MDPI - Publisher of Open Access Journals

23 pages, 11432 KiB

Open AccessArticle

Chaos Regulation via Complex Nonlinear Feedback and Its Implementation Based on FPAA

by Jitong Xu, Chunbiao Li, Xiaoliang Cen, Xin Zhang and Lin Chai

Symmetry 2025, 17(2), 212; https://doi.org/10.3390/sym17020212 - 31 Jan 2025

Cited by 1 | Viewed by 711

Complex nonlinear feedback is a key factor in the generation of chaos. In many cases, complex nonlinear functions have a higher probability for chaos producing, and correspondingly new bifurcations may be triggered in the dynamical system. Due to the difficulty in circuit implementation [...] Read more.

Complex nonlinear feedback is a key factor in the generation of chaos. In many cases, complex nonlinear functions have a higher probability for chaos producing, and correspondingly new bifurcations may be triggered in the dynamical system. Due to the difficulty in circuit implementation of complex nonlinear feedback, researchers often introduce simple nonlinear constraints to study the occurrence and evolution of chaos. In fact, the impact of complex nonlinear feedback on chaotic dynamics deserves further investigation. In this work, complex nonlinear feedback is introduced into an offset-boostable chaotic system as an example to observe and analyze its regulatory effect on the dynamics. Complex nonlinear feedback may destroy the property of symmetry of a system; therefore, we examine the evolution of chaotic attractors under the corresponding feedback and the functional transformation between bifurcation and non-bifurcation parameters as well. By fully utilizing the flexible configuration advantages of Field Programmable Analog Array (FPAA), arbitrary complex nonlinear functions are implemented to verify the chaotic dynamics. Full article

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

► Show Figures

Figure 1

16 pages, 19985 KiB

Open AccessArticle

A Chaotic Jerk Oscillator with Complete Control via Fractional Exponentiation and Its Experimental Analog Circuit Realization

by Menghui Shen, Chunbiao Li, Xiaoliang Cen, Manyu Zhao, Yuanxiao Xu and Ludovico Minati

Symmetry 2025, 17(2), 174; https://doi.org/10.3390/sym17020174 - 24 Jan 2025

Cited by 1 | Viewed by 795

Abstract

By introducing fractional exponentiation into a three-dimensional chaotic system, a jerk system with only six terms is designed. It has the property of total amplitude control, where a single non-bifurcation parameter can directly rescale all system variables without affecting the dynamics. It also [...] Read more.

By introducing fractional exponentiation into a three-dimensional chaotic system, a jerk system with only six terms is designed. It has the property of total amplitude control, where a single non-bifurcation parameter can directly rescale all system variables without affecting the dynamics. It also features two-dimensional offset boosting, where a single parameter can realize direct offset boosting while another provides interlocked cross-dimensional offset boosting. Furthermore, this jerk system has a parameter-dominated symmetric attractor, which means that symmetric attractors appear successively as the parameter changes from positive to negative. Circuit experiments confirm the feasibility of analog fractional exponentiation using the 444 circuit and the complete control, including amplitude control and offset boosting, of the resulting system. The proposed circuit may facilitate applications of chaotic signal generators where signal versatility is important and exemplifies the generative potential of analog fractional exponentiation. Full article

(This article belongs to the Section Engineering and Materials)

► Show Figures

Figure 1

15 pages, 4965 KiB

Open AccessArticle

Symmetric Pseudo-Multi-Scroll Attractor and Its Application in Mobile Robot Path Planning

by Yongxin Li, Chunbiao Li, Wanning Yu, Tengfei Lei and Rita Yi Man Li

Symmetry 2024, 16(7), 868; https://doi.org/10.3390/sym16070868 - 9 Jul 2024

Cited by 1 | Viewed by 1132

Abstract

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset [...] Read more.

