-plus-closed-loop control, a nonlinear coupling strategy is designed to realize the anti-synchronization in the fractional-order Rucklidge chaotic systems and proved by the stability theory of fractional-order differential equations. In addition, using the frequency-domain approximation and circuit theory in the Laplace domain, the corresponding electronic circuit experiments are performed for both uncoupled and coupled fractional-order Rucklidge systems. Finally, our circuit implementation including the fractance module may provide an effective method for generating chaotic encrypted signals, which could be applied to secure communication and data encryption. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

15 pages, 4713 KiB  
Article
Dynamic Analysis and Audio Encryption Application in IoT of a Multi-Scroll Fractional-Order Memristive Hopfield Neural Network
by Fei Yu, Qiulin Yu, Huifeng Chen, Xinxin Kong, Abdulmajeed Abdullah Mohammed Mokbel, Shuo Cai and Sichun Du
Fractal Fract. 2022, 6(7), 370; https://doi.org/10.3390/fractalfract6070370 - 30 Jun 2022
Cited by 31 | Viewed by 1790
Abstract
Fractional-order chaotic systems are widely used in the field of encryption because of its initial value sensitivity and historical memory. In this paper, the fractional-order definition of Caputo is introduced based on a nonideal flux-controlled memristive Hopfield neural network model, when changing the [...] Read more.
Fractional-order chaotic systems are widely used in the field of encryption because of its initial value sensitivity and historical memory. In this paper, the fractional-order definition of Caputo is introduced based on a nonideal flux-controlled memristive Hopfield neural network model, when changing the parameters of the fractional-order memristive Hopfield neural network (FMHNN) can generate a different amount of multi-scroll attractors. Some dynamical behaviors are investigated by numerical simulation, especially analyzed coexistence and bifurcation under different orders and different coupling strengths. The results show that the chaotic system of FMHNN has abundant dynamic behaviors. In addition, a chaotic audio encryption scheme under a Message Queueing Telemetry Transport (MQTT) protocol is proposed and implemented by Raspberry Pi; the audio encryption system based on FMHNN has a broad future in intelligent home and other IoT applications. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

13 pages, 1460 KiB  
Article
Finite-Time Projective Synchronization and Parameter Identification of Fractional-Order Complex Networks with Unknown External Disturbances
by Shuguo Wang, Song Zheng and Linxiang Cui
Fractal Fract. 2022, 6(6), 298; https://doi.org/10.3390/fractalfract6060298 - 29 May 2022
Cited by 3 | Viewed by 1472
Abstract
This paper is devoted to exploring the finite-time projective synchronization (FTPS) of fractional-order complex dynamical networks (FOCDNs) with unknown parameters and external disturbances. Based on the stability theory of fractional-order differential systems, synchronization criteria between drive-response networks were obtained and both the uncertain [...] Read more.
This paper is devoted to exploring the finite-time projective synchronization (FTPS) of fractional-order complex dynamical networks (FOCDNs) with unknown parameters and external disturbances. Based on the stability theory of fractional-order differential systems, synchronization criteria between drive-response networks were obtained and both the uncertain parameters and external disturbances were identified or conquered simultaneously. Moreover, the upper limit of the settling-time function was obtained. Finally, a numerical example was given to verify the effectiveness of the results. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

9 pages, 1978 KiB  
Article
The Effect of Learning Rate on Fractal Image Coding Using Artificial Neural Networks
by Rashad A. Al-Jawfi
Fractal Fract. 2022, 6(5), 280; https://doi.org/10.3390/fractalfract6050280 - 23 May 2022
Cited by 4 | Viewed by 1473
Abstract
The amount by which the artificial neural network weights are updated during the training process is called the learning rate. More precisely, the learning rate is an adjustable parameter used in training neural networks in which small values, often in the interval [0, [...] Read more.
The amount by which the artificial neural network weights are updated during the training process is called the learning rate. More precisely, the learning rate is an adjustable parameter used in training neural networks in which small values, often in the interval [0, 1], are handled. The learning rate determines how quickly the model updates its weights to adapt to the problem. Smaller learning rates require more training periods due to small changes to the weights per refresh cycle, while larger learning rates lead to faster changes and require fewer training periods. In this paper, the effect of changing the learning rate value in the artificial neural network designed to solve the inverse problem of fractals was studied. Some results were obtained showing the impact of this change, whether when using large values of the learning rate or small values based on the type of fractal shape required to identify the recursive functions that generate it. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

12 pages, 1817 KiB  
Article
Synchronization of Incommensurate Fractional-Order Chaotic Systems Based on Linear Feedback Control
by Fei Qi, Jianfeng Qu, Yi Chai, Liping Chen and António M. Lopes
Fractal Fract. 2022, 6(4), 221; https://doi.org/10.3390/fractalfract6040221 - 14 Apr 2022
Cited by 16 | Viewed by 2033
Abstract
The dynamics and synchronization of fractional-order (FO) chaotic systems have received much attention in recent years. However, the research are focused mostly on FO commensurate systems. This paper addresses the synchronization of incommensurate FO (IFO) chaotic systems. By employing the comparison principle for [...] Read more.
The dynamics and synchronization of fractional-order (FO) chaotic systems have received much attention in recent years. However, the research are focused mostly on FO commensurate systems. This paper addresses the synchronization of incommensurate FO (IFO) chaotic systems. By employing the comparison principle for FO systems with multi-order and the linear feedback control method, a sufficient condition for ensuring the synchronization of IFO chaotic systems is developed in terms of linear matrix inequalities (LMIs). Such synchronization condition relies just on the system parameters, and is easily verify and implemented. Two typical FO chaotic systems, named the IFO Genesio-Tesi system and Hopfied neural networks are selected to demonstrate the effectiveness and feasibility of the proposed method. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

8 pages, 1214 KiB  
Article
Fractals Generated via Numerical Iteration Method
by Wadia Faid Hassan Al-shameri and Mohamed El Sayed
Fractal Fract. 2022, 6(4), 196; https://doi.org/10.3390/fractalfract6040196 - 31 Mar 2022
Viewed by 2047
Abstract
In this research article, a modified algorithm for the generation of a fractal pattern resulting from the iteration of an algebraic function using the numerical iteration method is presented. This fractal pattern shows the dynamical behavior of the numerical iterations. A nonstandard convergence [...] Read more.
In this research article, a modified algorithm for the generation of a fractal pattern resulting from the iteration of an algebraic function using the numerical iteration method is presented. This fractal pattern shows the dynamical behavior of the numerical iterations. A nonstandard convergence test of the displayable fractal pattern was applied. Full article
(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-18
Back to TopTop