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Chaotic Systems and Their Applications

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

Deadline for manuscript submissions: closed (20 September 2023) | Viewed by 21346

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Prof. Dr. Chunhua Wang

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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

College of Computer Science and Electronic Engineering, Hunan University, Changsha, 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”. 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 finding 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. Noted that applications are not limited to the topics mentioned. In conclusion, potential topics include but are not limited to:

Chaos theory;
Chaotic systems and circuits;
Memristor-based circuits
Neural networks based on memristor
Chaotic encryption and chaotic communication
Fractional-order chaotic systems;
Chaos synchronization;
Complex systems and complex networks
Other non-linear systems and circuits
Chaos-based other applications

Prof. Dr. Chunhua Wang
Dr. Hairong Lin
Guest Editors

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Keywords

chaos
chaotic system
neural network
memristor
nonlinear dynamics
synchronous control
circuit implementation

Published Papers (12 papers)

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Research

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17 pages, 13020 KiB

Open AccessArticle

A Fractional-Order Memristive Two-Neuron-Based Hopfield Neuron Network: Dynamical Analysis and Application for Image Encryption

by Jayaraman Venkatesh, Alexander N. Pchelintsev, Anitha Karthikeyan, Fatemeh Parastesh and Sajad Jafari

Mathematics 2023, 11(21), 4470; https://doi.org/10.3390/math11214470 - 28 Oct 2023

Cited by 3 | Viewed by 794

Abstract

This paper presents a study on a memristive two-neuron-based Hopfield neural network with fractional-order derivatives. The equilibrium points of the system are identified, and their stability is analyzed. Bifurcation diagrams are obtained by varying the magnetic induction strength and the fractional-order derivative, revealing significant changes in the system dynamics. It is observed that lower fractional orders result in an extended bistability region. Also, chaos is only observed for larger magnetic strengths and fractional orders. Additionally, the application of the fractional-order model for image encryption is explored. The results demonstrate that the encryption based on the fractional model is efficient with high key sensitivity. It leads to an encrypted image with high entropy, neglectable correlation coefficient, and uniform distribution. Furthermore, the encryption system shows resistance to differential attacks, cropping attacks, and noise pollution. The Peak Signal-to-Noise Ratio (PSNR) calculations indicate that using a fractional derivative yields a higher PSNR compared to an integer derivative. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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20 pages, 7561 KiB

Open AccessArticle

A K-SVD Based Compressive Sensing Method for Visual Chaotic Image Encryption

by Zizhao Xie, Jingru Sun, Yiping Tang, Xin Tang, Oluyomi Simpson and Yichuang Sun

Mathematics 2023, 11(7), 1658; https://doi.org/10.3390/math11071658 - 30 Mar 2023

Cited by 9 | Viewed by 1595

Abstract

The visually secure image encryption scheme is an effective image encryption method, which embeds an encrypted image into a visual image to realize a secure and secret image transfer. This paper proposes a merging compression and encryption chaos image visual encryption scheme. First, a dictionary matrix D is constructed with the plain image by the K-SVD algorithm, which can encrypt the image while sparsing. Second, an improved Zeraoulia-Sprott chaotic map and logistic map are employed to generate three S-Boxes, which are used to complete scrambling, diffusion, and embedding operations. The secret keys of this scheme contain the initial value of the chaotic system and the dictionary matrix D, which significantly increases the key space, plain image correlation, and system security. Simulation shows the proposed image encryption scheme can resist most attacks and, compared with the existing scheme, the proposed scheme has a larger key space, higher plain image correlation, and better image restoration quality, improving image encryption processing efficiency and security. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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11 pages, 1265 KiB

