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Chaotic Systems and Their Applications, 2nd Edition
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Chaotic Systems and Their Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: 31 May 2025 | Viewed by 4437

Special Issue Editors

Prof. Dr. Chunhua Wang

E-Mail Website
Guest Editor
College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China
Interests: fractional-order systems; chaotic circuits; memristor-based chaos; neural networks; neural networks and brain-inspired computing; chaotic image encryption
Special Issues, Collections and Topics in MDPI journals
Dr. Hairong Lin

E-Mail Website
Guest Editor
School of Electronic Information, Central South University, Changsha 410083, China
Interests: memristor; memristive neural networks; chaotic system and circuit; image encryption; neuromorphic engineering
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We invite you to submit your latest theoretical and applied research in the field of chaos to the Special Issue entitled “Chaotic Systems and Their Applications, 2nd Edition”. The aim of the Special Issue is to promote the development and application of chaos theory in the areas of mathematics, physics, computer, information, economics, engineering, artificial intelligence, and so on. Any innovative study of theoretical and applied developments in chaos is highly welcome. In addition, research papers that focus on new chaos phenomena, constructing new chaotic systems, and proposing new chaos applications are also welcome. We are looking forward to receiving research manuscripts on chaos, chaotic systems, memristor, neural networks, bifurcation, nonlinear dynamics, synchronous control, equilibrium points, stability, nonlinear circuits, complex systems, fractional-order systems, and chaos-based applications. Note that appications are not limited to the topics mentioned.

Prof. Dr. Chunhua Wang
Dr. Hairong Lin
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. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Published Papers (5 papers)

