混沌动力学

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[1] arXiv:2508.02786 [中文pdf, pdf, html, 其他]

标题： 混沌过渡与锥形弹球 显示英文标题

标题： Transition to chaos with conical billiards

Lara Braverman, David R. Nelson

评论： 19+8页，25图

主题： 混沌动力学 (nlin.CD) ; 软凝聚态物理 (cond-mat.soft)

我们借鉴了几何光学和经典弹道动力学的思想，考虑在圆锥体上以恒定速度运动的粒子轨迹，在倾斜平面与圆锥体相交形成的椭圆边界上发生镜面反射，倾斜角为$\gamma$。 我们研究了动态作为$\gamma$和圆锥体缺陷角$\chi$的函数，该缺陷角控制顶点的尖锐程度，其中正高斯曲率的点源集中于此。 我们在 ($\gamma, \chi$) 平面上发现了区域，根据初始条件，轨迹可能（A）采样整个圆锥底面并避开顶点区域；（B）仅采样底面的一部分，同时再次避开顶点；或（C）更均匀地采样整个圆锥表面，暗示遍历性。 未倾斜圆锥的特殊情况仅显示类型 A 的轨迹，在接近顶点的距离形成环形焦散线。 然而，我们观察到当$\chi$和$\gamma$足够大时，出现了由类型（C）轨迹主导的复杂混沌动力学转变。 一个总结由镜面反射中断的测地线段轨迹的庞加莱映射，为可视化混沌转变提供了一种强有力的手段。 随后，我们分析了圆锥弹道系统通往混沌的路径与其他保面积保守映射之间的相似性和差异性。 显示英文摘要

We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane, with tilt angle $\gamma$. We explore the dynamics as a function of $\gamma$ and the cone deficit angle $\chi$ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the ($\gamma, \chi$) plane where, depending on the initial conditions, either (A) the trajectories sample the entire cone base and avoid the apex region; (B) sample only a portion of the base region while again avoiding the apex; or (C) sample the entire cone surface much more uniformly, suggestive of ergodicity. The special case of an untilted cone displays only type A trajectories which form a ring caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently large $\chi$ and $\gamma$. A Poincar\'e map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps.

[2] arXiv:2508.03228 [中文pdf, pdf, 其他]

标题： 非线性动力学的稀疏识别应用于声学大物体的悬浮 显示英文标题

标题： Sparse identification of nonlinear dynamics applied to the levitation of acoustically large objects

Mehdi Akbarzadeh, Ben Halkon, Sebastian Oberst

主题： 混沌动力学 (nlin.CD)

许多关于声辐射力的研究集中在表征声场的行为上。 然而，物体的动态响应，特别是那些大于波长的物体，仍研究不足。 在此，我们通过推导受声辐射力和外部激励作用下被捕获球形物体的非线性运动方程来填补这一空白，其中Gorkov公式无法提供准确的结果。 使用稀疏识别非线性动力系统（SINDy），我们从通过Gorkov公式和外部激励获得的解析时间序列数据中推导出相应的非线性运动方程，并且与解析解相比，系数值的误差小于0.05%。我们使用TinyLev悬浮器进行实验，施加外部激励以生成声学大粒子的时间序列。 然后，应用SINDy从实验数据中重建控制方程，以观察外部激励幅度如何影响声学大物体的动力学行为。 我们的研究结果表明，SINDy可以有效地用作从复杂数据中推导控制方程的工具，以改进和精炼理论发展；在本例中，对于Gorkov公式无法提供准确预测的声学大物体。 显示英文摘要

Many studies on acoustic radiation forces focus on characterizing the behavior of acoustic fields. However, the dynamic response of objects, particularly those larger than the wavelength, remains underexplored. Here we bridge this gap by deriving nonlinear equations of motion for a trapped spherical object under acoustic radiation forces and external excitation, where the Gorkov formulation fails to provide accurate results. Using Sparse Identification of Nonlinear Dynamical Systems (SINDy), we derive the corresponding nonlinear equation of motion from analytical time series data obtained through the Gorkov formulation and external excitation for acoustically small objects, and which recovers the governing equation with less than 0.05% error in coefficient values compared to the analytical solution.. We conduct experiments with the TinyLev levitator with external excitation to generate time series for acoustically large particles. Then, SINDy is applied to reconstruct governing equations from experimental data to see how external excitation amplitude influences the dynamics of acoustically large objects. Our findings demonstrate that the SINDy can effectively be used as a tool for deriving governing equations from complex data to improve and refine theoretical developments; in the present case, for acoustically large objects, where the Gorkov formulation fails to provide an accurate prediction.

交叉提交 (展示 2 之 2 条目 )

[3] arXiv:2508.01801 (交叉列表自 nlin.AO) [中文pdf, pdf, html, 其他]

标题： 非线性科学中的主题简介 显示英文标题

标题： Introduction to Focus Issue: Topics in Nonlinear Science

Elizabeth Bradley, Adilson E. Motter, Louis M. Pecora

评论： 特别专辑：非线性科学专题：献给大卫·K·坎贝尔80岁生日

期刊参考： 混沌 35，070402 (2025)

主题： 适应性与自组织系统 (nlin.AO) ; 无序系统与神经网络 (cond-mat.dis-nn)

非线性科学在《Chaos》期刊创刊以来的35年里取得了显著发展。 本期专题，献给其创刊主编David K. Campbell80岁生日，汇集了关于有影响力主题的精选贡献，其中许多主题得到了Campbell自身研究计划和领导角色的推动。 这些主题包括网络动力学、机器学习、量子和材料系统、混沌与分形、局域态以及生命系统中的新现象和方法发展，文献综述、原创贡献和未来研究的展望之间有着良好的平衡。 显示英文摘要

Nonlinear science has evolved significantly over the 35 years since the launch of the journal Chaos. This Focus Issue, dedicated to the 80th Birthday of its founding editor-in-chief, David K. Campbell, brings together a selection of contributions on influential topics, many of which were advanced by Campbell's own research program and leadership role. The topics include new phenomena and method development in the realms of network dynamics, machine learning, quantum and material systems, chaos and fractals, localized states, and living systems, with a good balance of literature review, original contributions, and perspectives for future research.

[4] arXiv:2508.02935 (交叉列表自 nlin.PS) [中文pdf, pdf, html, 其他]

标题： 使用动力系统理论量化渐近Lenia中的复杂性 显示英文标题

标题： Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia

Ivan Yevenko, Hiroki Kojima, Chrystopher L. Nehaniv

评论： 8页，包括参考文献。3张图表。已提交并被ALIFE 2025会议接受，但尚未发表

主题： 模式形成与孤子 (nlin.PS) ; 混沌动力学 (nlin.CD) ; 细胞自动机与格子气体 (nlin.CG)

连续细胞自动机（CCAs）从离散查找表演变为连续偏微分方程（PDE）公式，以寻找新的复杂形式。 尽管在定性行为方面有所创新，分析方法却落后了，这强化了涌现复杂性难以简单解释的观点。 在本文中，我们证明了渐近Lenia的PDE公式可以通过动力系统理论进行严格的分析。 我们将对称性、吸引子、李雅普诺夫指数和分形维数的概念应用于数学上表征复杂行为。 我们的贡献包括：（1）对四种不同的解类（孤子、旋转器、周期性和混沌模式）的数学解释，（2）存在具有分形维数$>4$的全局吸引子的条件，（3）将Kaplan-Yorke维数识别为CCAs的有效复杂度度量，（4）一种高效的开源实现，用于计算CCAs的李雅普诺夫指数和协变李雅普诺夫向量。 最后，我们确定了在更广泛的CCAs类别中实现复杂行为的最小属性集。 这个框架为理解和测量人工生命系统中的复杂性提供了基础。 显示英文摘要

Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts of symmetries, attractors, Lyapunov exponents, and fractal dimensions to characterize complex behaviors mathematically. Our contributions include: (1) a mathematical explanation for the four distinct solution classes (solitons, rotators, periodic and chaotic patterns), (2) conditions for the existence of a global attractor with fractal dimension $>4$, (3) identification of Kaplan-Yorke dimension as an effective complexity measure for CCAs, and (4) an efficient open-source implementation for calculating Lyapunov exponents and the covariant Lyapunov vectors for CCAs. We conclude by identifying the minimal set of properties that enable complex behavior in a broader class of CCAs. This framework provides a foundation for understanding and measuring complexity in artificial life systems.

替换提交 (展示 2 之 2 条目 )

[5] arXiv:2308.01514 (替换) [中文pdf, pdf, html, 其他]

标题： 具有布罗迪间距分布的2 X 2相关随机矩阵模型类 显示英文标题

标题： A class of 2 X 2 correlated random-matrix models with Brody spacing distribution

Jamal Sakhr

评论： 期刊接受的稿件（见下面的期刊参考文献）

期刊参考： 物理学年鉴 480, 170080 (2025)

主题： 数学物理 (math-ph) ; 混沌动力学 (nlin.CD) ; 量子物理 (quant-ph)

