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 <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · TaylorIntegration.jl</title><meta name="title" content="Home · TaylorIntegration.jl"/><meta property="og:title" content="Home · TaylorIntegration.jl"/><meta property="twitter:title" content="Home · TaylorIntegration.jl"/><meta name="description" content="Documentation for TaylorIntegration.jl."/><meta property="og:description" content="Documentation for TaylorIntegration.jl."/><meta property="twitter:description" content="Documentation for TaylorIntegration.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script data-outdated-warner src="assets/warner.js"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 href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" 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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></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>TaylorIntegration.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 class="is-active"><a class="tocitem" href>Home</a></li><li><span class="tocitem">Background</span><ul><li><a class="tocitem" href="taylor_method/">Taylor&#39;s method</a></li><li><a class="tocitem" href="lyapunov_spectrum/">Lyapunov spectrum</a></li><li><a class="tocitem" href="jet_transport/">Jet transport</a></li></ul></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="simple_example/">Infinity in finite time</a></li><li><a class="tocitem" href="kepler/">The Kepler problem</a></li><li><a class="tocitem" href="lorenz_lyapunov/">Lyapunov spectrum of Lorenz system</a></li><li><a class="tocitem" href="pendulum/">Jet transport: the simple pendulum</a></li><li><a class="tocitem" href="root_finding/">Poincaré maps</a></li><li><a class="tocitem" href="common/">Interoperability with <code>DifferentialEquations.jl</code></a></li></ul></li><li><a class="tocitem" href="taylorize/">Optimizing: <code>@taylorize</code></a></li><li><a class="tocitem" href="api/">API</a></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>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/PerezHz/TaylorIntegration.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/PerezHz/TaylorIntegration.jl/blob/main/docs/src/index.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="TaylorIntegration.jl"><a class="docs-heading-anchor" href="#TaylorIntegration.jl">TaylorIntegration.jl</a><a id="TaylorIntegration.jl-1"></a><a class="docs-heading-anchor-permalink" href="#TaylorIntegration.jl" title="Permalink"></a></h1><p>ODE integration using Taylor&#39;s method, and more, in <a href="http://julialang.org">Julia</a>.</p><hr/><h3 id="Authors"><a class="docs-heading-anchor" href="#Authors">Authors</a><a id="Authors-1"></a><a class="docs-heading-anchor-permalink" href="#Authors" title="Permalink"></a></h3><ul><li><a href="https://www.linkedin.com/in/perezhz">Jorge A. Pérez</a>, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)</li><li><a href="http://www.cicc.unam.mx/~benet/">Luis Benet</a>, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)</li></ul><h3 id="License"><a class="docs-heading-anchor" href="#License">License</a><a id="License-1"></a><a class="docs-heading-anchor-permalink" href="#License" title="Permalink"></a></h3><p><code>TaylorIntegration</code> is licensed under the <a href="https://github.com/PerezHz/TaylorIntegration.jl/blob/main/LICENSE.md">MIT &quot;Expat&quot; license</a>.</p><h3 id="Installation"><a class="docs-heading-anchor" href="#Installation">Installation</a><a id="Installation-1"></a><a class="docs-heading-anchor-permalink" href="#Installation" title="Permalink"></a></h3><p>TaylorIntegration.jl is a <a href="http://pkg.julialang.org">registered package</a>, and is simply installed by running</p><pre><code class="language-julia hljs">pkg&gt; add TaylorIntegration</code></pre><h3 id="Supporting-and-citing"><a class="docs-heading-anchor" href="#Supporting-and-citing">Supporting and citing</a><a id="Supporting-and-citing-1"></a><a class="docs-heading-anchor-permalink" href="#Supporting-and-citing" title="Permalink"></a></h3><p>This package is developed as part of academic research. If you would like to help supporting it, please star the repository as such metrics may help us secure funding. If you use this software, we would be grateful if you could cite our work as follows (Bibtex entry can be found <a href="https://zenodo.org/record/2562353/export/hx">here</a>):</p><pre><code class="nohighlight hljs">J.A. Pérez-Hernández and L. Benet
 TaylorIntegration.jl: Taylor Integration in Julia
 https://github.com/PerezHz/TaylorIntegration.jl
