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26 pages, 10500 KB

Open AccessArticle

Lossless Frequency-Domain Image Encryption via 3D Exponential Hyper-Chaotic Map and Integer Lifting Wavelet Transform

by Xiangqun Shi, Yifan Su, Xiaole Yang, Wei Feng, Xian Zhang, Zhenhua Chen, Guangjun Wen and Heping Wen

Axioms 2026, 15(5), 315; https://doi.org/10.3390/axioms15050315 - 28 Apr 2026

Viewed by 319

Abstract

To resolve the inherent conflict between high robustness and strict reversibility in frequency-domain image encryption, as well as to eliminate data expansion caused by floating-point errors, this paper presents a novel lossless frequency-domain image encryption scheme via 3D exponential hyper-chaos and integer lifting [...] Read more.

To resolve the inherent conflict between high robustness and strict reversibility in frequency-domain image encryption, as well as to eliminate data expansion caused by floating-point errors, this paper presents a novel lossless frequency-domain image encryption scheme via 3D exponential hyper-chaos and integer lifting wavelet transform (ILWT). Firstly, a 3D hyper-chaotic exponential sine map (3D-HESM) is constructed by introducing nonlinear exponential coupling, providing a high-entropy keystream source with wider chaotic ranges than traditional maps. Secondly, to guarantee lossless reconstruction, the ILWT is employed to diffuse image coefficients in the frequency domain. By integrating modular arithmetic into the lifting steps, this transform confines coefficients within the finite integer ring, effectively solving the data expansion problem while maintaining perfect mathematical reversibility. Thirdly, an adaptive key generation protocol is designed by fusing SHA-512 with Singular Value Decomposition (SVD). Leveraging the geometric stability of singular values, this mechanism establishes a balance between extreme sensitivity to plaintext alterations and tolerance to channel noise. Experimental results and security analyses demonstrate that the proposed scheme achieves a vast key space and resists differential attacks. Furthermore, it exhibits superior robustness against data cropping and noise interference compared to state-of-the-art methods, validating its suitability for secure and lossless image transmission. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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26 pages, 4266 KB

Open AccessArticle

A Chaos-Based Image Encryption Algorithm via Integrated Cellular Automata and Tent Map Systems

by Yuanyuan Huang, Zixi Zhou, Diqing Liang, Fei Yu and Jie Jin

Axioms 2026, 15(5), 304; https://doi.org/10.3390/axioms15050304 - 23 Apr 2026

Viewed by 234

Abstract

This study proposes a novel image encryption algorithm based on a two-dimensional discrete chaotic system that integrates cellular automata (CA) with a tent map. The algorithm addresses security vulnerabilities in digital image transmission and storage across open networks or cloud environments. It employs [...] Read more.

This study proposes a novel image encryption algorithm based on a two-dimensional discrete chaotic system that integrates cellular automata (CA) with a tent map. The algorithm addresses security vulnerabilities in digital image transmission and storage across open networks or cloud environments. It employs a three-phase encryption process: coordinate permutation, spatial permutation, and diffusion. Sequential application of Arnold’s coordinate scrambling, maze traversal-based spatial rearrangement, and a CA-driven diffusion mechanism enhances robustness against noise, differential attacks, and partial cropping. A Dynamic CA–Tent Map (DCA–TM) hybrid chaotic system is designed to overcome periodicity and limited key space issues inherent in conventional chaotic encryption. The permutation stage is refined into coordinate and spatial phases to achieve comprehensive pixel randomization. During diffusion, CA rules are selected dynamically based on the iteration counts of the initial parameters, yielding an adaptive encryption system with a variable key space. Performance evaluations—including Lyapunov exponent tests, bifurcation analysis, information entropy measurement, and pixel correlation assessment—confirm the strong chaotic behavior and high security of the proposed scheme. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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29 pages, 15216 KB

Open AccessArticle

Equivariant Hopf Bifurcation of a Delayed Reaction–Diffusion Predator-Prey Model with Stage Structures on a Circular Domain

by Ruitong Gao, Xiaofeng Xu and Ming Liu

Axioms 2026, 15(3), 174; https://doi.org/10.3390/axioms15030174 - 28 Feb 2026

Viewed by 319

Abstract

This paper mainly studies the equivariant Hopf bifurcation of a delayed reaction–diffusion predator–prey model with stage structures on a two-dimensional circular domain. Firstly, we calculate the existence of steady-state solutions, and then analyze the existence of Hopf and equivariant Hopf bifurcation for the [...] Read more.