Open AccessArticle

Building Fixed Point-Free Maps with Memristor

by Othman Abdullah Almatroud and Viet-Thanh Pham

Mathematics 2023, 11(6), 1319; https://doi.org/10.3390/math11061319 - 9 Mar 2023

Cited by 10 | Viewed by 1276

Abstract

A memristor is a two-terminal passive electronic device that exhibits memory of resistance. It is essentially a resistor with memory, hence the name “memristor”. The unique property of memristors makes them useful in a wide range of applications, such as memory storage, neuromorphic computing, reconfigurable logic circuits, and especially chaotic systems. Fixed point-free maps or maps without fixed points, which are different from normal maps due to the absence of fixed points, have been explored recently. This work proposes an approach to build fixed point-free maps by connecting a cosine term and a memristor. Four new fixed point-free maps displaying chaos are reported to illustrate this approach. The dynamics of the proposed maps are verified by iterative plots, bifurcation diagram, and Lyapunov exponents. Because such chaotic maps are highly sensitive to the initial conditions and parameter variations, they are suitable for developing novel lightweight random number generators. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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15 pages, 19037 KiB

Open AccessArticle

Dynamic Analysis and FPGA Implementation of a New, Simple 5D Memristive Hyperchaotic Sprott-C System

by Fei Yu, Wuxiong Zhang, Xiaoli Xiao, Wei Yao, Shuo Cai, Jin Zhang, Chunhua Wang and Yi Li

Mathematics 2023, 11(3), 701; https://doi.org/10.3390/math11030701 - 30 Jan 2023

Cited by 28 | Viewed by 2142

Abstract

In this paper, we first present a simple seven-term 4D hyperchaotic system based on the classical Sprott-C 3D chaotic system. This novel system is inspired by the simple 4D hyperchaotic system based on Sprott-B proposed by A. T. Sheet (2022). We discuss the phenomenon of premature divergence brought about by the improper choice of coupling parameters in that paper and describe the basic properties of the new system with phase diagrams, Lyapunov exponential spectra and bifurcation diagrams. Then, we find that the dynamical behaviors of the system suffer from the limitation of the control parameters and cannot represent the process of motion in detail. To improve the system, we expand the dimensionality and add the control parameters and memristors. A 5D memristive hyperchaotic system with hidden attractors is proposed, and the basic dynamical properties of the system, such as its dissipation, equilibrium point, stability, Lyapunov exponential spectra and bifurcation diagram, are analyzed. Finally, the hardware circuits of the 4D Sprott-C system and the 5D memristive hyperchaotic system were realized by a field programmable gate array (FPGA) and verified by an experiment. The experimental results are consistent with the numerical simulation results obtained in MATLAB, which demonstrates the feasibility and potential of the system. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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11 pages, 6066 KiB

Open AccessArticle

A Switchable Chaotic Oscillator with Multiscale Amplitude/Frequency Control

by Zhoubin Sheng, Chunbiao Li, Yikai Gao, Zhinan Li and Lin Chai

Mathematics 2023, 11(3), 618; https://doi.org/10.3390/math11030618 - 26 Jan 2023

Cited by 4 | Viewed by 1122

Abstract

For the wide frequency spectrum of chaotic signals, it is difficult to realize chaotic signal conditioning. Therefore, researchers turn to the exploration of chaotic systems with independent non-bifurcation control for easy chaos modification. In this paper, a system with only one non-quadratic term is modified for providing multiscale amplitude/frequency control. By adjusting the feedback with an odd higher degree term, a switchable chaotic oscillator is obtained, which provides the different scales of amplitude/frequency control with the chaotic signal. Multisim-based circuit simulation shows the efficiency and convenience of chaotic signal control. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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18 pages, 8353 KiB

Open AccessArticle

A Color Image Encryption Algorithm Based on Hash Table, Hilbert Curve and Hyper-Chaotic Synchronization

by Xiaoyuan Wang, Xinrui Zhang, Meng Gao, Yuanze Tian, Chunhua Wang and Herbert Ho-Ching Iu

Mathematics 2023, 11(3), 567; https://doi.org/10.3390/math11030567 - 21 Jan 2023

Cited by 16 | Viewed by 1927

Abstract

Chaotic systems, especially hyper-chaotic systems are suitable for digital image encryption because of their complex properties such as pseudo randomness and extreme sensitivity. This paper proposes a new color image encryption algorithm based on a hyper-chaotic system constructed by a tri-valued memristor. The encryption process is based on the structure of permutation-diffusion, and the transmission of key information is realized through hyper-chaotic synchronization technology. In this design, the hash value of the plaintext image is used to generate the initial key the permutation sequence with the Hash table structure based on the hyper-chaotic sequence is used to implement pixel-level and bit-level permutation operations. Hilbert curves combining with the ciphertext feedback mechanism are applied to complete the diffusion operation. A series of experimental analyses have been applied to measure the novel algorithm, and the results show that the scheme has excellent encryption performance and can resist a variety of attacks. This method can be applied in secure image communication fields. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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13 pages, 5216 KiB

