Research

19 pages, 2526 KiB

Open AccessArticle

Bifurcation, Hidden Chaos, Entropy and Control in Hénon-Based Fractional Memristor Map with Commensurate and Incommensurate Orders

by Mayada Abualhomos, Abderrahmane Abbes, Gharib Mousa Gharib, Abdallah Shihadeh, Maha S. Al Soudi, Ahmed Atallah Alsaraireh and Adel Ouannas

Mathematics 2023, 11(19), 4166; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194166 - 05 Oct 2023

Cited by 1 | Viewed by 849

Abstract

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting [...] Read more.

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting bifurcation diagrams, and applying the sample entropy test to assess the complexity and validate the chaotic characteristics. However, since the proposed fractional map has no fixed points, the outcomes reveal that the map can exhibit a wide range of hidden dynamical behaviors. This phenomenon significantly augments the complexity of the fractal structure inherent to the chaotic attractors. Moreover, we introduce nonlinear controllers designed for stabilizing and synchronizing the proposed fractional Hénon-based memristor map. The research emphasizes the system’s sensitivity to fractional-order parameters, resulting in the emergence of distinct dynamic patterns. The memristor-based chaotic map exhibits rich and intricate behavior, making it a captivating and significant area of investigation. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

29 pages, 54206 KiB

Open AccessArticle

A Novel Eighth-Order Hyperchaotic System and Its Application in Image Encryption

by Hanshuo Qiu, Xiangzi Zhang, Huaixiao Yue and Jizhao Liu

Mathematics 2023, 11(19), 4099; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194099 - 28 Sep 2023

Cited by 1 | Viewed by 1139

Abstract

With the advancement in information and communication technologies (ICTs), the widespread dissemination and sharing of digital images has raised concerns regarding privacy and security. Traditional methods of encrypting images often suffer from limitations such as a small key space and vulnerability to brute-force [...] Read more.

With the advancement in information and communication technologies (ICTs), the widespread dissemination and sharing of digital images has raised concerns regarding privacy and security. Traditional methods of encrypting images often suffer from limitations such as a small key space and vulnerability to brute-force attacks. To address these issues, this paper proposes a novel eighth-order hyperchaotic system. This hyperchaotic system exhibits various dynamic behaviors, including hyperchaos, sub-hyperchaos, and chaos. The encryption scheme based on this system offers a key space larger than 2²³³⁸. Through a comprehensive analysis involving histogram analysis, key space analysis, correlation analysis, entropy analysis, key sensitivity analysis, differential attack analysis, and cropping attack analysis, it is demonstrated that the proposed system is capable of resisting statistical attacks, brute force attacks, differential attacks, and cropping attacks, thereby providing excellent security performance. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

10 pages, 1078 KiB

Open AccessArticle

Design of High-Dimensional Maps with Sine Terms

by Othman Abdullah Almatroud, Viet-Thanh Pham, Giuseppe Grassi, Mohammad Alshammari, Sahar Albosaily and Van Van Huynh

Mathematics 2023, 11(17), 3725; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173725 - 30 Aug 2023

Cited by 2 | Viewed by 670

Abstract

The use of the advancements in memristor technology to construct chaotic maps has garnered significant research attention in recent years. The combination of memristors and nonlinear terms provides an effective approach to proposing novel maps. In this study, we have leveraged memristors and [...] Read more.

The use of the advancements in memristor technology to construct chaotic maps has garnered significant research attention in recent years. The combination of memristors and nonlinear terms provides an effective approach to proposing novel maps. In this study, we have leveraged memristors and sine terms to develop three-dimensional maps, capable of processing special fixed points. Additionally, we have conducted an in depth study of a specific example (TDMM

_{1}

map) to demonstrate its dynamics, feasibility, and application for lightweight encryption. Notably, our general approach could be extended to develop higher-dimensional maps, including four- and five-dimensional ones, thereby opening up the possibility to create numerous higher-dimensional maps. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

14 pages, 4377 KiB

Open AccessArticle

Hopf Bifurcation, Periodic Solutions, and Control of a New 4D Hyperchaotic System

by Yu Liu, Yan Zhou and Biyao Guo

Mathematics 2023, 11(12), 2699; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122699 - 14 Jun 2023

Cited by 1 | Viewed by 1140

Abstract

In this paper, a new four-dimensional (4D) hyperchaotic biplane system is designed and presented. The dynamical properties of this new system are studied by means of tools such as bifurcation diagrams, Lyapunov exponents and phase diagrams. The Hopf bifurcation and periodic solutions of [...] Read more.