This paper mainly studies the equivariant Hopf bifurcation of a delayed reaction–diffusion predator–prey model with stage structures on a two-dimensional circular domain. Firstly, we calculate the existence of steady-state solutions, and then analyze the existence of Hopf and equivariant Hopf bifurcation for the model according to bifurcation theory. Secondly, we calculate the normal form of the equivariant Hopf bifurcation. Finally, we conduct numerical simulations to verify the conclusion. And through simulation, we obtain a spatially homogeneous periodic solution, and spatially inhomogeneous periodic solution including rotating waves and standing waves on a two-dimensional circular domain, which shows rich dynamic properties on a two-dimensional space. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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19 pages, 21031 KB

Open AccessArticle

Bifurcation Analysis of a Semilinear Generalized Friction System with Time-Delayed Feedback Control

by Haicheng Liu, Yanfeng Li and Xuejiao Liu

Axioms 2026, 15(1), 25; https://doi.org/10.3390/axioms15010025 - 28 Dec 2025

Viewed by 321

Abstract

In this paper, we investigate a semilinear parabolic friction system with time-delay feedback control and diffusion. This model more accurately describes the coupled dynamic behavior between vibrations induced by time-delayed control forces and the diffusion-driven evolution of material surface properties in practical friction [...] Read more.

In this paper, we investigate a semilinear parabolic friction system with time-delay feedback control and diffusion. This model more accurately describes the coupled dynamic behavior between vibrations induced by time-delayed control forces and the diffusion-driven evolution of material surface properties in practical friction processes. Through eigenvalue analysis, it is proven that the system’s stability does not vary monotonically with parameters. Instead, as the time delay varies, the system undergoes a finite number of alternating switches between stability and instability, before eventually losing its stability. The established stability criteria and bifurcation formulae can provide a predictive basis for and strategies to avoid the frictional vibration caused by time-delayed feedback in mechanical systems, providing significant guidance for vibration-reducing design and control parameter optimization in equipment such as braking systems and precision machine tools. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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33 pages, 11429 KB

Open AccessArticle

Two-Dimensional Coupling-Enhanced Cubic Hyperchaotic Map with Exponential Parameters: Construction, Analysis, and Application in Hierarchical Significance-Aware Multi-Image Encryption

by Wei Feng, Zixian Tang, Xiangyu Zhao, Zhentao Qin, Yao Chen, Bo Cai, Zhengguo Zhu, Kun Qian and Heping Wen

Axioms 2025, 14(12), 901; https://doi.org/10.3390/axioms14120901 - 6 Dec 2025

Cited by 18 | Viewed by 834

Abstract

As digital images proliferate across open networks, securing them against unauthorized access has become imperative. However, many recent image encryption algorithms are limited by weak chaotic dynamics and inadequate cryptographic design. To overcome these, we propose a new 2D coupling-enhanced cubic hyperchaotic map [...] Read more.

As digital images proliferate across open networks, securing them against unauthorized access has become imperative. However, many recent image encryption algorithms are limited by weak chaotic dynamics and inadequate cryptographic design. To overcome these, we propose a new 2D coupling-enhanced cubic hyperchaotic map with exponential parameters (2D-CCHM-EP). By incorporating exponential terms and strengthening interdependence among state variables, the 2D-CCHM-EP exhibits strict local expansiveness, effectively suppresses periodic windows, and achieves robust hyperchaotic behavior, validated both theoretically and numerically. It outperforms several recent chaotic maps in key metrics, yielding significantly higher Lyapunov exponents and Kolmogorov–Sinai entropy, and passes all NIST SP 800-22 randomness tests. Leveraging the 2D-CCHM-EP, we further develop a hierarchical significance-aware multi-image encryption algorithm (MIEA-CPHS). The core of MIEA-CPHS is a hierarchical significance-aware encryption strategy that decomposes input images into high-, medium-, and low-significance layers, which undergo three, two, and one round of vector-level adaptive encryption operations. An SHA-384-based hash of the fused data dynamically generates a 48-bit adaptive control parameter, enhancing plaintext sensitivity and enabling integrity verification. Comprehensive security analyses confirm the exceptional performance of MIEA-CPHS: near-zero inter-pixel correlation (<0.0016), near-ideal Shannon entropy (>7.999), and superior plaintext sensitivity (NPCR

\approx 99.61 %

, UACI

\approx 33.46 %

). Remarkably, the hierarchical design and vectorized operations achieve an average encryption throughput of 87.6152 Mbps, striking an outstanding balance between high security and computational efficiency. This makes MIEA-CPHS highly suitable for modern high-throughput applications such as secure cloud storage and real-time media transmission. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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25 pages, 1419 KB

Open AccessArticle

Hopf Bifurcation Analysis of a Phagocyte–Bacteria Diffusion Model with Delay in Crohn’s Disease

by Yu Sui and Ruizhi Yang

Axioms 2025, 14(12), 861; https://doi.org/10.3390/axioms14120861 - 24 Nov 2025

Viewed by 490

Abstract

Dysbiosis of the gut microbiota and dysregulated immune responses are key pathological features in both the onset and progression of Crohn’s disease. We propose a phagocyte–bacteria diffusion model with a time delay to explore their dynamic interactions and impact on the progression of [...] Read more.

Dysbiosis of the gut microbiota and dysregulated immune responses are key pathological features in both the onset and progression of Crohn’s disease. We propose a phagocyte–bacteria diffusion model with a time delay to explore their dynamic interactions and impact on the progression of Crohn’s disease. We first supplement the proof of the positivity, boundedness, existence, uniqueness, and global stability of the solutions for the ordinary differential system without time delay. Then we examine the stability of the positive equilibrium point and the occurrence of a Hopf bifurcation. By applying normal form and center manifold theory, we determine the direction of the bifurcation and the stability of the bifurcating periodic solution. Numerical simulations are used to verify the theoretical results. We find that the time delay significantly slows the system’s approach to a steady state. With a fixed delay, increased intestinal permeability prolongs the stabilization time. Conversely, with fixed intestinal permeability, a larger delay renders the system more prone to oscillations. Furthermore, a higher maximum engulfment rate by phagocytes reduces bacterial biomass but prolongs stabilization, whereas an increased phagocyte death rate shortens it. Additionally, an elevated bacterial growth rate increases both the bacterial biomass and the stabilization time. These results enhance our understanding of the dynamic equilibrium in immune systems. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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16 pages, 708 KB

Open AccessArticle

Total Decoupling of 2D Lattice Vibration

by Nan Jiang, Qizhi Zhang and Jianwei Wang

Axioms 2025, 14(11), 781; https://doi.org/10.3390/axioms14110781 - 24 Oct 2025

Viewed by 595

Abstract

Lattice structures find broad application in aerospace, automotive, biomedical, and energy systems owing to their exceptional structural stability. These systems typically exhibit complex internal couplings that facilitate vibration propagation across the entire network. The primary objective of this study is to achieve total [...] Read more.

Lattice structures find broad application in aerospace, automotive, biomedical, and energy systems owing to their exceptional structural stability. These systems typically exhibit complex internal couplings that facilitate vibration propagation across the entire network. The primary objective of this study is to achieve total decoupling of 2D lattice vibration system, which involves eliminating all inter-subsystem interactions while preserving spectrum. Building upon prior research, we develop structure-preserving isospectral transformation flow (SPITF) framework to address this challenge. Two principle results are established: first, the equations of motion are systematically derived for lattice vibration systems; second, total decoupling is successfully realized for such systems. Numerical experiments validate the decoupling capability of lattice vibration systems. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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22 pages, 981 KB

Open AccessArticle

Analysis of the Dynamic Properties of a Discrete Epidemic Model Affected by Media Coverage

by Yanfang Liang and Wenlong Wang

Axioms 2025, 14(9), 681; https://doi.org/10.3390/axioms14090681 - 4 Sep 2025

Cited by 1 | Viewed by 929

Abstract

This study investigates the dynamic behaviors of the discrete epidemic model influenced by media coverage through integrated analytical and numerical approaches. The primary objective is to quantitatively assess the impact of media coverage on disease outbreak patterns using mathematical modeling. Firstly, the Euler [...] Read more.

This study investigates the dynamic behaviors of the discrete epidemic model influenced by media coverage through integrated analytical and numerical approaches. The primary objective is to quantitatively assess the impact of media coverage on disease outbreak patterns using mathematical modeling. Firstly, the Euler method is used to discretize the model (2), and the periodic solution is strictly analyzed. Secondly, the coefficients and conditions of restricted flip and Neimark–Sacker bifurcation are studied by using the center manifold theorem and bifurcation theory. By calculating the largest Lyapunov exponent near the critical bifurcation point, the occurrence of chaos and limit cycles is proved. On this basis, the chaotic control of the system is carried out by using state feedback and hybrid control. Under certain conditions, the chaos and bifurcation of the system can be stabilized by control strategies. Numerical simulations further reveal bifurcation dynamics, chaotic behaviors, and control technologies. Our results show that media coverage is a key factor in regulating the intensity of disease transmission and chaos. The control technology can effectively prevent the large-scale outbreak of epidemic diseases. Importantly, enhanced media coverage can effectively promote public awareness and defensive behaviors, thereby contributing to the mitigation of disease transmission. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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37 pages, 45303 KB

Open AccessArticle

Dynamic Analysis and Application of 6D Multistable Memristive Chaotic System with Wide Range of Hyperchaotic States

by Fei Yu, Yumba Musoya Gracia, Rongyao Guo, Zhijie Ying, Jiarong Xu, Wei Yao, Jie Jin and Hairong Lin

Axioms 2025, 14(8), 638; https://doi.org/10.3390/axioms14080638 - 15 Aug 2025

Cited by 27 | Viewed by 1735

Abstract

In this study, we present a novel, six-dimensional, multistable, memristive, hyperchaotic system model demonstrating two positive Lyapunov exponents. With the maximum Lyapunov exponents surpassing 21, the developed system shows pronounced hyperchaotic behavior. The dynamical behavior was analyzed through phase portraits, bifurcation diagrams, and [...] Read more.

In this study, we present a novel, six-dimensional, multistable, memristive, hyperchaotic system model demonstrating two positive Lyapunov exponents. With the maximum Lyapunov exponents surpassing 21, the developed system shows pronounced hyperchaotic behavior. The dynamical behavior was analyzed through phase portraits, bifurcation diagrams, and Lyapunov exponent spectra. Parameter b was a key factor in regulating the dynamical behavior of the system, mainly affecting the strength and direction of the influence of

z_{1}

on

z_{2}

. It was found that when the system parameter b was within a wide range of

[13, 300]

, the system remained hyperchaotic throughout. Analytical establishment of multistability mechanisms was achieved through invariance analysis of the state variables under specific coordinate transformations. Furthermore, offset boosting control was realized by strategically modulating the fifth state variable,

z_{5}

. The FPGA-based experimental results demonstrated that attractors observed via an oscilloscope were in close agreement with numerical simulations. To validate the system’s reliability for cybersecurity applications, we designed a novel image encryption method utilizing this hyperchaotic model. The information entropy of the proposed encryption algorithm was closer to the theoretical maximum value of 8. This indicated that the system can effectively disrupt statistical patterns. Experimental outcomes confirmed that the proposed image encryption method based on the hyperchaotic system exhibits both efficiency and reliability. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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