Editorial

4 pages, 181 KiB

Open AccessEditorial

Special Issue Editorial “Chaotic Systems and Nonlinear Dynamics”

by Giuseppe Grassi

Symmetry 2022, 14(6), 1137; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061137 - 31 May 2022

Cited by 1 | Viewed by 990

Referring to chaotic systems, it is well-known that they are nonlinear dynamical systems, which are distinguished by sensitive dependence on initial conditions and by having evolution through phase space that appears to be quite random [...] Full article

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

Research

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20 pages, 5044 KiB

Open AccessArticle

Memristive Structure-Based Chaotic System for PRNG

by Serhii Haliuk, Oleh Krulikovskyi, Dmytro Vovchuk and Fernando Corinto

Symmetry 2022, 14(1), 68; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010068 - 04 Jan 2022

Cited by 14 | Viewed by 1774

Abstract

This paper suggests an approach to generate pseudo-random sequences based on the discrete-time model of the simple memristive chaotic system. We show that implementing Euler’s and Runge–Kutta’s methods for the simulation solutions gives the possibility of obtaining chaotic sequences that maintain general properties [...] Read more.

This paper suggests an approach to generate pseudo-random sequences based on the discrete-time model of the simple memristive chaotic system. We show that implementing Euler’s and Runge–Kutta’s methods for the simulation solutions gives the possibility of obtaining chaotic sequences that maintain general properties of the original chaotic system. A preliminary criterion based on the binary sequence balance estimation is proposed and applied to separate any binary representation of the chaotic time sequences into random and non-random parts. This gives us the possibility to delete obviously non-random sequences prior to the post-processing. The investigations were performed for arithmetic with both fixed and floating points. In both cases, the obtained sequences successfully passed the NIST SP 800-22 statistical tests. The utilization of the unidirectional asymmetric coupling of chaotic systems without full synchronization between them was suggested to increase the performance of the chaotic pseudo-random number generator (CPRNG) and avoid identical sequences on different outputs of the coupled systems. The proposed CPRNG was also implemented and tested on FPGA using Euler’s method and fixed-point arithmetic for possible usage in different applications. The FPGA implementation of CPRNG supports a generation speed up to 1.2 Gbits/s for a clock frequency of 50 MHz. In addition, we presented an example of the application of CPRNG to symmetric image encryption, but nevertheless, one is suitable for the encryption of any binary source. Full article

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

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23 pages, 2009 KiB

Open AccessArticle

Authenticated Encryption Based on Chaotic Neural Networks and Duplex Construction

by Nabil Abdoun, Safwan El Assad, Thang Manh Hoang, Olivier Deforges, Rima Assaf and Mohamad Khalil

Symmetry 2021, 13(12), 2432; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122432 - 16 Dec 2021

Cited by 3 | Viewed by 2325

Abstract

In this paper, we propose, implement and analyze an Authenticated Encryption with Associated Data Scheme (AEADS) based on the Modified Duplex Construction (MDC) that contains a chaotic compression function (CCF) based on our chaotic neural network revised (CNNR). Unlike the standard duplex construction [...] Read more.

In this paper, we propose, implement and analyze an Authenticated Encryption with Associated Data Scheme (AEADS) based on the Modified Duplex Construction (MDC) that contains a chaotic compression function (CCF) based on our chaotic neural network revised (CNNR). Unlike the standard duplex construction (SDC), in the MDC there are two phases: the initialization phase and the duplexing phase, each contain a CNNR formed by a neural network with single layer, and followed by a set of non-linear functions. The MDC is implemented with two variants of width, i.e., 512 and 1024 bits. We tested our proposed scheme against the different cryptanalytic attacks. In fact, we evaluated the key and the message sensitivity, the collision resistance analysis and the diffusion effect. Additionally, we tested our proposed AEADS using the different statistical tests such as NIST, Histogram, chi-square, entropy, and correlation analysis. The experimental results obtained on the security performance of the proposed AEADS system are notable and the proposed system can then be used to protect data and authenticate their sources. Full article

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

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7 pages, 322 KiB

Open AccessArticle

Homotopic Parametric Continuation Method for Determining Stationary States of Chemical Reactors with Dispersion

by Marek Berezowski and Marcin Lawnik

Symmetry 2021, 13(12), 2324; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122324 - 04 Dec 2021

Cited by 2 | Viewed by 1301

Abstract

Physical processes occurring in devices with distributed variables and a turbulent tide with a dispersion of mass and heat are often modeled using systems of nonlinear equations. Solving such a system is sometimes impossible in an analytical manner. The iterative methods, such as [...] Read more.

Physical processes occurring in devices with distributed variables and a turbulent tide with a dispersion of mass and heat are often modeled using systems of nonlinear equations. Solving such a system is sometimes impossible in an analytical manner. The iterative methods, such as Newton’s method, are not always sufficiently effective in such cases. In this article, a combination of the homotopy method and the parametric continuation method was proposed to solve the system of nonlinear differential equations. These methods are symmetrical, i.e., the calculations can be made by increasing or decreasing the value of the parameters. Thanks to this approach, the determination of all roots of the system does not require any iterative method. Moreover, when the solutions of the system are close to each other, the proposed method easily determines all of them. As an example of the method use a mathematical model of a non-adiabatic catalytic pseudohomogeneous tubular chemical reactor with longitudinal dispersion was chosen. Full article

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

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24 pages, 11921 KiB

Open AccessArticle

A New Conservative Hyperchaotic System-Based Image Symmetric Encryption Scheme with DNA Coding

by Qing Lu, Linlan Yu and Congxu Zhu

Symmetry 2021, 13(12), 2317; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122317 - 04 Dec 2021

Cited by 11 | Viewed by 1444

Abstract

In the current paper, a new conservative hyperchaotic system is proposed. We make a quantitative analysis of the complexity of the conservative hyperchaotic system from several different aspects, such as phase diagrams, bifurcation diagrams, Lyapunov exponents, and Kaplan–Yorke dimension. The complexity of chaotic [...] Read more.

In the current paper, a new conservative hyperchaotic system is proposed. We make a quantitative analysis of the complexity of the conservative hyperchaotic system from several different aspects, such as phase diagrams, bifurcation diagrams, Lyapunov exponents, and Kaplan–Yorke dimension. The complexity of chaotic time series is tested with various measurement tools, such as the scale index, the multiscale sample entropy and approximate entropy, TESTU01, and NIST test. In addition, a novel hyperchao-based image encryption scheme with dynamic DNA coding is proposed. The encryption algorithm consists of line-by-line scrambling and diffusion of DNA encoding characters. The dynamic DNA coding mechanism is introduced by using the chaotic sequence. The generation of the intermediate secret keys is related to the sum of the image DNA code, and the ciphertext feedback mechanism of the DNA encoding image is introduced in the diffusion procedure. Simulation experiments and various security analyses show that this algorithm has a good effect on encryption, high time efficiency, and can effectively resist brute force attacks, statistical attacks, chosen-plaintext attacks, and differential attacks. Full article

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

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12 pages, 23174 KiB

Open AccessArticle

On Dynamic Investigations of Cournot Duopoly Game: When Firms Want to Maximize Their Relative Profits

by Sameh Askar

Symmetry 2021, 13(12), 2235; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122235 - 23 Nov 2021

Cited by 2 | Viewed by 1250

Abstract

This paper studies a Cournot duopoly game in which firms produce homogeneous goods and adopt a bounded rationality rule for updating productions. The firms are characterized by an isoelastic demand that is derived from a simple quadratic utility function with linear total costs. [...] Read more.