The symmetric multi-scroll strange attractor has shown great potential in chaos-based applications due to its high complexity in phase space. Here, the approach of symmetrization is employed for attractor doubling to generate pseudo-multi-scroll attractors in a discrete map, where a carefully selected offset constant is the key to organizing coexisting attractors. By choosing the Hénon map to generate the pseudo-multi-scroll attractor and implementing the digital circuit on a microcontroller, this study fills a significant gap in the research on discrete chaotic systems. The complexity performance is further validated using a pseudo-random number generator, demonstrating substantial academic contributions to the field of chaos theory. Additionally, a pseudo-multi-scroll attractor-based squirrel search algorithm is first developed, showcasing its practical application in mobile robot path planning. This work not only advances the theoretical understanding of chaotic systems but also provides practical methods for implementation in digital systems, offering valuable insights for policy-making in advanced robotic systems and intelligent manufacturing. Full article

(This article belongs to the Section Computer)

► Show Figures

Figure 1

22 pages, 20786 KiB

Open AccessArticle

Neural Chaotic Oscillation: Memristive Feedback, Symmetrization, and Its Application in Image Encryption

by Keyu Huang, Chunbiao Li, Yongxin Li, Tengfei Lei and Haiyan Fu

Electronics 2024, 13(11), 2138; https://doi.org/10.3390/electronics13112138 - 30 May 2024

Cited by 1 | Viewed by 1248

Abstract

The symmetry of neuron discharging has some relationship with the electrophysiological characteristics and dynamic behavior of a neuron, and has a close relation with the symmetry of ion channels, current balance, neuron type, synaptic transmission, and network effects. Among them, the feedback and [...] Read more.

The symmetry of neuron discharging has some relationship with the electrophysiological characteristics and dynamic behavior of a neuron, and has a close relation with the symmetry of ion channels, current balance, neuron type, synaptic transmission, and network effects. Among them, the feedback and interactions in the network have a particularly direct impact on the symmetrical discharge of a neuron element. This work introduces a memristor as a synapse into a neuron cell, taking the membrane potential back to ion channels, and therefore various symmetric firing behaviors of Hindmarsh–Rose (HR) neurons are observed, including chaos and various periodic firings. By further adjusting the feedback, coexisting symmetrical discharge of the neuron is achieved. Furthermore, the impact of frequency variations on the memristor synapse is analyzed, and thus the operating regimes of memristor and resistor are classified and discussed. Circuit simulations prove the neural chaotic firings along with their symmetrized discharging processes, demonstrating the effectiveness of symmetrical control of chaotic discharge. Finally, applying the symmetrical system to DNA image encryption can effectively protect the security of images. Full article

(This article belongs to the Special Issue Recent Advances in Chaotic Systems and Their Security Applications, 2nd edition)

► Show Figures

Figure 1

27 pages, 12047 KiB

Open AccessFeature PaperReview

Symmetric Strange Attractors: A Review of Symmetry and Conditional Symmetry

by Chunbiao Li, Zhinan Li, Yicheng Jiang, Tengfei Lei and Xiong Wang

Symmetry 2023, 15(8), 1564; https://doi.org/10.3390/sym15081564 - 10 Aug 2023

Cited by 24 | Viewed by 4277

Abstract

A comprehensive review of symmetry and conditional symmetry is made from the core conception of symmetry and conditional symmetry. For a dynamical system, the structure of symmetry means its robustness against the polarity change of some of the system variables. Symmetric systems typically [...] Read more.

A comprehensive review of symmetry and conditional symmetry is made from the core conception of symmetry and conditional symmetry. For a dynamical system, the structure of symmetry means its robustness against the polarity change of some of the system variables. Symmetric systems typically show symmetrical dynamics, and even when the symmetry is broken, symmetric pairs of coexisting attractors are born, annotating the symmetry in another way. The polarity balance can be recovered through combinations of the polarity reversal of system variables, and furthermore, it can also be restored by the offset boosting of some of the system variables if the variables lead to the polarity reversal of their functions. In this case, conditional symmetry is constructed, giving a chance for a dynamical system outputting coexisting attractors. Symmetric strange attractors typically represent the flexible polarity reversal of some of the system variables, which brings more alternatives of chaotic signals and more convenience for chaos application. Symmetric and conditionally symmetric coexisting attractors can also be found in memristive systems and circuits. Therefore, symmetric chaotic systems and systems with conditional symmetry provide sufficient system options for chaos-based applications. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers)

► Show Figures

Figure 1

17 pages, 7764 KiB

Open AccessFeature PaperArticle

Multiple Alternatives of Offset Boosting in a Symmetric Hyperchaotic Map

by Xizhai Ge, Chunbiao Li, Yongxin Li, Chuang Zhang and Changyuan Tao

Symmetry 2023, 15(3), 712; https://doi.org/10.3390/sym15030712 - 13 Mar 2023

Cited by 10 | Viewed by 3839

Abstract

The offset as the average value of a variable plays an important role in signal processing and system design. Offset boosting can be realized by a non-bifurcation parameter or an initial condition. In this work, symmetric coexisting attractors with opposite polarity and a [...] Read more.

