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</script></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img alt="ModelingToolkit.jl logo" src="../../assets/logo.png"/></a><div class="docs-package-name"><span class="docs-autofit">ModelingToolkit.jl</span></div><form action="../../search/" class="docs-search"><input class="docs-search-query" id="documenter-search-query" name="q" placeholder="Search docs" type="text"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><span class="tocitem">Symbolic Modeling Tutorials</span><ul><li><a class="tocitem" href="../ode_modeling/">Composing Ordinary Differential Equations</a></li><li><a class="tocitem" href="../acausal_components/">Acausal Component-Based Modeling the RC Circuit</a></li><li><a class="tocitem" href="../higher_order/">Automatic Transformation of Nth Order ODEs to 1st Order ODEs</a></li><li><a class="tocitem" href="../tearing_parallelism/">Exposing More Parallelism By Tearing Algebraic Equations in ODESystems</a></li><li class="is-active"><a class="tocitem" href="">Modeling Nonlinear Systems</a></li><li><a class="tocitem" href="../optimization/">Modeling Optimization Problems</a></li><li><a class="tocitem" href="../stochastic_diffeq/">Modeling with Stochasticity</a></li><li><a class="tocitem" href="../nonlinear_optimal_control/">Nonlinear Optimal Control</a></li></ul></li><li><span class="tocitem">ModelingToolkitize Tutorials</span><ul><li><a class="tocitem" href="../../mtkitize_tutorials/modelingtoolkitize/">Automatically Accelerating ODEProblem Code</a></li><li><a class="tocitem" href="../../mtkitize_tutorials/modelingtoolkitize_index_reduction/">Automated Index Reduction of DAEs</a></li></ul></li><li><span class="tocitem">Basics</span><ul><li><a class="tocitem" href="../../basics/AbstractSystem/">The AbstractSystem Interface</a></li><li><a class="tocitem" href="../../basics/ContextualVariables/">Contextual Variable Types</a></li><li><a class="tocitem" href="../../basics/Composition/">Composing Models and Building Reusable Components</a></li><li><a class="tocitem" href="../../basics/Validation/">Model Validation and Units</a></li><li><a class="tocitem" href="../../basics/DependencyGraphs/">Dependency Graphs</a></li><li><a class="tocitem" href="../../basics/FAQ/">Frequently Asked Questions</a></li></ul></li><li><span class="tocitem">System Types</span><ul><li><a class="tocitem" href="../../systems/ODESystem/">ODESystem</a></li><li><a class="tocitem" href="../../systems/SDESystem/">SDESystem</a></li><li><a class="tocitem" href="../../systems/JumpSystem/">JumpSystem</a></li><li><a class="tocitem" href="../../systems/NonlinearSystem/">NonlinearSystem</a></li><li><a class="tocitem" href="../../systems/OptimizationSystem/">OptimizationSystem</a></li><li><a class="tocitem" href="../../systems/ControlSystem/">ControlSystem</a></li><li><a class="tocitem" href="../../systems/ReactionSystem/">ReactionSystem</a></li><li><a class="tocitem" href="../../systems/PDESystem/">PDESystem</a></li></ul></li><li><a class="tocitem" href="../../comparison/">Comparison of ModelingToolkit vs Equation-Based Modeling Languages</a></li><li><a class="tocitem" href="../../internals/">Internal Details</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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Symbolic Modeling Tutorials</a></li><li class="is-active"><a href="">Modeling Nonlinear Systems</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="">Modeling Nonlinear Systems</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/SciML/ModelingToolkit.jl/blob/master/docs/src/tutorials/nonlinear.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" href="#" id="documenter-settings-button" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" href="#" id="documenter-sidebar-button"></a></div></header><article class="content" id="documenter-page"><h1 id="Modeling-Nonlinear-Systems-1"><a class="docs-heading-anchor" href="#Modeling-Nonlinear-Systems-1">Modeling Nonlinear Systems</a><a class="docs-heading-anchor-permalink" href="#Modeling-Nonlinear-Systems-1" 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'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 => 1.0,
y => 0.0,
z => 0.0]
ps = [
σ => 10.0
ρ => 26.0
β => 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></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></p><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div><p></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> on <span class="colophon-date" title="Wednesday 17 March 2021 15:51">Wednesday 17 March 2021</span>. Using Julia version 1.5.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></HTML>