Chaos Solitons & Fractals最新文献_第7页

A method for ground surface temperature prediction and its applications by solving heat conduction problems under complex periodic boundary conditions 求解复杂周期边界条件下热传导问题的地表温度预测方法及其应用

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-04 DOI: 10.1016/j.chaos.2025.117723

Yuqin Zhao , Gaosheng Li , Fenglei Han , Ying Lai , Xiangtian Xu

In the present study, the calculation of ground surface temperature (GST) is formulated as a heat conduction problem under complex-periodic thermal boundary conditions involving solar radiation heat flux, convective heat transfer and radiative heat transfer (RHT). By solving the heat conduction problem, we develop a quantitative functional relationship between surface heat fluxes and GST, proposing a novel method for calculating GST. Comprehensive verification and comparison with other methods of GST, demonstrate the accuracy of the method in predicting GST under different environmental and material conditions. This method clarifies the main factors affecting GST and quantifies the surface heat flux ratio and heat balance ratio under different heat fluxes. The results show that the surface heat flux from RHT is approximately 5.67

ε

(surface radiance) times the ground-air temperature difference, and RHT contributes nearly 50 % to the total heat transfer at higher heat flux and lower wind speed, highlighting the need to include RHT effects in predictive models of GST to reduce potential error. In addition, this work also confirms that the dynamic change of vegetation coverage also plays a critical role in the GST, so it emphasizes the need to consider this variable in the GST prediction models to enhance the accuracy.

在本研究中，地表温度（GST）的计算被表述为复杂周期热边界条件下的热传导问题，涉及太阳辐射热通量、对流换热和辐射换热（RHT）。通过求解热传导问题，建立了表面热通量与GST之间的定量函数关系，提出了一种计算GST的新方法。综合验证和与其他GST方法的比较，证明了该方法在不同环境和物质条件下预测GST的准确性。该方法明确了影响GST的主要因素，量化了不同热通量下的地表热流比和热平衡比。结果表明，RHT的地表热通量约为地表辐射强度的5.67ε（地表辐射强度）倍于地空温差，在高热流密度和低风速条件下，RHT对总换热的贡献接近50%，因此需要在GST预测模型中考虑RHT效应，以减少潜在误差。此外，本工作还证实了植被覆盖度的动态变化在GST中也起着至关重要的作用，因此强调在GST预测模型中需要考虑这一变量以提高精度。

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引用次数: 0

Dynamic evolution of Nyquist-pulse solitons from a mode-locked fiber oscillator 锁模光纤振荡器中奈奎斯特脉冲孤子的动态演化

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-04 DOI: 10.1016/j.chaos.2025.117711

Ziyi Pu, Xinyao Li, Wenjuan Wu, Binjie Zuo, Chao Zhang, Shuzhen Zou, Chaojian He, Song Yang, Xuechun Lin

Nyquist pulses, characterized by their sinc-shaped temporal and rectangular spectral profiles, have found widespread applications in high-speed optical communication and optical storage. Traditional methods for generating Nyquist pulses, which typically rely on active spectral modulation, often experience deviations from the ideal rectangular spectrum and encounter increased system complexity. In this work, the direct generation of Nyquist-pulse solitons (NPSs) from a passively mode-locked fiber oscillator is numerically demonstrated. The dynamic evolution of the pulses is systematically investigated, highlighting the pivotal role of gain in regulating pulse dynamics. By tuning the gain saturation energy (Eₛₐₜ), controllable transitions among dissipative solitons, unstable states, and NPSs are revealed. At Eₛₐₜ = 13.5 pJ, a NPS with a pulse duration of 3.6 ps is achieved, exhibiting 99.96 % fidelity to the ideal sinc function. Furthermore, the spectrum of NPSs evolves from concave-topped to flat-topped and ultimately to convex-topped with increasing Eₛₐₜ, unveiling a gain-driven nonlinear shaping mechanism dominated by self-phase modulation (SPM) and spectral filtering. These findings demonstrate that gain not only defines the accessible pulse regimes but also serves as a precise control parameter for tailoring both the temporal and spectral characteristics of NPSs. This establishes a pathway for high-fidelity NPSs generation directly from the oscillator and lay a theoretical foundation for advanced applications in photonic neural networks, quantum key distribution, and ultrafast laser science.