The offset as the average value of a variable plays an important role in signal processing and system design. Offset boosting can be realized by a non-bifurcation parameter or an initial condition. In this work, symmetric coexisting attractors with opposite polarity and a 2D hyperchaotic map with multiple modes of offset boosting are proposed, where the offset can be controlled both by the initial condition and system parameter, and as a result, multiple alternatives of offset boosting and offset competition show up. Consequently, the final offset is determined eventually by the balance of two factors. The theoretical findings are verified through the hardware experiment based on the STM32. Finally, a pseudo-random number generator (PRNG) is constructed based on the newly proposed hyperchaotic map, demonstrating its high performance in engineering applications. Full article

(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits 2022)

► Show Figures

Figure 1

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 10 | Viewed by 2091

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 [...] Read more.

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)

► Show Figures

Figure 1

11 pages, 4156 KiB

Open AccessArticle

A 2D Hyperchaotic Map: Amplitude Control, Coexisting Symmetrical Attractors and Circuit Implementation

by Xuejiao Zhou, Chunbiao Li, Xu Lu, Tengfei Lei and Yibo Zhao

Symmetry 2021, 13(6), 1047; https://doi.org/10.3390/sym13061047 - 10 Jun 2021

Cited by 12 | Viewed by 2355

Abstract

An absolute value function was introduced for chaos construction, where hyperchaotic oscillation was found with amplitude rescaling. The nonlinear absolute term brings the convenience for amplitude control. Two regimes of amplitude control including total and partial amplitude control are discussed, where the attractor [...] Read more.

An absolute value function was introduced for chaos construction, where hyperchaotic oscillation was found with amplitude rescaling. The nonlinear absolute term brings the convenience for amplitude control. Two regimes of amplitude control including total and partial amplitude control are discussed, where the attractor can be rescaled separately by two independent coefficients. Symmetrical pairs of coexisting attractors are captured by corresponding initial conditions. Circuit implementation by the platform STM32 is consistent with the numerical exploration and the theoretical observation. This finding is helpful for promoting discrete map application, where amplitude control is realized in an easy way and coexisting symmetrical sequences with opposite polarity are obtained. Full article

(This article belongs to the Special Issue Chaotic Systems and Nonlinear Dynamics)

► Show Figures

Figure 1

14 pages, 6904 KiB

Open AccessFeature PaperArticle

Asymmetry Evolvement and Controllability of a Symmetric Hyperchaotic Map

by Sixiao Kong, Chunbiao Li, Haibo Jiang, Yibo Zhao and Yanling Wang

Symmetry 2021, 13(6), 1039; https://doi.org/10.3390/sym13061039 - 9 Jun 2021

Cited by 3 | Viewed by 2305

Abstract

Trigonometric functions were used to construct a 2-D symmetrical hyperchaotic map with infinitely many attractors. The regime of multistability depends on the periodicity of the trigonometric function, which is closely related to the initial condition. For this trigonometric nonlinearity and the introduction of [...] Read more.

Trigonometric functions were used to construct a 2-D symmetrical hyperchaotic map with infinitely many attractors. The regime of multistability depends on the periodicity of the trigonometric function, which is closely related to the initial condition. For this trigonometric nonlinearity and the introduction of an offset controller, the initial condition triggers a specific multistability evolvement, in which infinitely countless symmetric and asymmetric attractors are produced. Initial condition-triggered offset boosting is explored, combined with constant controlled offset regulation. Furthermore, this symmetric map gives the sequences in various types of asymmetric attractors, in which the polarity balance is maintained by the initial condition and a negative coefficient due to the trigonometric function. Finally, as determined through the hardware implementation of STM32, the corresponding results agree with the numerical simulation. Full article

(This article belongs to the Special Issue Chaotic Systems and Nonlinear Dynamics)

► Show Figures

Figure 1

10 pages, 6900 KiB

Open AccessFeature PaperArticle

Symmetry Evolution in Chaotic System

by Chunbiao Li, Jiayu Sun, Tianai Lu and Tengfei Lei

Symmetry 2020, 12(4), 574; https://doi.org/10.3390/sym12040574 - 5 Apr 2020

Cited by 17 | Viewed by 2626

Abstract

A comprehensive exploration of symmetry and conditional symmetry is made from the evolution of symmetry. Unlike other chaotic systems of conditional symmetry, in this work it is derived from the symmetric diffusionless Lorenz system. Transformation from symmetry and asymmetry to conditional symmetry is [...] Read more.

