Journal of Computational Dynamics最新文献_第8页

Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor 平面二次哈密顿系统的Kahan离散几何。Ⅱ。具有线性泊松张量的系统

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Journal of Computational Dynamics Pub Date : 2018-11-13 DOI: 10.3934/jcd.2019020

M. Petrera, Y. Suris

{"title":"Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor","authors":"M. Petrera, Y. Suris","doi":"10.3934/jcd.2019020","DOIUrl":"https://doi.org/10.3934/jcd.2019020","url":null,"abstract":"Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper `Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation' by P. van der Kamp et al., it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form $ell(x,y)$, let $B_1,B_2$ be any two distinct points on the line $ell(x,y)=-c$, and let $B_3,B_4$ be any two distinct points on the line $ell(x,y)=c$. Set $B_0=tfrac{1}{2}(B_1+B_3)$ and $B_5=tfrac{1}{2}(B_2+B_4)$; these points lie on the line $ell(x,y)=0$. Finally, let $B_infty$ be the point at infinity on this line. Let $mathfrak E$ be the pencil of conics with the base points $B_1,B_2,B_3,B_4$. Then the composition of the $B_infty$-switch and of the $B_0$-switch on the pencil $mathfrak E$ is the Kahan discretization of a Hamiltonian vector field $f=ell(x,y)begin{pmatrix}partial H/partial y -partial H/partial x end{pmatrix}$ with a quadratic Hamilton function $H(x,y)$. This birational map $Phi_f:mathbb C P^2dashrightarrowmathbb C P^2$ has three singular points $B_0,B_2,B_4$, while the inverse map $Phi_f^{-1}$ has three singular points $B_1,B_3,B_5$.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"5 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2018-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76367615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 5

Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production $ M_t/M_t/ 1 $分布的连续逼近及其在生产中的应用

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Journal of Computational Dynamics Pub Date : 2018-07-18 DOI: 10.3934/jcd.2020010

D. Armbruster, Simone Gottlich, S. Knapp

引用次数: 1

Geometrical properties of the mean-median map 中位数图的几何性质

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Journal of Computational Dynamics Pub Date : 2018-06-26 DOI: 10.3934/jcd.2020004

Jonathan Hoseana, F. Vivaldi

引用次数: 3

Generalised Manin transformations and QRT maps 广义Manin变换和QRT映射

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Journal of Computational Dynamics Pub Date : 2018-06-14 DOI: 10.3934/JCD.2021009

Peter H. van der Kamp, D. McLaren, G. Quispel

引用次数: 10

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations 控制Fokker-Planck和Liouville-von Neumann方程的模型简化

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Journal of Computational Dynamics Pub Date : 2017-06-26 DOI: 10.3934/jcd.2020001

P. Benner, T. Breiten, C. Hartmann, B. Schmidt

引用次数: 1

Evaluating the accuracy of the dynamic mode decomposition 评价动态模态分解的精度

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Journal of Computational Dynamics Pub Date : 2016-11-21 DOI: 10.3934/jcd.2020002

Hao Zhang, Scott T. M. Dawson, C. Rowley, Eric A. Deem, L. Cattafesta

{"title":"Evaluating the accuracy of the dynamic mode decomposition","authors":"Hao Zhang, Scott T. M. Dawson, C. Rowley, Eric A. Deem, L. Cattafesta","doi":"10.3934/jcd.2020002","DOIUrl":"https://doi.org/10.3934/jcd.2020002","url":null,"abstract":"Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation to the Koopman operator, an infinite-dimensional linear operator defined for (nonlinear) dynamical systems. This work proposes a new criterion to estimate the accuracy of DMD on a mode-by-mode basis, by estimating how closely each individual DMD eigenfunction approximates the corresponding Koopman eigenfunction. This approach does not require any prior knowledge of the system dynamics or the true Koopman spectral decomposition. The method may be applied to extensions of DMD (i.e., extended/kernel DMD), which are applicable to a wider range of problems. The accuracy criterion is first validated against the true error with a synthetic system for which the true Koopman spectral decomposition is known. We next demonstrate how this proposed accuracy criterion can be used to assess the performance of various choices of kernel when using the kernel method for extended DMD. Finally, we show that our proposed method successfully identifies modes of high accuracy when applying DMD to data from experiments in fluids, in particular particle image velocimetry of a cylinder wake and a canonical separated boundary layer.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"28 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86131916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 14

Approximation of Lyapunov functions from noisy data 李雅普诺夫函数在噪声数据中的近似

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Journal of Computational Dynamics Pub Date : 2016-01-07 DOI: 10.3934/jcd.2020003

P. Giesl, B. Hamzi, M. Rasmussen, K. Webster

引用次数: 29

Algebraic structure of aromatic B-series 芳香b系的代数结构

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Journal of Computational Dynamics Pub Date : 2015-05-08 DOI: 10.3934/jcd.2019010

Geir Bogfjellmo

引用次数: 11