In this paper, a new four-dimensional (4D) hyperchaotic biplane system is designed and presented. The dynamical properties of this new system are studied by means of tools such as bifurcation diagrams, Lyapunov exponents and phase diagrams. The Hopf bifurcation and periodic solutions of this hyperchaotic system are solved analytically. In addition, a new hyperchaotic control strategy is applied, and a comparative analysis of the controlled system is performed. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

30 pages, 434 KiB

Open AccessArticle

Investigating Symmetric Soliton Solutions for the Fractional Coupled Konno–Onno System Using Improved Versions of a Novel Analytical Technique

by Humaira Yasmin, Noufe H. Aljahdaly, Abdulkafi Mohammed Saeed and Rasool Shah

Mathematics 2023, 11(12), 2686; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122686 - 13 Jun 2023

Cited by 35 | Viewed by 2096

Abstract

The present research investigates symmetric soliton solutions for the Fractional Coupled Konno–Onno System (FCKOS) by using two improved versions of an Extended Direct Algebraic Method (EDAM) i.e., modified EDAM (mEDAM) and r+mEDAM. By obtaining precise analytical solutions, this research explores the characteristics [...] Read more.

The present research investigates symmetric soliton solutions for the Fractional Coupled Konno–Onno System (FCKOS) by using two improved versions of an Extended Direct Algebraic Method (EDAM) i.e., modified EDAM (mEDAM) and r+mEDAM. By obtaining precise analytical solutions, this research explores the characteristics and behaviours of symmetric solitons in FCKOS. Further, the amplitude, shape and propagation behaviour of some solitons are visualized by means of a 3D graph. This investigation fosters a more thorough comprehension of non-linear wave phenomena in considered systems and offers helpful insights towards soliton behavior in it. The outcomes reveal that the recommended techniques are successful in constructing symmetric soliton solutions for complex models like the FCKOS. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

20 pages, 24527 KiB

Open AccessArticle

Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation

by Weipeng Lyu, Shaolong Li, Zhenyang Chen and Qinsheng Bi

Mathematics 2023, 11(11), 2486; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112486 - 28 May 2023

Cited by 3 | Viewed by 917

Abstract

As a kind of dynamical system with a particular nonlinear structure, a multi-time scale nonlinear system is one of the essential directions of the current development of nonlinear dynamics theory. Multi-time scale nonlinear systems in practical applications are often complex forms of coupling [...] Read more.

As a kind of dynamical system with a particular nonlinear structure, a multi-time scale nonlinear system is one of the essential directions of the current development of nonlinear dynamics theory. Multi-time scale nonlinear systems in practical applications are often complex forms of coupling of high-dimensional and high codimension characteristics, leading to various complex bursting oscillation behaviors and bifurcation characteristics in the system. For exploring the complex bursting dynamics caused by high codimension bifurcation, this paper considers the normal form of the vector field with triple zero bifurcation. Two kinds of codimension-2 bifurcation that may lead to complex bursting oscillations are discussed in the two-parameter plane. Based on the fast–slow analysis method, by introducing the slow variable

W = A sin (ω t)

, the evolution process of the motion trajectory of the system changing with W was investigated, and the dynamical mechanism of several types of bursting oscillations was revealed. Finally, by varying the frequency of the slow variable, a class of chaotic bursting phenomena caused by the period-doubling cascade is deduced. Developing related work has played a positive role in deeply understanding the nature of various complex bursting phenomena and strengthening the application of basic disciplines such as mechanics and mathematics in engineering practice. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

21 pages, 31507 KiB

Open AccessArticle

On the Fractional-Order Complex Cosine Map: Fractal Analysis, Julia Set Control and Synchronization

by A. A. Elsadany, A. Aldurayhim, H. N. Agiza and Amr Elsonbaty

Mathematics 2023, 11(3), 727; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030727 - 01 Feb 2023

Cited by 1 | Viewed by 1090

Abstract

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia [...] Read more.