A comprehensive exploration of symmetry and conditional symmetry is made from the evolution of symmetry. Unlike other chaotic systems of conditional symmetry, in this work it is derived from the symmetric diffusionless Lorenz system. Transformation from symmetry and asymmetry to conditional symmetry is examined by constant planting and dimension growth, which proves that the offset boosting of some necessary variables is the key factor for reestablishing polarity balance in a dynamical system. Full article

(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)

► Show Figures

Figure 1

17 pages, 5422 KiB

Open AccessFeature PaperArticle

A Symmetric Controllable Hyperchaotic Hidden Attractor

by Xin Zhang, Chunbiao Li, Tengfei Lei, Zuohua Liu and Changyuan Tao

Symmetry 2020, 12(4), 550; https://doi.org/10.3390/sym12040550 - 4 Apr 2020

Cited by 20 | Viewed by 3027

Abstract

By introducing a simple feedback, a hyperchaotic hidden attractor is found in the newly proposed Lorenz-like chaotic system. Some variables of the equilibria-free system can be controlled in amplitude and offset by an independent knob. A circuit experiment based on Multisim is consistent [...] Read more.

By introducing a simple feedback, a hyperchaotic hidden attractor is found in the newly proposed Lorenz-like chaotic system. Some variables of the equilibria-free system can be controlled in amplitude and offset by an independent knob. A circuit experiment based on Multisim is consistent with the theoretic analysis and numerical simulation. Full article

(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)

► Show Figures

Figure 1

14 pages, 9380 KiB

Open AccessArticle

Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems

by Shaobo He, Chunbiao Li, Kehui Sun and Sajad Jafari

Entropy 2018, 20(8), 556; https://doi.org/10.3390/e20080556 - 27 Jul 2018

Cited by 48 | Viewed by 4109

Abstract

Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many [...] Read more.

Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many chaotic attractors. The analysis results show that complexity of this class of chaotic systems is determined by the initial conditions. Meanwhile, the values of MMPE are independent of the scale factor, which is different from the algorithm of MMLZC. The analysis proposed here is helpful as a reference for the application of the self-reproducing systems. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

► Show Figures

Figure 1

23 pages, 8699 KiB

Open AccessArticle

A New Chaotic System with a Self-Excited Attractor: Entropy Measurement, Signal Encryption, and Parameter Estimation

by Guanghui Xu, Yasser Shekofteh, Akif Akgül, Chunbiao Li and Shirin Panahi

Entropy 2018, 20(2), 86; https://doi.org/10.3390/e20020086 - 27 Jan 2018

Cited by 86 | Viewed by 7873

Abstract

In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In [...] Read more.

In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In addition, we provide a parameter estimation method to extract chaotic model parameters from the real data of the chaotic circuit. The parameter estimation method is based on the attractor distribution modeling in the state space, which is compatible with the chaotic system characteristics. Here, a Gaussian mixture model (GMM) is used as a main part of cost function computations in the parameter estimation method. To optimize the cost function, we also apply two recent efficient optimization methods: WOA (Whale Optimization Algorithm), and MVO (Multi-Verse Optimizer) algorithms. The results show the success of the parameter estimation procedure. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

► Show Figures

Figure 1

15 pages, 5274 KiB

Open AccessArticle

A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design

by Qiang Lai, Akif Akgul, Chunbiao Li, Guanghui Xu and Ünal Çavuşoğlu

Entropy 2018, 20(1), 12; https://doi.org/10.3390/e20010012 - 27 Dec 2017

Cited by 97 | Viewed by 8665

Abstract

This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is [...] Read more.

This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is its multiple attractors caused by different initial values. With the change of parameters, the system experiences mono-stability, bi-stability, mono-periodicity, bi-periodicity, one strange attractor, and two coexisting strange attractors. The complex dynamic behaviors of the system are revealed by analyzing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the system. Using the new chaotic system, an S-Box is developed for cryptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

► Show Figures

Figure 1

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (14)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI