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TutorialDeIntegracionNumerica4.html
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TutorialDeIntegracionNumerica4.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>TutorialDeIntegracionNumerica4</title>
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<h1 id="Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-con-forzamiento-temporal-estocástico">Tutorial de solución de ecuaciones diferenciales ordinarias con forzamiento temporal estocástico<a class="anchor-link" href="#Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-con-forzamiento-temporal-estocástico">¶</a></h1><h3 id="Lourdes-Martín-Aguilar,-César-Flores-López-&-Marco-Herrera-Valdez">Lourdes Martín Aguilar, César Flores López & Marco Herrera Valdez<a class="anchor-link" href="#Lourdes-Martín-Aguilar,-César-Flores-López-&-Marco-Herrera-Valdez">¶</a></h3><h4 id="Facultad-de-Ciencias,-Universidad-Nacional-Autónoma-de-México">Facultad de Ciencias, Universidad Nacional Autónoma de México<a class="anchor-link" href="#Facultad-de-Ciencias,-Universidad-Nacional-Autónoma-de-México">¶</a></h4>
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<p>Importamos los módulos necesarios</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sc</span>
<span class="kn">import</span> <span class="nn">matplotlib.pylab</span> <span class="k">as</span> <span class="nn">gr</span>
<span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="k">import</span> <span class="n">odeint</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<p>Definimos un diccionario</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="o">=</span><span class="nb">dict</span><span class="p">()</span>
<span class="n">p</span><span class="o">=</span><span class="p">{</span><span class="s2">"a"</span><span class="p">:</span><span class="mf">1.0</span><span class="p">,</span> <span class="s2">"tau"</span><span class="p">:</span><span class="mf">1.0</span><span class="p">,</span><span class="s2">"x0"</span><span class="p">:</span><span class="mf">0.5</span><span class="p">}</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"timeMax"</span><span class="p">]</span><span class="o">=</span><span class="mf">10.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"timeStep"</span><span class="p">]</span><span class="o">=</span><span class="mi">1</span><span class="n">e</span><span class="o">-</span><span class="mi">3</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">]</span><span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">p</span><span class="p">[</span><span class="s2">"timeMax"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s2">"timeStep"</span><span class="p">]);</span>
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<p>Definimos nuestra función</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">linearForcedEq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">):</span>
<span class="n">dx</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"a"</span><span class="p">]</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">]</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">](</span><span class="n">t</span><span class="p">)</span>
<span class="k">return</span> <span class="n">dx</span>
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<p>Se aplica el forzamiento estocástico a la función en dos modalidades (normal y uniforme).</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.3</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mf">0.0</span>
<span class="n">o0</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearForcedEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">],</span><span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
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<p>Proceso estocástico con distribución uniforme alrededor de 0</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Proceso estocástico con distribución uniforme</span>
<span class="n">uUnif</span><span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span> <span class="o">+</span> <span class="n">sc</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">]))</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sc</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">xp</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">fp</span><span class="o">=</span><span class="n">uUnif</span><span class="p">)</span>
<span class="n">oUnif</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearForcedEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">],</span><span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
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<p>Proceso estocástico con distribución normal alrededor de 0</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">uNorm</span><span class="o">=</span> <span class="mf">0.3</span><span class="o">*</span><span class="n">sc</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">]))</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sc</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">xp</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">fp</span><span class="o">=</span><span class="n">uNorm</span><span class="p">)</span>
<span class="n">oNorm</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearForcedEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">],</span><span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="c1">#</span>
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<p>Se grafican los tres resultados: no estocástico (gris), estocástico uniforme (rojo) y estocástico normal(azul).</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">gr</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
<span class="n">gr</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">o0</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">10.0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">r"$(t,x(t))$"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">oUnif</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">r"$(t,x_</span><span class="si">{unif}</span><span class="s2">(t))$"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">oNorm</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">r"$(t,x_</span><span class="si">{norm}</span><span class="s2">(t))$"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">ion</span><span class="p">();</span> <span class="n">gr</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">]</span><span class="o">=</span><span class="mf">1.5</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="mf">0.0</span>
<span class="n">o0</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearForcedEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">],</span><span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="n">uUnif</span><span class="o">=</span> <span class="o">-</span><span class="mf">0.5</span> <span class="o">+</span> <span class="n">sc</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">]))</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sc</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">xp</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">fp</span><span class="o">=</span><span class="n">uUnif</span><span class="p">)</span>
<span class="n">oUnif</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearForcedEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">],</span><span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="n">uNorm</span><span class="o">=</span> <span class="mf">0.3</span><span class="o">*</span><span class="n">sc</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">]))</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">:</span> <span class="n">sc</span><span class="o">.</span><span class="n">interp</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">xp</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">fp</span><span class="o">=</span><span class="n">uNorm</span><span class="p">)</span>
<span class="n">oNorm</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearForcedEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"x0"</span><span class="p">],</span><span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="c1">#</span>
<span class="n">gr</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
<span class="n">gr</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">o0</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'k'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">10.0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">r"$(t,x(t))$"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">oUnif</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">r"$(t,x_</span><span class="si">{unif}</span><span class="s2">(t))$"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">oNorm</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'b'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">r"$(t,x_</span><span class="si">{norm}</span><span class="s2">(t))$"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">ion</span><span class="p">();</span> <span class="n">gr</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
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