In this paper, we introduce a generalized complex discrete fractional-order cosine map. Dynamical analysis of the proposed complex fractional order map is examined. The existence and stability characteristics of the map’s fixed points are explored. The existence of fractal Mandelbrot sets and Julia sets, as well as their fractal properties, are examined in detail. Several detailed simulations illustrate the effects of the fractional-order parameter, as well as the values of the map constant and exponent. In addition, complex domain controllers are constructed to control Julia sets produced by the proposed map or to achieve synchronization of two Julia sets in master/slave configurations. We identify the more realistic synchronization scenario in which the master map’s parameter values are unknown. Finally, numerical simulations are employed to confirm theoretical results obtained throughout the work. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

20 pages, 1598 KiB

Open AccessArticle

Exact Solutions and Non-Traveling Wave Solutions of the (2+1)-Dimensional Boussinesq Equation

by Lihui Gao, Chunxiao Guo, Yanfeng Guo and Donglong Li

Mathematics 2022, 10(14), 2522; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142522 - 20 Jul 2022

Cited by 2 | Viewed by 1155

Abstract

By the extended

(\frac{G^{'}}{G})

method and the improved tanh function method, the exact solutions of the (2+1) dimensional Boussinesq equation are studied. Firstly, with the help of the solutions of the nonlinear ordinary differential equation, we obtain the new [...] Read more.

By the extended

(\frac{G^{'}}{G})

method and the improved tanh function method, the exact solutions of the (2+1) dimensional Boussinesq equation are studied. Firstly, with the help of the solutions of the nonlinear ordinary differential equation, we obtain the new traveling wave exact solutions of the equation by the homogeneous equilibrium principle and the extended

(\frac{G^{'}}{G})

method. Secondly, by constructing the new ansatz solutions and applying the improved tanh function method, many non-traveling wave exact solutions of the equation are given. The solutions mainly include hyperbolic, trigonometric and rational functions, which reflect different types of solutions for nonlinear waves. Finally, we discuss the effects of these solutions on the formation of rogue waves according to the numerical simulation. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

13 pages, 1759 KiB

Open AccessArticle

Chaotic Dynamics of Non-Autonomous Nonlinear System for a Sandwich Plate with Truss Core

by Dongmei Zhang and Feng Li

Mathematics 2022, 10(11), 1889; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111889 - 31 May 2022

Cited by 1 | Viewed by 1191

Abstract

This paper deals with the multi-pulse chaotic dynamics of a sandwich plate with truss core under transverse and in-plane excitations. In order to analyze the complicated nonlinear behaviors of the sandwich plate model by means of the improved extended Melnikov technique, the two-degrees [...] Read more.

This paper deals with the multi-pulse chaotic dynamics of a sandwich plate with truss core under transverse and in-plane excitations. In order to analyze the complicated nonlinear behaviors of the sandwich plate model by means of the improved extended Melnikov technique, the two-degrees non-autonomous system is transformed into an appropriate form. Through theoretical analysis, the sufficient conditions for the existence of multi-pulse homoclinic orbits and the criterion for the occurrence of chaotic motion are obtained. The damping coefficients and transverse excitation parameters are considered as the control parameters to discuss chaotic behaviors of the sandwich plate system. Numerical results and the maximal Lyapunov exponents are performed to further test the existence of the multi-pulse jumping orbits. The theoretical predictions and numerical results demonstrate that the chaos phenomena may exist in the parametrical excited sandwich plate. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

11 pages, 2889 KiB

Open AccessArticle

Complex Periodic Mixed-Mode Oscillation Patterns in a Filippov System

by Chun Zhang and Qiaoxia Tang

Mathematics 2022, 10(5), 673; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050673 - 22 Feb 2022

Cited by 2 | Viewed by 1244

Abstract

The main task of this article is to study the patterns of mixed-mode oscillations and non-smooth behaviors in a Filippov system with external excitation. Different types of periodic spiral crossing mixed-mode oscillation patterns, i.e., “cusp-F

^{-}

/fold-F

^{-}

” oscillation, “cusp-F

^{-}

/two-fold/two-fold/fold-F [...] Read more.

The main task of this article is to study the patterns of mixed-mode oscillations and non-smooth behaviors in a Filippov system with external excitation. Different types of periodic spiral crossing mixed-mode oscillation patterns, i.e., “cusp-F

^{-}

/fold-F

^{-}

” oscillation, “cusp-F

^{-}

/two-fold/two-fold/fold-F

^{-}

” oscillation and “two-fold/fold-F

^{-}

” oscillation, are explored. Based on the analysis of the equilibrium and tangential singularities of the fast subsystem, spiral crossing oscillation around the tangential singularities is investigated. Meanwhile, by combining the fast and slow analysis methods, we can observe that the cusp, two-fold and fold-cusp singularities play an important role in generating all kinds of complex mixed-mode oscillations. Full article

(This article belongs to the Special Issue Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Recent Advances in Chaos, Fractal and Complex Dynamics in Nonlinear Systems

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (10 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI