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README.md.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>README</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
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extensions: [],
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extensions: ["AMSmath.js","AMSsymbols.js","noErrors.js","noUndefined.js"]
},
showMathMenu: false
});
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<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js">
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<body class='markdown-preview'><h1>Laboratorium Specjalistyczne: rezonans stochastyczny</h1>
<h3>Mateusz Kaczorek, Dominik Stańczak</h3>
<h2>Problem</h2>
<p>Rezonans stochastyczny to zjawisko polegające na wzmacnianiu sygnału okresowego poprzez
dodanie do niego składowej stochastycznej. W ćwiczeniu zrealizowaliśmy ten efekt
poprzez symulację ruchu układu w potencjale bistabilnym z dodaną składową
czasową, sinusoidalną.</p>
<h2>Symulacja ruchu</h2>
<p>Bardzo masywna cząstka porusza się w potencjale oscylatora Duffinga z okresową siłą wymuszającą</p>
<span class="math"><script type="math/tex; mode=display">V(x) = a x^4 - b x^2 - c x\sin(\omega t)
</script></span>
<p>Z dodatkiem losowej siły o gaussowskim rozkładzie i wariancji <span class="math"><script type="math/tex">D</script></span>.</p>
<p>Ruch cząstki symulujemy przy użyciu zmodyfikowanego schematu Eulerowskiego dla równań stochastycznych
przy założeniu, że cząstka jest bardzo masywna i przetłumiona.
co ostatecznie pozwala nam zapisać schemat aktualizacji pozycji cząstki
między kolejnymi aktualizacjami jako:</p>
<span class="math"><script type="math/tex; mode=display">x_{n+1} = x_n - \nabla V(x_n) dt + \sqrt{2 D dt} \chi
</script></span>
<p>gdzie <span class="math"><script type="math/tex">\chi</script></span> jest zmienną losową z rozkładu normalnego, “normalną”
(o odchyleniu standardowym równym jeden i wartości średniej równej 0).</p>
<p>W tej symulacji całkowaliśmy równania dla następujących parametrów:</p>
<table>
<thead>
<tr>
<th>a</th>
<th>b</th>
<th>c</th>
<th>dt</th>
<th>Liczba kroków w okresie</th>
<th>Liczba okresów na symulację</th>
<th><span class="math"><script type="math/tex">\omega = 1/T</script></span></th>
</tr>
</thead>
<tbody>
<tr>
<td>0.25</td>
<td>0.5</td>
<td>0.25</td>
<td>0.005</td>
<td><span class="math"><script type="math/tex">2^{10}</script></span></td>
<td><span class="math"><script type="math/tex">2^{14}</script></span></td>
<td>0.1953125</td>
</tr>
</tbody>
</table>
<p>Poniższe symulacje przeprowadziliśmy dla 200 różnych wartości wariancji liczby losowej <span class="math"><script type="math/tex">D</script></span> z zakresu od <span class="math"><script type="math/tex">10^{-2}</script></span> do <span class="math"><script type="math/tex">10^1</script></span>, rozłożonych równomiernie w skali logarytmicznej.</p>
<h2>Analiza fourierowska</h2>
<p>W celu analizy trajektorie ruchu w każdej symulacji zamieniliśmy na wartości binarne:</p>
<ul>
<li><span class="math"><script type="math/tex">1</script></span> gdy <span class="math"><script type="math/tex">x \geq 0</script></span></li>
<li><span class="math"><script type="math/tex">-1</script></span> gdy <span class="math"><script type="math/tex">x < 0</script></span></li>
</ul>
<p>Następnie przeprowadziliśmy analizę fourierowską spektrum mocy danych, używając transformaty Fouriera. Przykłady kilku z tych transformat przedstawiamy na wykresach poniżej.</p>
<p>Transformaty wygładziliśmy, uśredniając 51 najbliższych punktów z kwadratu modułu transformaty Fouriera (widma mocy). Następnie odjęliśmy od widma mocy uśrednioną wartość i dla każdej symulacji odczytaliśmy wartość otrzymanej “względnej” amplitudy dla składowej odpowiadającej okresowej sile wymuszającej.</p>
<p><img src="/home/dominik/Code/Stochastic/plots/0.0174263338600965.png?v=1467183926156" alt="0.0174263338600965.png"></p>
<p><img src="/home/dominik/Code/Stochastic/plots/0.10234114021054533.png?v=1467183926157" alt="0.10234114021054533.png"></p>
<p><img src="/home/dominik/Code/Stochastic/plots/0.22739657523579276.png?v=1467183926158" alt="0.22739657523579276.png"></p>
<p><img src="/home/dominik/Code/Stochastic/plots/10.0.png?v=1467183926158" alt="10.0.png"></p>
<h2>Zebranie danych z 200 symulacji</h2>
<p>Wartości składowej odpowiadającej sile wymuszającej zebrane są, w skali log-log, na poniższym wykresie.</p>
<p><img src="/home/dominik/Code/Stochastic/ostateczny_wykres.png?v=1467183926158" alt="Wykres zbierający dane"></p>
<h2>Wnioski</h2>
<p>W trakcie realizacji ćwiczenia dowiedzieliśmy się wiele:</p>
<ul>
<li>o samym istnieniu zjawiska rezonansu stochastycznego, jego zastosowaniach np. w badaniu zmian klimatycznych oraz <a href="http://scitation.aip.org/content/aip/journal/pop/6/8/10.1063/1.873595">w niestabilnych plazmach</a></li>
<li>jak całkować numerycznie stochastyczne równania różniczkowe, uwzględniając - w pierwszym przybliżeniu - człony proporcjonalne do <span class="math"><script type="math/tex">\sqrt{dt}</script></span></li>
<li>jak radzić sobie z długimi szeregami czasowymi (obsługa formatu do zapisu danych liczbowych <code>.hdf5</code>)</li>
<li>jak analizować dane z szeregów czasowych poprzez analizę Fourierowską i jak radzić sobie z szumem w takich symulacjach</li>
</ul></body>
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