Open AccessArticle

Dynamic Behavior Analysis and Synchronization of Memristor-Coupled Heterogeneous Discrete Neural Networks

by Minglin Ma, Kangling Xiong, Zhijun Li and Yichuang Sun

Mathematics 2023, 11(2), 375; https://doi.org/10.3390/math11020375 - 10 Jan 2023

Cited by 37 | Viewed by 2391

Abstract

Continuous memristors have been widely studied in recent years; however, there are few studies on discrete memristors in the field of neural networks. In this paper, a four-stable locally active discrete memristor (LADM) is proposed as a synapse, which is used to connect a two-dimensional Chialvo neuron and a three-dimensional KTZ neuron, and construct a simple heterogeneous discrete neural network (HDNN). Through a bifurcation diagram and Lyapunov exponents diagram, the period and chaotic regions of the discrete neural network model are shown. Through numerical analysis, it was found that the chaotic region and periodic region of the neural network based on DLAM are significantly improved. In addition, coexisting chaos and chaos attractors, coexisting periodic and chaotic attractors, and coexisting periodic and periodic attractors will appear when the initial value of the LADM is changed. Coupled by a LADM synapse, two heterogeneous discrete neurons are gradually synchronized by changing the coupling strength. This paper lays a good foundation for the future analysis of LADMs and the related research of discrete neural networks coupled by LADMs. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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13 pages, 6364 KiB

Open AccessArticle

Dynamic Analysis and DSP Implementation of Memristor Chaotic Systems with Multiple Forms of Hidden Attractors

by Zhenggang Guo, Junjie Wen and Jun Mou

Mathematics 2023, 11(1), 24; https://doi.org/10.3390/math11010024 - 21 Dec 2022

Cited by 14 | Viewed by 1477

Abstract

In this paper, a new six dimensional memristor chaotic system is designed by combining the chaotic system with a memristor. By analyzing the phase diagram of the chaotic attractors, eleven different attractors are found, including a multi-wing attractor and symmetric attractors. By analyzing the equilibrium point of the system, it is proven that the system has the property of a hidden chaotic attractor. The dynamic behavior of the system when the three parameters change is analyzed by means of LEs and a Bifurcation diagram. Other phenomenon, such as chaos degradation, coexistence of multiple attractors and bias boosting, are also found. Finally, the simulation on the DSP platform also verifies the accuracy of the chaotic system simulation. The theoretical analysis and simulation results show that the system has rich dynamical characteristics; therefore, it is suitable for secure communication and image encryption and other fields. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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16 pages, 5830 KiB

Open AccessArticle

LQR Chaos Synchronization for a Novel Memristor-Based Hyperchaotic Oscillator

by Qifeng Fu, Xuemei Xu and Chuwen Xiao

Mathematics 2023, 11(1), 11; https://doi.org/10.3390/math11010011 - 20 Dec 2022

Cited by 1 | Viewed by 1080

Abstract

In a three-dimensional dissipative chaotic system circuit, by superimposing a cubic magnetron-type memristor and connecting a feedback circuit, a new four-dimensional synchronous controlling system is established. The control parameters have a significant impact on the system, and the system displays rich dynamic features such as hyperchaos, chaos, and period states. At the same time, the synchronization scheme for the chaotic system is designed based on the linear quadratic regulator (LQR), which effectively improves the system response speed and reduces the complexity of the synchronous controlling system. Further, numerical verification is carried out. Finally, a detailed verification of the chaotic system’s dynamic characteristics is performed by hardware simulation. Simulation results and performance analysis show that the proposed method has synchronous controlling performance. Compared to some existing synchronous controlling schemes, this method is more widely applicable. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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16 pages, 960 KiB