奈奎斯特脉冲以其正弦和矩形的谱线为特征，在高速光通信和光存储中得到了广泛的应用。产生奈奎斯特脉冲的传统方法通常依赖于有源频谱调制，通常会偏离理想的矩形频谱，并且会增加系统的复杂性。在这项工作中，从被动锁模光纤振荡器直接产生奈奎斯特脉冲孤子（nps）进行了数值演示。系统地研究了脉冲的动态演变，突出了增益在调节脉冲动力学中的关键作用。通过调节增益饱和能（Eₛl），揭示了耗散孤子、不稳定态和nps之间的可控跃迁。在Eₛ= 13.5 pJ时，实现了脉冲持续时间为3.6 ps的NPS，与理想sinc函数的保真度为99.96%。此外，nps的光谱随着Eₛ- l的增加，从凹顶到平顶再到凸顶，揭示了由自相位调制（SPM）和光谱滤波主导的增益驱动的非线性整形机制。这些发现表明，增益不仅定义了可达的脉冲范围，而且还作为精确的控制参数，用于裁剪nps的时间和光谱特性。这为直接从振荡器产生高保真nps建立了途径，为光子神经网络、量子密钥分发和超快激光科学的高级应用奠定了理论基础。

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引用次数: 0

Stationary pattern emergence in reaction–diffusion systems with unstable equilibria and Lienard kinetics 具有不稳定平衡和Lienard动力学的反应扩散系统中出现的固定模式

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-03 DOI: 10.1016/j.chaos.2025.117719

José L. Aragón

Pattern formation in reaction–diffusion systems is traditionally understood to arise from Turing instabilities when a stable equilibrium becomes unstable due to diffusion. Here we investigate reaction–diffusion systems with chemical kinetics transformable into Lienard-type equations, enabling stationary spatial patterns to emerge when the equilibrium point of the diffusion-less system is unstable and loses stability through a Hopf bifurcation. Three examples were fully explored: the BVAM model, the van der Pol oscillator, and the Briggs–Rauscher chemical kinetics. Linear and nonlinear analyses were performed for the three cases, and the predictions were tested using one- and two-dimensional numerical simulations. Our results demonstrate that oscillatory chemical kinetics can generate stable spatial patterns, expanding the classical framework of pattern formation. These findings suggest new avenues for exploring pattern formation in oscillatory systems and have implications for understanding oscillation quenching by diffusion in chemical and biological contexts.

传统上认为，反应扩散系统中的模式形成是由图灵不稳定性引起的，当一个稳定的平衡由于扩散而变得不稳定时。本文研究了化学动力学可转换为lienard型方程的反应扩散系统，当无扩散系统的平衡点不稳定并通过Hopf分岔失去稳定性时，可以出现平稳的空间模式。三个例子充分探讨：BVAM模型，范德波尔振荡器，和布里格斯-劳舍尔化学动力学。对这三种情况进行了线性和非线性分析，并利用一维和二维数值模拟对预测结果进行了验证。我们的研究结果表明，振荡化学动力学可以产生稳定的空间模式，扩展了模式形成的经典框架。这些发现为探索振荡系统的模式形成提供了新的途径，并对理解化学和生物学背景下扩散引起的振荡猝灭具有重要意义。

引用次数: 0

Proof of an open problem on the maximization of the Euler–Sombor index in chemical unicyclic graphs 化学单环图中Euler-Sombor指标最大化的一个开放问题的证明

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-03 DOI: 10.1016/j.chaos.2025.117720

Sultan Ahmad , Kinkar Chandra Das

A topological index is a numerical property of a molecular graph that reflects its structural features. The geometric interpretation and the capacity of topological indices to distinguish between molecular structures have made them an important focus of current research. In this line, numerous degree-based indices have been introduced in recent years. Among these, the Euler–Sombor index, derived from Euler’s approximation formula for the perimeter of an ellipse, has attracted particular attention. For a graph