-DOI:[10.5281/zenodo.2562352](https://doi.org/10.5281/zenodo.2562352)</code></pre><h3 id="Acknowledgments"><a class="docs-heading-anchor" href="#Acknowledgments">Acknowledgments</a><a id="Acknowledgments-1"></a><a class="docs-heading-anchor-permalink" href="#Acknowledgments" title="Permalink"></a></h3><p>We acknowledge financial support from the DGAPA-PAPIIT (UNAM) grant IG-100616. LB acknowledges support through a <em>Cátedra Marcos Moshinsky</em> (2013).</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="taylor_method/">Taylor&#39;s method »</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 1.5.0 on <span class="colophon-date" title="Thursday 1 August 2024 15:48">Thursday 1 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+DOI:[10.5281/zenodo.2562352](https://doi.org/10.5281/zenodo.2562352)</code></pre><h3 id="Acknowledgments"><a class="docs-heading-anchor" href="#Acknowledgments">Acknowledgments</a><a id="Acknowledgments-1"></a><a class="docs-heading-anchor-permalink" href="#Acknowledgments" title="Permalink"></a></h3><p>We acknowledge financial support from the DGAPA-PAPIIT (UNAM) grant IG-100616. LB acknowledges support through a <em>Cátedra Marcos Moshinsky</em> (2013).</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="taylor_method/">Taylor&#39;s method »</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 1.5.0 on <span class="colophon-date" title="Thursday 1 August 2024 15:57">Thursday 1 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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         \delta y
       \end{array}
     \right).
-\end{aligned}\]</p><p>The first terms in the expressions for <span>$\mathbf{x}_1$</span> and <span>$\mathbf{x}_2$</span> above correspond to the result of an Euler integration step using the initial conditions only. The other terms are the (linear) corrections which involve the small displacements <span>$\delta x$</span> and <span>$\delta y$</span>.</p><p>In general, for differential equations involving non-linear terms, the resulting expansions in <span>$\delta x$</span> and <span>$\delta y$</span> will reflect aspects of the non-linearities of the ODEs. Clearly, jet transport techniques allow to address stability properties beyond the linear case, though memory constraints may play a role. See <a href="../pendulum/#pendulum">this example</a> illustrating the implementation for the simple pendulum, and <a href="../root_finding/#jettransport2">this one</a> illustrating the construction of a Poincaré map with Jet transport techniques.</p><h3 id="refsJT"><a class="docs-heading-anchor" href="#refsJT">References</a><a id="refsJT-1"></a><a class="docs-heading-anchor-permalink" href="#refsJT" title="Permalink"></a></h3><p>[1] D. Pérez-Palau, Josep J. Masdemont, Gerard Gómez, 2015, Celest. Mech. Dyn. Astron. 123, 239.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../lyapunov_spectrum/">« Lyapunov spectrum</a><a class="docs-footer-nextpage" href="../simple_example/">Infinity in finite time »</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 1.5.0 on <span class="colophon-date" title="Thursday 1 August 2024 15:47">Thursday 1 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><p>The first terms in the expressions for <span>$\mathbf{x}_1$</span> and <span>$\mathbf{x}_2$</span> above correspond to the result of an Euler integration step using the initial conditions only. The other terms are the (linear) corrections which involve the small displacements <span>$\delta x$</span> and <span>$\delta y$</span>.</p><p>In general, for differential equations involving non-linear terms, the resulting expansions in <span>$\delta x$</span> and <span>$\delta y$</span> will reflect aspects of the non-linearities of the ODEs. Clearly, jet transport techniques allow to address stability properties beyond the linear case, though memory constraints may play a role. See <a href="../pendulum/#pendulum">this example</a> illustrating the implementation for the simple pendulum, and <a href="../root_finding/#jettransport2">this one</a> illustrating the construction of a Poincaré map with Jet transport techniques.</p><h3 id="refsJT"><a class="docs-heading-anchor" href="#refsJT">References</a><a id="refsJT-1"></a><a class="docs-heading-anchor-permalink" href="#refsJT" title="Permalink"></a></h3><p>[1] D. Pérez-Palau, Josep J. Masdemont, Gerard Gómez, 2015, Celest. Mech. Dyn. Astron. 123, 239.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../lyapunov_spectrum/">« Lyapunov spectrum</a><a class="docs-footer-nextpage" href="../simple_example/">Infinity in finite time »</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 1.5.0 on <span class="colophon-date" title="Thursday 1 August 2024 15:56">Thursday 1 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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