Open AccessArticle

Event-Based Impulsive Control for Heterogeneous Neural Networks with Communication Delays

by Yilin Li, Chengbo Yi, Jianwen Feng and Jingyi Wang

Mathematics 2022, 10(24), 4836; https://doi.org/10.3390/math10244836 - 19 Dec 2022

Viewed by 1109

Abstract

The quasi-synchronization for a class of general heterogeneous neural networks is explored by event-based impulsive control strategy. Compared with the traditional average impulsive interval (AII) method, instead, an event-triggered mechanism (ETM) is employed to determine the impulsive instants, in which case the subjectivity of selecting the controlling sequence can be eliminated. In addition, considering the fact that communication delay is inevitable between the allocation and execution of instructions in practice, we further nominate an ETM centered on communication delays and aperiodic sampling, which is more accessible and affordable, yet can straightforwardly avoid Zeno behavior. Hence, on the basis of the novel event-triggered impulsive control strategy, quasi-synchronization of heterogeneous neural network model is investigated and some general conditions are also achieved. Finally, two numerical simulations are afforded to validate the efficacy of theoretical results. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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16 pages, 11065 KiB

Open AccessArticle

A Memristor-Based Colpitts Oscillator Circuit

by Ling Zhou, Zhenzhen You, Xiaolin Liang and Xiaowu Li

Mathematics 2022, 10(24), 4820; https://doi.org/10.3390/math10244820 - 19 Dec 2022

Cited by 3 | Viewed by 1528

Abstract

This paper investigates a simple memristor emulator consisting of a diode bridge and a capacitor. It exhibits pinched hysteresis loops, and what is more striking is the higher frequency, as it operates up to greater than 5 MHz. Based on the proposed memristor, a higher-frequency Colpitts circuit was established. According to the mathematical model of the system, the system only possesses one unstable equilibrium point. Period doubling bifurcation, reverse periodic doubling bifurcation, different types of periodic and chaotic orbits, transient chaos, coexisting bifurcations and offset boosting are depicted. More interestingly, it has coexisting multiple attractors with different topologies, such as a chaotic attractor accompanied with periodic orbits, period-1 orbits with bicuspid structure and periodic-2 orbits with tridentate structure. Moreover, a hardware circuit using discrete components was fabricated and experimental measurements were consistent with the MATLAB numerical results, further confirming the real feasibility of the proposed circuit. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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Review

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18 pages, 1637 KiB

Open AccessReview

A Review of Chaotic Systems Based on Memristive Hopfield Neural Networks

by Hairong Lin, Chunhua Wang, Fei Yu, Jingru Sun, Sichun Du, Zekun Deng and Quanli Deng

Mathematics 2023, 11(6), 1369; https://doi.org/10.3390/math11061369 - 11 Mar 2023

Cited by 49 | Viewed by 2722

Abstract

Since the Lorenz chaotic system was discovered in 1963, the construction of chaotic systems with complex dynamics has been a research hotspot in the field of chaos. Recently, memristive Hopfield neural networks (MHNNs) offer great potential in the design of complex, chaotic systems because of their special network structures, hyperbolic tangent activation function, and memory property. Many chaotic systems based on MHNNs have been proposed and exhibit various complex dynamical behaviors, including hyperchaos, coexisting attractors, multistability, extreme multistability, multi-scroll attractors, multi-structure attractors, and initial-offset coexisting behaviors. A comprehensive review of the MHNN-based chaotic systems has become an urgent requirement. In this review, we first briefly introduce the basic knowledge of the Hopfiled neural network, memristor, and chaotic dynamics. Then, different modeling methods of the MHNN-based chaotic systems are analyzed and discussed. Concurrently, the pioneering works and some recent important papers related to MHNN-based chaotic systems are reviewed in detail. Finally, we survey the progress of MHNN-based chaotic systems for application in various scenarios. Some open problems and visions for the future in this field are presented. We attempt to provide a reference and a resource for both chaos researchers and those outside the field who hope to apply chaotic systems in a particular application. Full article

(This article belongs to the Special Issue Chaotic Systems and Their Applications)

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