Γ

, the Euler–Sombor index (abbreviated as

E U

–index) is defined as:

E U (Γ) = \sum_{ν_{i} ν_{j} \in E (Γ)} \sqrt{d_{i}^{2} + d_{j}^{2} + d_{i} d_{j}},

where

E (Γ)

denotes the edge set and

d_{i}

is the degree of a vertex

ν_{i}

in

Γ

. Quite recently, Khanra and Das (2025) posed a problem on characterizing chemical unicyclic graphs with respect to the

E U

–index in terms of graph order, addressing both the maximizing and minimizing cases. This problem was subsequently discussed by Das et al. (in press), where the minimizing case was completely resolved, while the maximizing case remained open. In this paper, we present a complete characterization of the maximizing problem and identify the corresponding extremal graphs.

拓扑指数是分子图反映其结构特征的数值性质。拓扑指数的几何解释和区分分子结构的能力使其成为当前研究的一个重要热点。在这方面，近年来引入了许多基于学位的指数。其中，欧拉-松博尔指数（Euler - sombor index）引起了特别的关注，该指数是由欧拉近似公式导出的椭圆周长。对于图Γ， Euler-Sombor指数（简称EU - index）定义为：EU（Γ）=∑νiνj∈E（Γ）di2+dj2+didj，其中E（Γ）表示边集，di表示Γ中顶点νi的度。最近，Khanra和Das（2025）提出了一个关于eu -指数的化学单环图的图序特征问题，解决了最大化和最小化的情况。Das等人随后讨论了这个问题（已发表），其中最小化的情况完全解决了，而最大化的情况仍未解决。在本文中，我们给出了最大化问题的完整表征，并确定了相应的极值图。

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引用次数: 0

Vibration responses of graphene oxide powder reinforced hyperelastic neo-Hookean cylindrical shells under time-varying axial velocity 时变轴向速度下氧化石墨烯粉末增强超弹性新hookean圆柱壳的振动响应

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-03 DOI: 10.1016/j.chaos.2025.117706

J. Zhang , Y.F. Zhang , W. Zhang

Hyperelastic materials are widely used in the fields of aerospace and medical engineering, such as oil transfer hoses and drug delivery systems, both of which are one kind of soft materials. The graphene oxide powder (GOP) with the advantages of high strength is commonly used as a reinforcing material in soft structures. The vibration behaviors of GOP reinforced hyperelastic cylindrical shells under time-varying axial velocity are investigated for the first time. Considering both geometric and material nonlinearities and three distribution types of the GOP including Hyper-UD, Hyper-O and Hyper-X, the dynamic equations of the hyperelastic cylindrical shells under time-varying axial velocity are constructed by the shell theory, hyperelastic neo-Hookean model, Halpin-Tsai model and Lagrange equations. The natural frequencies of the linearized system for the GOP reinforced hyperelastic cylindrical shells under constant axial velocity are presented under varying parameters. The harmonic balance method (HBM) is employed to derive the relations between the frequencies and amplitudes under different parameter values. Numerical results indicate that certain values of axially constant velocity, weight fractions of GOP and structural parameters can enable the shell to escape the unstable regions. The hardening behaviors appear in this system. Furthermore, the time-varying axial velocity significantly affects the vibration responses of the GOP reinforced hyperelastic cylindrical shells, the incorporation of GOP makes the chaotic motions appear the hysteresis phenomena.

