{"title":"论多维仓本模型的数值积分","authors":"Marcus A. M. de Aguiar","doi":"10.1007/s13538-024-01493-z","DOIUrl":null,"url":null,"abstract":"<p>The Kuramoto model, describing the synchronization dynamics of coupled oscillators, has been generalized in many ways over the past years. One recent extension of the model replaces the oscillators, originally characterized by a single phase, by particles with <span>\\(D-1\\)</span> internal phases, represented by a point on the surface of the unit D-sphere. Particles are then more easily represented by <i>D</i>-dimensional unit vectors than by <span>\\(D-1\\)</span> spherical angles. However, numerical integration of the state equations should ensure that the propagated vectors remain unit and that particles rotate on the sphere as predicted by the dynamical equations. As discussed in (Lee et al. in Journal of Statistical Mechanics: Theory and Experiment 2023(4):043403, 2023), integration of the three-dimensional Kuramoto model using Euler’s method with time step <span>\\(\\Delta t\\)</span> not only changes the norm of the vectors but produces a small rotation of the particles around the wrong axis. Importantly, the error in the axis’ direction does not vanish in the limit <span>\\(\\Delta t \\rightarrow 0\\)</span>. Therefore, instead of displacing the unit vectors in the direction of the velocity, one should perform a sequence of direct small rotations, as dictated by the equations of motion. This keeps the particles on the sphere at all times, ensuring exact norm preservation, and rotates the particles around the proper axis for small <span>\\(\\Delta t\\)</span> (Lee et al. in Journal of Statistical Mechanics: Theory and Experiment 2023(4):043403, 2023). Here, I propose an alternative way to do such integration by rotations in 3D that can be generalized to more dimensions using Cayley-Hamilton’s theorem. Explicit formulas are provided for 2, 3, and 4 dimensions. I also compare the results with the fourth-order Runge–Kutta method, which seems to provide accurate results even requiring renormalization of the vectors after each integration step.</p>","PeriodicalId":499,"journal":{"name":"Brazilian Journal of Physics","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Numerical Integration of the Multidimensional Kuramoto Model\",\"authors\":\"Marcus A. 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As discussed in (Lee et al. in Journal of Statistical Mechanics: Theory and Experiment 2023(4):043403, 2023), integration of the three-dimensional Kuramoto model using Euler’s method with time step <span>\\\\(\\\\Delta t\\\\)</span> not only changes the norm of the vectors but produces a small rotation of the particles around the wrong axis. Importantly, the error in the axis’ direction does not vanish in the limit <span>\\\\(\\\\Delta t \\\\rightarrow 0\\\\)</span>. Therefore, instead of displacing the unit vectors in the direction of the velocity, one should perform a sequence of direct small rotations, as dictated by the equations of motion. This keeps the particles on the sphere at all times, ensuring exact norm preservation, and rotates the particles around the proper axis for small <span>\\\\(\\\\Delta t\\\\)</span> (Lee et al. in Journal of Statistical Mechanics: Theory and Experiment 2023(4):043403, 2023). Here, I propose an alternative way to do such integration by rotations in 3D that can be generalized to more dimensions using Cayley-Hamilton’s theorem. Explicit formulas are provided for 2, 3, and 4 dimensions. I also compare the results with the fourth-order Runge–Kutta method, which seems to provide accurate results even requiring renormalization of the vectors after each integration step.</p>\",\"PeriodicalId\":499,\"journal\":{\"name\":\"Brazilian Journal of Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2024-05-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Brazilian Journal of Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1007/s13538-024-01493-z\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s13538-024-01493-z","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
过去几年中,描述耦合振荡器同步动力学的仓本模型以多种方式得到了推广。该模型最近的一个扩展是用具有 \(D-1\)内部相位的粒子取代了最初以单一相位为特征的振荡器,后者由单位 D 球表面上的一个点表示。这样,粒子就更容易用 D 维单位矢量来表示,而不是用 \(D-1\) 球角来表示。然而,状态方程的数值积分应确保传播矢量保持单位,并确保粒子按照动力学方程的预测在球面上旋转。正如(李等人在《统计力学学报:理论与实验 2023(2023)》中)所讨论的那样:Theory and Experiment 2023(4):043403, 2023)中所讨论的那样,使用欧拉法以时间步长(\Δ t\)对三维仓本模型进行积分,不仅会改变矢量的法线,还会产生粒子绕错误轴线的微小旋转。重要的是,轴线方向的误差在极限((\Delta t \rightarrow 0))下不会消失。因此,我们不应该沿着速度方向移动单位向量,而是应该按照运动方程的要求,直接进行一连串小的旋转。这样就能使粒子始终保持在球面上,确保精确的规范保持,并使粒子绕适当的小轴(\Δ t\ )旋转(Lee 等人在 Journal of Statistical Mechanics:理论与实验》2023(4):043403,2023)。在此,我提出了一种在三维空间中通过旋转进行积分的替代方法,该方法可以使用 Cayley-Hamilton 定理推广到更多维度。我提供了 2 维、3 维和 4 维的明确公式。我还将结果与四阶 Runge-Kutta 方法进行了比较,后者似乎能提供精确的结果,即使需要在每一步积分后对向量进行重正化。
On the Numerical Integration of the Multidimensional Kuramoto Model
The Kuramoto model, describing the synchronization dynamics of coupled oscillators, has been generalized in many ways over the past years. One recent extension of the model replaces the oscillators, originally characterized by a single phase, by particles with \(D-1\) internal phases, represented by a point on the surface of the unit D-sphere. Particles are then more easily represented by D-dimensional unit vectors than by \(D-1\) spherical angles. However, numerical integration of the state equations should ensure that the propagated vectors remain unit and that particles rotate on the sphere as predicted by the dynamical equations. As discussed in (Lee et al. in Journal of Statistical Mechanics: Theory and Experiment 2023(4):043403, 2023), integration of the three-dimensional Kuramoto model using Euler’s method with time step \(\Delta t\) not only changes the norm of the vectors but produces a small rotation of the particles around the wrong axis. Importantly, the error in the axis’ direction does not vanish in the limit \(\Delta t \rightarrow 0\). Therefore, instead of displacing the unit vectors in the direction of the velocity, one should perform a sequence of direct small rotations, as dictated by the equations of motion. This keeps the particles on the sphere at all times, ensuring exact norm preservation, and rotates the particles around the proper axis for small \(\Delta t\) (Lee et al. in Journal of Statistical Mechanics: Theory and Experiment 2023(4):043403, 2023). Here, I propose an alternative way to do such integration by rotations in 3D that can be generalized to more dimensions using Cayley-Hamilton’s theorem. Explicit formulas are provided for 2, 3, and 4 dimensions. I also compare the results with the fourth-order Runge–Kutta method, which seems to provide accurate results even requiring renormalization of the vectors after each integration step.
期刊介绍:
The Brazilian Journal of Physics is a peer-reviewed international journal published by the Brazilian Physical Society (SBF). The journal publishes new and original research results from all areas of physics, obtained in Brazil and from anywhere else in the world. Contents include theoretical, practical and experimental papers as well as high-quality review papers. Submissions should follow the generally accepted structure for journal articles with basic elements: title, abstract, introduction, results, conclusions, and references.