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{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-12T18:28:30","documenter_version":"1.8.0"}} |
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@article{toda1967vibration, | ||
title={Vibration of a chain with nonlinear interaction}, | ||
author={Toda, Morikazu}, | ||
journal={Journal of the Physical Society of Japan}, | ||
volume={22}, | ||
number={2}, | ||
pages={431--436}, | ||
year={1967}, | ||
publisher={The Physical Society of Japan} | ||
} | ||
|
||
@book{arnold1978mathematical, | ||
title={Mathematical methods of classical mechanics}, | ||
author={Arnold, Vladimir Igorevich}, | ||
volume={60}, | ||
year={1978}, | ||
series={Graduate Texts in Mathematics}, | ||
publisher={Springer Verlag}, | ||
address={Berlin} | ||
} | ||
|
||
@book{hairer2006geometric, | ||
title={Geometric Numerical integration: structure-preserving algorithms for ordinary differential equations}, | ||
author={Hairer, Ernst and Lubich, Christian and Wanner, Gerhard}, | ||
year={2006}, | ||
publisher={Springer}, | ||
address={Berlin} | ||
} | ||
|
||
@article{bajars2023locally, | ||
title={Locally-symplectic neural networks for learning volume-preserving dynamics}, | ||
author={Baj{\=a}rs, J{\=a}nis}, | ||
journal={Journal of Computational Physics}, | ||
volume={476}, | ||
pages={111911}, | ||
year={2023}, | ||
publisher={Elsevier} | ||
} | ||
|
||
@article{buchfink2023symplectic, | ||
title={Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder}, | ||
author={Buchfink, Patrick and Glas, Silke and Haasdonk, Bernard}, | ||
journal={SIAM Journal on Scientific Computing}, | ||
volume={45}, | ||
number={2}, | ||
pages={A289--A311}, | ||
year={2023}, | ||
publisher={SIAM} | ||
} |
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<!DOCTYPE html> | ||
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>ABC Flow · GeometricProblems.jl</title><meta name="title" content="ABC Flow · GeometricProblems.jl"/><meta property="og:title" content="ABC Flow · GeometricProblems.jl"/><meta property="twitter:title" content="ABC Flow · GeometricProblems.jl"/><meta name="description" content="Documentation for GeometricProblems.jl."/><meta property="og:description" content="Documentation for GeometricProblems.jl."/><meta property="twitter:description" content="Documentation for GeometricProblems.jl."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link 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href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script><link href="../assets/extra_styles.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">GeometricProblems.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../diagnostics/">Diagnostics</a></li><li class="is-active"><a class="tocitem" href>ABC Flow</a><ul class="internal"><li><a class="tocitem" href="#Library-functions"><span>Library functions</span></a></li></ul></li><li><a class="tocitem" href="../coupled_harmonic_oscillator/">Coupled Harmonic Oscillator</a></li><li><a class="tocitem" href="../double_pendulum/">Double Pendulum</a></li><li><a class="tocitem" href="../harmonic_oscillator/">Harmonic Oscillator</a></li><li><a class="tocitem" href="../henon_heiles/">Hénon-Heiles System</a></li><li><a class="tocitem" href="../kepler_problem/">Kepler Problem</a></li><li><a class="tocitem" href="../linear_wave/">Linear Wave Equation</a></li><li><a class="tocitem" href="../lorenz_attractor/">Lorenz Attractor</a></li><li><a class="tocitem" href="../lotka_volterra_2d/">Lotka-Volterra 2d</a></li><li><a class="tocitem" href="../lotka_volterra_3d/">Lotka-Volterra 3d</a></li><li><a class="tocitem" href="../lotka_volterra_4d/">Lotka-Volterra 4d</a></li><li><a class="tocitem" href="../massless_charged_particle/">Massless Charged Particle</a></li><li><a class="tocitem" href="../pendulum/">Mathematical Pendulum</a></li><li><a class="tocitem" href="../nonlinear_oscillators/">Nonlinear Oscillators</a></li><li><a class="tocitem" href="../point_vortices/">Point Vortices</a></li><li><a class="tocitem" href="../inner_solar_system/">Inner Solar System</a></li><li><a class="tocitem" href="../outer_solar_system/">Outer Solar System</a></li><li><a class="tocitem" href="../rigid_body/">Rigid body</a></li><li><a class="tocitem" href="../toda_lattice/">Toda Lattice</a></li><li><span class="tocitem">Initial conditions</span><ul><li><a class="tocitem" href="../initial_condition/">bump</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>ABC Flow</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>ABC Flow</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaGNI/GeometricProblems.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/JuliaGNI/GeometricProblems.jl/blob/main/docs/src/abc_flow.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="ABC-Flow"><a class="docs-heading-anchor" href="#ABC-Flow">ABC Flow</a><a id="ABC-Flow-1"></a><a class="docs-heading-anchor-permalink" href="#ABC-Flow" title="Permalink"></a></h1><p>The ABC flow (see [<a href="#hairer2006geometric">1</a>]) is described by a divergence-free differential equation whose flow strongly depends on the initial condition. </p><pre><code class="language-julia hljs">using GeometricIntegrators: integrate, ImplicitMidpoint | ||
using GeometricProblems.ABCFlow | ||
using Plots | ||
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||
ensemble_solution = integrate(odeensemble(), ImplicitMidpoint()) | ||
|
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p = plot() | ||
for solution in ensemble_solution | ||
plot!(p, solution.q[:, 1], solution.q[:, 2], solution.q[:, 3]) | ||
end | ||
p</code></pre><img src="f66208c8.svg" alt="Example block output"/><h2 id="Library-functions"><a class="docs-heading-anchor" href="#Library-functions">Library functions</a><a id="Library-functions-1"></a><a class="docs-heading-anchor-permalink" href="#Library-functions" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="GeometricProblems.ABCFlow" href="#GeometricProblems.ABCFlow"><code>GeometricProblems.ABCFlow</code></a> — <span class="docstring-category">Module</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><p><strong>ABC Flow</strong></p><p class="math-container">\[\begin{aligned} | ||
\dot{x} = A\sin(z) + C\cos(y) \\ | ||
\dot{y} = B\sin(x) + A\cos(z) \\ | ||
\dot{z} = C\sin(y) + B\cos(x) | ||
\end{aligned}\]</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/JuliaGNI/GeometricProblems.jl/blob/c7794d8add481f4b879f73b59ea894d579a12c8b/src/abc_flow.jl#L1-L11">source</a></section></article><div class="citation canonical"><dl><dt>[1]</dt><dd><div id="hairer2006geometric">E. Hairer, C. Lubich and G. Wanner. <em>Geometric Numerical integration: structure-preserving algorithms for ordinary differential equations</em> (Springer, Berlin, 2006).</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../diagnostics/">« Diagnostics</a><a class="docs-footer-nextpage" href="../coupled_harmonic_oscillator/">Coupled Harmonic Oscillator »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Thursday 12 December 2024 18:28">Thursday 12 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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