超弹性材料广泛应用于航空航天和医学工程领域，如输油软管和药物输送系统，都是一种软质材料。氧化石墨烯粉体具有高强度的优点，是软结构中常用的增强材料。首次研究了轴向速度时变作用下GOP加筋超弹性圆柱壳的振动特性。考虑几何非线性和材料非线性，以及超弹性轴向速度的三种分布类型（Hyper-UD、Hyper-O和Hyper-X），利用壳理论、超弹性neo-Hookean模型、Halpin-Tsai模型和Lagrange方程，建立了轴向速度随时间变化的超弹性圆柱壳动力学方程。给出了等轴向速度下GOP加筋超弹性圆柱壳线性化系统在不同参数下的固有频率。采用谐波平衡法（HBM）推导了不同参数值下频率与幅值的关系。数值结果表明，一定的轴向等速、GOP重量分数和结构参数可以使壳脱离不稳定区域。该体系出现了硬化行为。随时间变化的轴向速度对GOP加筋超弹性圆柱壳的振动响应有显著影响，GOP的加入使混沌运动出现滞后现象。

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引用次数: 0

An evolutionary paradox: Disease-driven social avoidance promotes cooperation while also increasing infection 进化悖论：疾病驱动的社会回避促进了合作，同时也增加了感染

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-03 DOI: 10.1016/j.chaos.2025.117716

Liang Zhang , Jianwei Wang

Social interactions constitute the fundamental environment underlying both the evolution of cooperation and disease transmission. However, traditional research has largely treated these two processes as isolated systems, overlooking their potential interplay. In reality, when individuals can voluntarily choose to engage in social interactions, any change in social participation — triggered by either process — reshapes the contact network and creates a feedback loop, weaving the two processes into a tightly coupled coevolutionary system. In this work, we propose a coupled dynamics framework for the evolution of cooperation and disease transmission under voluntary participation, with two variant models. The results reveal a thought-provoking paradox: moderate, disease-driven social avoidance, while intuitively expected to suppress disease transmission, can unexpectedly lead to an expanded infection size while simultaneously enhancing the level of cooperation. This paradox arises due to the unique, highly mixed spatial distribution between cooperators and loners. Further analysis shows that this phenomenon is robust across a wide range of parameters but can be naturally alleviated when people’s sensitivity to perceiving the risk of disease transmission is sufficiently high. This study uncovers a subtle interplay between cooperation evolution and disease transmission in the real world, implying non-intuitive and potentially conflicting dynamics between cooperation and public health goals under certain conditions. Simultaneously, it highlights the importance of interdisciplinary joint modeling for understanding complex social systems.

社会互动构成了合作进化和疾病传播的基本环境。然而，传统研究在很大程度上将这两个过程视为孤立的系统，忽视了它们潜在的相互作用。实际上，当个人可以自愿选择参与社会互动时，社会参与的任何变化——由任何一个过程触发——都会重塑联系网络，并创造一个反馈循环，将这两个过程编织成一个紧密耦合的共同进化系统。在这项工作中，我们提出了一个耦合的动态框架，自愿参与下的合作和疾病传播的演变，有两个变体模型。研究结果揭示了一个发人深省的悖论：适度的、疾病驱动的社会回避，虽然直观上被认为会抑制疾病传播，但却意外地导致了感染规模的扩大，同时提高了合作水平。这种悖论的产生是由于合作者和独行侠之间独特的、高度混合的空间分布。进一步的分析表明，这种现象在广泛的参数范围内都是稳定的，但当人们对感知疾病传播风险的敏感性足够高时，这种现象可以自然缓解。本研究揭示了现实世界中合作进化与疾病传播之间的微妙相互作用，暗示了在某些条件下合作与公共卫生目标之间的非直觉和潜在冲突动态。同时，它强调了跨学科联合建模对理解复杂社会系统的重要性。

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引用次数: 0

From Chaos to compression: Emergent simplicity in semiclassical density matrices 从混沌到压缩：半经典密度矩阵中的突现简单性

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-03 DOI: 10.1016/j.chaos.2025.117702

A. Plastino , A.M. Kowalski

We revisit a semiclassical Hamiltonian model in which quantum operators

(\hat{x}, \hat{p})

are coupled to classical canonical variables

(A, P_{A})

, a framework relevant to strong-field interactions such as meson pair production. Conventional wisdom associates chaos with proliferating disorder and growing entropy. Here we show the opposite: in the strict classical limit, the chaotic dynamics is represented not by a maximally mixed density matrix but by a pure state, indicating drastic information compression. Rather than amplifying complexity, the irregular detail of chaotic trajectories collapses into a minimal representation, exemplifying the principle of emergent simplicity. This result highlights how semiclassical systems can transform apparent disorder into compact information structures, providing a concrete and transparent realization of compression mechanisms that also appear in statistical mechanics and many-body quantum systems.