This paper studies a Cournot duopoly game in which firms produce homogeneous goods and adopt a bounded rationality rule for updating productions. The firms are characterized by an isoelastic demand that is derived from a simple quadratic utility function with linear total costs. The two competing firms in this game seek the optimal quantities of their production by maximizing their relative profits. The model describing the game’s evolution is a two-dimensional nonlinear discrete map and has only one equilibrium point, which is a Nash point. The stability of this point is discussed and it is found that it loses its stability by two different ways, through flip and Neimark–Sacker bifurcations. Because of the asymmetric structure of the map due to different parameters, we show by means of global analysis and numerical simulation that the nonlinear, noninvertible map describing the game’s evolution can give rise to many important coexisting stable attractors (multistability). Analytically, some investigations are performed and prove the existence of areas known in literature with lobes. Full article

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

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12 pages, 3297 KiB

Open AccessArticle

Dynamics Analysis and Synchronous Control of Fractional-Order Entanglement Symmetrical Chaotic Systems

by Tengfei Lei, Beixing Mao, Xuejiao Zhou and Haiyan Fu

Symmetry 2021, 13(11), 1996; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13111996 - 21 Oct 2021

Cited by 5 | Viewed by 1285

Abstract

In this paper, the Adomian decomposition method (ADM) semi-analytical solution algorithm is applied to solve a fractional-order entanglement symmetrical chaotic system. The dynamics of the system are analyzed by the Lyapunov exponent spectrum, bifurcation diagrams, poincaré diagrams, and chaos diagrams. The results show [...] Read more.

In this paper, the Adomian decomposition method (ADM) semi-analytical solution algorithm is applied to solve a fractional-order entanglement symmetrical chaotic system. The dynamics of the system are analyzed by the Lyapunov exponent spectrum, bifurcation diagrams, poincaré diagrams, and chaos diagrams. The results show that the systems have rich dynamics. Meanwhile, sliding mode synchronizations of fractional-order chaotic systems are investigated theoretically and numerically. The results show the effectiveness of the proposed method and potential application value of fractional-order systems. Full article

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

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10 pages, 4174 KiB

Open AccessCommunication

Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

by Yiu-Yin Lee

Symmetry 2021, 13(7), 1188; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071188 - 01 Jul 2021

Cited by 4 | Viewed by 1690

Abstract

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two [...] Read more.

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated. Full article

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

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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://0-doi-org.brum.beds.ac.uk/10.3390/sym13061047 - 10 Jun 2021

Cited by 8 | Viewed by 1669

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)

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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://0-doi-org.brum.beds.ac.uk/10.3390/sym13061039 - 09 Jun 2021

Cited by 3 | Viewed by 1762

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)

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14 pages, 5879 KiB

Open AccessArticle

Complexity and Chimera States in a Network of Fractional-Order Laser Systems

by Shaobo He, Hayder Natiq, Santo Banerjee and Kehui Sun

Symmetry 2021, 13(2), 341; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13020341 - 20 Feb 2021

Cited by 11 | Viewed by 1790

Abstract

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation [...] Read more.

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation diagram. Numerical simulation outcomes demonstrate that the system generates chaos with the decreasing of q. Moreover, this paper designs the coupled fractional-order network of laser systems and subsequently obtains its numerical solution using ABM. These solutions have demonstrated chimera states of the proposed fractional-order laser network. Full article

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

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Review

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10 pages, 249 KiB

Open AccessEditor’s ChoiceReview

Chaos in the Real World: Recent Applications to Communications, Computing, Distributed Sensing, Robotic Motion, Bio-Impedance Modelling and Encryption Systems

by Giuseppe Grassi

Symmetry 2021, 13(11), 2151; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112151 - 11 Nov 2021

Cited by 28 | Viewed by 2407

Abstract

Most of the papers published so far in literature have focused on the theoretical phenomena underlying the formation of chaos, rather than on the investigation of potential applications of chaos to the real world. This paper aims to bridge the gap between chaos [...] Read more.

Most of the papers published so far in literature have focused on the theoretical phenomena underlying the formation of chaos, rather than on the investigation of potential applications of chaos to the real world. This paper aims to bridge the gap between chaos theory and chaos applications by presenting a survey of very recent applications of chaos. In particular, the manuscript covers the last three years by describing different applications of chaos as reported in the literature published during the years 2018 to 2020, including the matter related to the symmetry properties of chaotic systems. The topics covered herein include applications of chaos to communications, to distributed sensing, to robotic motion, to bio-impedance modelling, to hardware implementation of encryption systems, to computing and to random number generation. Full article

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

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