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on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Modeling-Nonlinear-Systems"><a class="docs-heading-anchor" href="#Modeling-Nonlinear-Systems">Modeling Nonlinear Systems</a><a id="Modeling-Nonlinear-Systems-1"></a><a class="docs-heading-anchor-permalink" href="#Modeling-Nonlinear-Systems" title="Permalink"></a></h1><p>In this example we will go one step deeper and showcase the direct function generation capabilities in ModelingToolkit.jl to build nonlinear systems. Let&#39;s say we wanted to solve for the steady state of the previous ODE. This is the nonlinear system defined by where the derivatives are zero. We use (unknown) variables for our nonlinear system.</p><pre><code class="language-julia">using ModelingToolkit, NonlinearSolve

@variables x y z
@parameters σ ρ β

# Define a nonlinear system
eqs = [0 ~ σ*(y-x),
       0 ~ x*(ρ-z)-y,
       0 ~ x*y - β*z]
ns = NonlinearSystem(eqs, [x,y,z], [σ,ρ,β])

guess = [x =&gt; 1.0,
         y =&gt; 0.0,
         z =&gt; 0.0]

ps = [
      σ =&gt; 10.0
      ρ =&gt; 26.0
      β =&gt; 8/3
      ]

prob = NonlinearProblem(ns,guess,ps)
sol = solve(prob,NewtonRaphson())</code></pre><p>We can similarly ask to generate the <code>NonlinearProblem</code> with the analytical Jacobian function:</p><pre><code class="language-julia">prob = NonlinearProblem(ns,guess,ps,jac=true)
sol = solve(prob,NewtonRaphson())</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../tearing_parallelism/">« Exposing More Parallelism By Tearing Algebraic Equations in ODESystems</a><a class="docs-footer-nextpage" href="../optimization/">Modeling Optimization Problems »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.3 on <span class="colophon-date" title="Tuesday 13 July 2021 11:09">Tuesday 13 July 2021</span>. Using Julia version 1.6.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>