我们重新审视了一个半经典哈密顿模型，其中量子算子（x,p）与经典正则变量（a，PA）耦合，这是一个与强场相互作用（如介子对产生）相关的框架。传统智慧将混乱与无序扩散和熵增长联系在一起。在这里，我们展示了相反的情况：在严格的经典极限下，混沌动力学不是由最大混合密度矩阵表示，而是由纯状态表示，表明剧烈的信息压缩。而不是放大复杂性，混乱轨迹的不规则细节坍塌成一个最小的表示，举例说明了涌现的简单性原则。这一结果突出了半经典系统如何将明显的无序转化为紧凑的信息结构，为统计力学和多体量子系统中也出现的压缩机制提供了具体和透明的实现。

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引用次数: 0

Mean first passage time of the symmetric noisy voter model with arbitrary initial and boundary conditions 具有任意初始和边界条件的对称噪声选民模型的平均首次通过时间

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-02 DOI: 10.1016/j.chaos.2025.117649

Rytis Kazakevičius, Aleksejus Kononovicius

Models of imitation and herding behavior often underestimate the role of individualistic actions and assume symmetric boundary conditions. However, real-world systems (e.g., electoral processes) frequently involve asymmetric boundaries. In this study, we explore how arbitrarily placed boundary conditions influence the mean first passage time in the symmetric noisy voter model, and how individualistic behavior amplifies this asymmetry. We derive exact analytical expressions for mean first passage time that accommodate any initial condition and two types of boundary configurations: (i) both boundaries absorbing, and (ii) one absorbing and one reflective. In both scenarios, mean first passage time exhibits a clear asymmetry with respect to the initial condition, shaped by the boundary placement and the rate of independent transitions. Symmetry in mean first passage time emerges only when absorbing boundaries are equidistant from the midpoint. Additionally, we show that Kramers’ law holds in both configurations when the rate of independent transitions is large. Our analytical results are in excellent agreement with numerical simulations, reinforcing the robustness of our findings.

模仿和羊群行为模型往往低估了个人主义行为的作用，并假设对称的边界条件。然而，现实世界的系统（例如，选举过程）经常涉及不对称边界。在本研究中，我们探讨了在对称噪声选民模型中，任意放置的边界条件如何影响平均首次通过时间，以及个人主义行为如何放大这种不对称性。我们导出了适用于任何初始条件和两种类型边界配置的平均首次通过时间的精确解析表达式：(i)两种边界吸收，（ii）一种吸收和一种反射。在这两种情况下，平均首次通过时间相对于初始条件表现出明显的不对称性，这是由边界位置和独立跃迁速率决定的。平均首次通过时间的对称性只有在吸收边界与中点等距时才会出现。此外，我们还证明，当独立跃迁速率较大时，Kramers定律在两种构型中都成立。我们的分析结果与数值模拟非常一致，加强了我们发现的稳健性。

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引用次数: 0

Co-evolution mechanism of peer effect and dynamic networks in multi-agent systems from a dual-game perspective 双博弈视角下多智能体系统同伴效应与动态网络的协同进化机制

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-02 DOI: 10.1016/j.chaos.2025.117631

Fengzhen Jiang , Yanfei Yang

This research develops a multi-agent evolutionary game model that integrates dynamic networks, the peer effect strategy update rule, and a dual-game framework including the Prisoner's Dilemma Game (PDG) and the Snowdrift Game (SDG). Unlike previous research that analyzes these elements separately, we systematically examine their synergistic effect on the evolution of cooperation. Through simulations on the Newman–Watts (NW) small-world network and the Barabási–Albert (BA) scale-free network, we reveal a novel cooperation-emergence mechanism: the synergy between network structural adaptation and peer effect substantially accelerates the diffusion of cooperation and stabilizes network evolution. The results reveal that the adjustment of network topology exerts a more significant impact on improving system stability compared with the strategy update, while peer effect provides necessary social reinforcement for cooperative clustering. After stabilization, residual defectors are predominantly isolated nodes, and SDG participants exhibit a higher survival rate. These findings enrich the theoretical understanding of cooperative dynamics in complex systems and offer guidance for promoting cooperation in real-world multi-agent environments.