引入了一类2x2随机矩阵模型，其中Brody分布是精确的本征值间距分布。 矩阵元素由受约束的有限和组成，该有限和是指数随机变量的不同幂次，这些幂次取决于Brody参数。 此处引入的随机矩阵与高斯正交系综（GOE）的矩阵有三个重要不同之处：矩阵元素不是独立同分布（即非IID）的，也不是高斯分布的，并且矩阵不一定是实数和/或对称的。 前两个特性来自于放弃经典的独立性假设，第三个特性则来自于在GOE构造中放弃量子力学条件。 特别是，本类模型中的厄米特性条件是本征值为实数的充分但非必要条件，并未被施加。 因此，具有实数或复数本征值的复数非厄米特2x2随机矩阵也可能具有介于泊松类和Wigner类之间的间距分布。 提供了不同类型随机矩阵的数值示例，包括具有实数或共轭复数本征值的复数对称矩阵。 讨论了各种推广和扩展，包括一种简单的修改，可以实现其他本征值间距统计类之间的交叉过渡。 作为新示例，介绍了半泊松和Ginibre间距统计之间的交叉过渡情况。 显示英文摘要

A class of 2x2 random-matrix models is introduced for which the Brody distribution is the exact eigenvalue spacing distribution. The matrix elements consist of constrained finite sums of an exponential random variable raised to various powers that depend on the Brody parameter. The random matrices introduced here differ from those of the Gaussian Orthogonal Ensemble (GOE) in three important ways: the matrix elements are not independent and identically distributed (i.e., not IID) nor Gaussian-distributed, and the matrices are not necessarily real and/or symmetric. The first two features arise from dropping the classical independence assumption, and the third feature stems from dropping the quantum-mechanical conditions imposed in the construction of the GOE. In particular, the hermiticity condition, which in the present class of models, is a sufficient but not necessary condition for the eigenvalues to be real, is not imposed. Consequently, complex non-Hermitian 2x2 random matrices with real or complex eigenvalues can also have spacing distributions that are intermediate between those of the Poisson and Wigner classes. Numerical examples are provided for different types of random matrices, including complex-symmetric matrices with real or complex-conjugate eigenvalues. Various generalizations and extensions are discussed including a simple modification that effectuates cross-over transitions between other classes of eigenvalue spacing statistics. The case of a cross-over transition between semi-Poisson and Ginibre spacing statistics is presented as a novel example.

[6] arXiv:2507.10863 (替换) [中文pdf, pdf, 其他]

标题： 非平衡双温度$(T_x, T_y)$诺斯-霍弗细胞模型中的混沌 显示英文标题

标题： Chaos in Nonequilibrium Two-Temperature $(T_x, T_y)$ Nosé-Hoover Cell Models

Hesam Arabzadeh, Carol Griswold Hoover, William Graham Hoover

主题： 统计力学 (cond-mat.stat-mech) ; 混沌动力学 (nlin.CD)

我们重新研究了一个嵌入在二维周期性2x2单元中的双温度Nosé-Hoover游动粒子，该单元在$(x,y) = (\pm 1, \pm 1)$处有四个光滑的排斥角落，以探索各向异性的热浴引起的混沌。 该模型在x和y方向上使用单独的热浴，从而实现对平衡的受控偏离。 通过积分完整的六维运动方程并计算完整的李雅普诺夫谱，我们确认了混沌，并以高数值精度量化了相空间收缩。 总收缩率被解释为熵产生，随着热浴各向异性非线性增长，并遵循超二次幂律，$\Lambda\propto -\delta^{2.44}$，偏离了线性响应理论，$\Lambda\propto -\delta^{2}$。 我们将这种行为与线性响应预测的四次修正进行比较，并讨论两种拟合的实证和理论意义。 我们的结果表明，即使在最小驱动系统中，非线性耗散标度也会自然出现。 近似的Kaplan-Yorke维数揭示了一个分形吸引子，随着$|T_x - T_y|$的增加而集中。 动量统计显示在强驱动下表现出显著的非高斯行为。 尽管该模型具有耗散性，但仍保持严格的时间可逆性，为微观可逆性与宏观熵产生共存提供了一个教学丰富的例子。 显示英文摘要

We revisit a two-temperature Nos\'e-Hoover wanderer particle embedded in a two-dimensional periodic 2x2 cell with four smooth repulsive corners at $(x,y) = (\pm 1, \pm 1)$ to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction with high numerical precision. The total contraction rate, interpreted as entropy production, grows nonlinearly with the thermostat anisotropy and follows a superquadratic power law, $\Lambda\propto -\delta^{2.44}$, deviating from linear-response theory, $\Lambda\propto -\delta^{2}$. We compare this behavior to a quartic correction to the linear-response prediction and discuss both fits in light of their empirical and theoretical implications. Our results demonstrate that nonlinear dissipation scaling emerges naturally even in minimal driven systems. The approximate Kaplan-Yorke dimension reveals a fractal attractor that concentrates as $|T_x - T_y|$ increases. Momentum statistics show significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.