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These transformations allow for symbolically changing the representation of the model to problems that are easier to numerically solve. One simple to demonstrate transformation is the <code>ode_order_lowering</code> transformation that sends an Nth order ODE to a 1st order ODE.</p><p>To see this, let's define a second order riff on the Lorenz equations. We utilize the derivative operator twice here to define the second order:</p><pre><code class="language-julia">using ModelingToolkit, OrdinaryDiffEq

@parameters t σ ρ β
@variables x(t) y(t) z(t)
D = Differential(t)

eqs = [D(D(x)) ~ σ*(y-x),
       D(y) ~ x*(ρ-z)-y,
       D(z) ~ x*y - β*z]

sys = ODESystem(eqs)</code></pre><p>Note that we could've used an alternative syntax for 2nd order, i.e. <code>D = Differential(t)^2</code> and then <code>E(x)</code> would be the second derivative, and this syntax extends to <code>N</code>-th order. Also, we can use <code>*</code> or <code>∘</code> to compose <code>Differential</code>s, like <code>Differential(t) * Differential(x)</code>.</p><p>Now let's transform this into the <code>ODESystem</code> of first order components. We do this by simply calling <code>ode_order_lowering</code>:</p><pre><code class="language-julia">sys = ode_order_lowering(sys)</code></pre><p>Now we can directly numerically solve the lowered system. Note that, following the original problem, the solution requires knowing the initial condition for <code>x'</code>, and thus we include that in our input specification:</p><pre><code class="language-julia">u0 = [D(x) =&gt; 2.0,
      x =&gt; 1.0,
      y =&gt; 0.0,
      z =&gt; 0.0]

p  = [σ =&gt; 28.0,
      ρ =&gt; 10.0,
      β =&gt; 8/3]

tspan = (0.0,100.0)
prob = ODEProblem(sys,u0,tspan,p,jac=true)
sol = solve(prob,Tsit5())
using Plots; plot(sol,vars=(x,y))</code></pre><p><img alt="Lorenz2" src="https://user-images.githubusercontent.com/1814174/79118645-744eb580-7d5c-11ea-9c37-13c4efd585ca.png"/></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../acausal_components/">« Acausal Component-Based Modeling the RC Circuit</a><a class="docs-footer-nextpage" href="../tearing_parallelism/">Exposing More Parallelism By Tearing Algebraic Equations in ODESystems »</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 9 June 2021 21:01">Wednesday 9 June 2021</span>. Using Julia version 1.6.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></HTML>