本研究建立了一个整合动态网络、同伴效应策略更新规则和囚徒困境博弈（PDG）和雪堆博弈（SDG）双重博弈框架的多智能体进化博弈模型。不同于以往的研究分别分析这些因素，我们系统地考察了它们对合作演化的协同效应。通过对Newman-Watts （NW）小世界网络和Barabási-Albert （BA）无标度网络的模拟，揭示了一种新的合作-涌现机制：网络结构适应和同伴效应的协同作用显著加速了合作扩散，稳定了网络演化。结果表明，网络拓扑调整对提高系统稳定性的影响比策略更新更显著，而对等效应为协同聚类提供了必要的社会强化。稳定后，剩余的叛逃者主要是孤立的节点，可持续发展目标参与者表现出更高的存活率。这些发现丰富了复杂系统中合作动力学的理论认识，并为促进现实多智能体环境中的合作提供了指导。

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引用次数: 0

μ-Stability of (η1,η2)-pseudo almost periodic solution for octonion-valued fuzzy BAM cellular neural networks with mixed delays 混合时滞八元数模糊BAM细胞神经网络(η 1, η 2) -伪概周期解的μ -稳定性

IF 5.6 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2025-12-02 DOI: 10.1016/j.chaos.2025.117665

Manel Amdouni

Complex-valued and quaternion-valued neural networks often struggle with inherently complex, high-dimensional data, such as hyperspectral images, volumetric data, or physical systems with seven degrees of freedom. To tackle these challenges, researchers have looked for a more advanced and expressive model: octonion-valued neural networks. Octonions extend quaternions and are represented as an eight-dimensional number system, providing a rich algebraic structure that enables the modeling and processing of highly multidimensional signals while capturing intricate relationships between components.

In this paper, we propose a class of fuzzy BAM cellular neural networks with mixed delays, where the inputs, outputs, weights, and biases are all octonions. Firstly, we will demonstrate the existence and uniqueness of a

(η_{1}, η_{2})

-pseudo almost periodic solution using the exponential dichotomy of linear equations, related inequalities, new sufficient conditions, and the contraction mapping fixed point theorem. Secondly, we will establish the

μ

-stability of octonion-valued fuzzy BAM cellular neural networks by constructing new Lyapunov functions. Finally, we will provide an example to illustrate the feasibility and effectiveness of our main results. Notably, we will employ the non-decomposition method to obtain the existence, uniqueness, and

μ

-stability of octonion-valued fuzzy BAM cellular neural networks.

复值和四元数值神经网络经常与固有的复杂高维数据作斗争，例如高光谱图像、体积数据或具有七个自由度的物理系统。为了应对这些挑战，研究人员寻找了一种更先进、更有表现力的模型：八元数值神经网络。八元数扩展四元数，并表示为一个八维数字系统，提供了丰富的代数结构，使建模和处理高度多维信号，同时捕获组件之间的复杂关系。在本文中，我们提出了一类具有混合延迟的模糊BAM细胞神经网络，其中输入、输出、权重和偏差都是八元数。首先，我们将利用线性方程的指数二分法、相关不等式、新的充分条件和收缩映射不动点定理证明(η1,η2)-伪概周期解的存在唯一性。其次，我们将通过构造新的Lyapunov函数来建立八元数值模糊BAM细胞神经网络的μ稳定性。最后，我们将提供一个实例来说明我们的主要结果的可行性和有效性。值得注意的是，我们将采用非分解方法来获得八元数值模糊BAM细胞神经网络的存在性、唯一性和μ稳定性。

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引用次数: 0