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Research

22 pages, 18396 KiB  
Article
A Dynamic Hill Cipher with Arnold Scrambling Technique for Medical Images Encryption
by Yuzhou Xi, Yu Ning, Jie Jin and Fei Yu
Mathematics 2024, 12(24), 3948; https://doi.org/10.3390/math12243948 - 15 Dec 2024
Viewed by 476
Abstract
Cryptography is one of the most important branches of information security. Cryptography ensures secure communication and data privacy, and it has been increasingly applied in healthcare and related areas. As a significant cryptographic method, the Hill cipher has attracted significant attention from experts [...] Read more.
Cryptography is one of the most important branches of information security. Cryptography ensures secure communication and data privacy, and it has been increasingly applied in healthcare and related areas. As a significant cryptographic method, the Hill cipher has attracted significant attention from experts and scholars. To enhance the security of the traditional Hill cipher (THC) and expand its application in medical image encryption, a novel dynamic Hill cipher with Arnold scrambling technique (DHCAST) is proposed in this work. Unlike the THC, the proposed DHCAST uses a time-varying matrix as its secret key, which greatly increases the security of the THC, and the new DHCAST is successfully applied in medical images encryption. In addition, the new DHCAST method employs the Zeroing Neural Network (ZNN) in its decryption to find the time-varying inversion key matrix (TVIKM). In order to enhance the efficiency of the ZNN for solving the TVIKM, a new fuzzy zeroing neural network (NFZNN) model is constructed, and the convergence and robustness of the NFZNN model are validated by both theoretical analysis and experiment results. Simulation experiments show that the convergence time of the NFZNN model is about 0.05 s, while the convergence time of the traditional Zeroing Neural Network (TZNN) model is about 2 s, which means that the convergence speed of the NFZNN model is about 400 times that of the TZNN model. Moreover, the Peak Signal to Noise Ratio (PSNR) and Number of Pixel Change Rate (NPCR) of the proposed DHCAST algorithm reach 9.51 and 99.74%, respectively, which effectively validates its excellent encryption quality and attack prevention ability. Full article
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)
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10 pages, 6834 KiB  
Article
A Rectified Linear Unit-Based Memristor-Enhanced Morris–Lecar Neuron Model
by Othman Abdullah Almatroud, Viet-Thanh Pham and Karthikeyan Rajagopal
Mathematics 2024, 12(19), 2970; https://doi.org/10.3390/math12192970 - 25 Sep 2024
Viewed by 588
Abstract
This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior [...] Read more.
This paper introduces a modified Morris–Lecar neuron model that incorporates a memristor with a ReLU-based activation function. The impact of the memristor on the dynamics of the ML neuron model is analyzed using bifurcation diagrams and Lyapunov exponents. The findings reveal chaotic behavior within specific parameter ranges, while increased magnetic strength tends to maintain periodic dynamics. The emergence of various firing patterns, including periodic and chaotic spiking as well as square-wave and triangle-wave bursting is also evident. The modified model also demonstrates multistability across certain parameter ranges. Additionally, the dynamics of a network of these modified models are explored. This study shows that synchronization depends on the strength of the magnetic flux, with synchronization occurring at lower coupling strengths as the magnetic flux increases. The network patterns also reveal the formation of different chimera states, such as traveling and non-stationary chimera states. Full article
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)
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16 pages, 4315 KiB  
Article
Synchronization of Bidirectionally Coupled Fractional-Order Chaotic Systems with Unknown Time-Varying Parameter Disturbance in Different Dimensions
by Chunli Zhang, Yangjie Gao, Junliang Yao and Fucai Qian
Mathematics 2024, 12(17), 2775; https://doi.org/10.3390/math12172775 - 8 Sep 2024
Viewed by 726
Abstract
In this article, the synchronization of bidirectionally coupled fractional-order chaotic systems with unknown time-varying parameter disturbance in different dimensions is investigated. The scale matrices are designed to address the problem of the synchronization for fractional-order chaotic systems across two different dimensions. Congelation of [...] Read more.
In this article, the synchronization of bidirectionally coupled fractional-order chaotic systems with unknown time-varying parameter disturbance in different dimensions is investigated. The scale matrices are designed to address the problem of the synchronization for fractional-order chaotic systems across two different dimensions. Congelation of variables is used to deal with the unknown time-varying parameter disturbance. Based on Lyapunov’s stability theorem, the synchronization controllers in different dimensions are obtained. At the same time, adaptive laws of the unknown disturbance can be designed. Benefiting from the proposed methods, we verify all the synchronization errors can converge to zero as time approaches infinity, regardless of whether in n-D or m-D synchronization, simultaneously ensuring that both control and estimation signals are bounded. Finally, simulation studies based on fractional-order financial systems are carried out to validate the effectiveness of the proposed synchronization method. Full article
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)
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22 pages, 2496 KiB  
Article
Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors
by Fei Yu, Shuai Xu, Yue Lin, Ting He, Chaoran Wu and Hairong Lin
Mathematics 2024, 12(14), 2227; https://doi.org/10.3390/math12142227 - 17 Jul 2024
Cited by 2 | Viewed by 868
Abstract
The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system’s [...] Read more.
The design of chaotic systems with complex dynamic behaviors has always been a key aspect of chaos theory in engineering applications. This study introduces a novel fractional-order system characterized by hidden dynamics, hyperchaotic behavior, and multi-scroll attractors. By employing fractional calculus, the system’s order is extended beyond integer values, providing a richer dynamic behavior. The system’s hidden dynamics are revealed through detailed numerical simulations and theoretical analysis, demonstrating complex attractors and bifurcations. The hyperchaotic nature of the system is verified through Lyapunov exponents and phase portraits, showing multiple positive exponents that indicate a higher degree of unpredictability and complexity. Additionally, the system’s multi-scroll attractors are analyzed, showcasing their potential for secure communication and encryption applications. The fractional-order approach enhances the system’s flexibility and adaptability, making it suitable for a wide range of practical uses, including secure data transmission, image encryption, and complex signal processing. Finally, based on the proposed fractional-order system, we designed a simple and efficient medical image encryption scheme and analyzed its security performance. Experimental results validate the theoretical findings, confirming the system’s robustness and effectiveness in generating complex chaotic behaviors. Full article
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)
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16 pages, 2990 KiB  
Article
On Chaos and Complexity Analysis for a New Sine-Based Memristor Map with Commensurate and Incommensurate Fractional Orders
by Tareq Hamadneh, Abderrahmane Abbes, Hassan Al-Tarawneh, Gharib Mousa Gharib, Wael Mahmoud Mohammad Salameh, Maha S. Al Soudi and Adel Ouannas
Mathematics 2023, 11(20), 4308; https://doi.org/10.3390/math11204308 - 16 Oct 2023
Cited by 6 | Viewed by 1120
Abstract
In this study, we expand a 2D sine map via adding the discrete memristor to introduce a new 3D fractional-order sine-based memristor map. Under commensurate and incommensurate orders, we conduct an extensive exploration and analysis of its nonlinear dynamic behaviors, employing diverse numerical [...] Read more.
In this study, we expand a 2D sine map via adding the discrete memristor to introduce a new 3D fractional-order sine-based memristor map. Under commensurate and incommensurate orders, we conduct an extensive exploration and analysis of its nonlinear dynamic behaviors, employing diverse numerical techniques, such as analyzing Lyapunov exponents, visualizing phase portraits, and plotting bifurcation diagrams. The results emphasize the sine-based memristor map’s sensitivity to fractional-order parameters, resulting in the emergence of distinct and diverse dynamic patterns. In addition, we employ the sample entropy (SampEn) method and C0 complexity to quantitatively measure complexity, and we also utilize the 0–1 test to validate the presence of chaos in the proposed fractional-order sine-based memristor map. Finally, MATLAB simulations are be executed to confirm the results provided. Full article
(This article belongs to the Special Issue Chaotic Systems and Their Applications, 2nd Edition)
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