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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="boost_numeric_odeint.tutorial.ensembles_of_oscillators"></a><a class="link" href="ensembles_of_oscillators.html" title="Ensembles of oscillators">Ensembles |
| of oscillators</a> |
| </h3></div></div></div> |
| <p> |
| Another important high dimensional system of coupled ordinary differential |
| equations is an ensemble of <span class="emphasis"><em>N</em></span> all-to-all coupled phase |
| oscillators <a class="link" href="../literature.html#synchronization_pikovsky_rosenblum">[9] </a>. |
| It is defined as |
| </p> |
| <p> |
| <span class="emphasis"><em>dφ<sub>​k</sub> / dt = ω<sub>​k</sub> + ε / N Σ<sub>​j</sub> sin( φ<sub>​j</sub> - φ<sub>​k</sub> )</em></span> |
| </p> |
| <p> |
| The natural frequencies <span class="emphasis"><em>ω<sub>​i</sub></em></span> of each oscillator follow |
| some distribution and <span class="emphasis"><em>ε</em></span> is the coupling strength. We |
| choose here a Lorentzian distribution for <span class="emphasis"><em>ω<sub>​i</sub></em></span>. Interestingly |
| a phase transition can be observed if the coupling strength exceeds a critical |
| value. Above this value synchronization sets in and some of the oscillators |
| oscillate with the same frequency despite their different natural frequencies. |
| The transition is also called Kuramoto transition. Its behavior can be analyzed |
| by employing the mean field of the phase |
| </p> |
| <p> |
| <span class="emphasis"><em>Z = K e<sup>i Θ</sup> = 1 / N Σ<sub>​k</sub>e<sup>i φ<sub>​k</sub></sup></em></span> |
| </p> |
| <p> |
| The definition of the system function is now a bit more complex since we |
| also need to store the individual frequencies of each oscillator. |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">vector</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span> |
| |
| |
| <span class="identifier">pair</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">calc_mean_field</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">size</span><span class="special">();</span> |
| <span class="keyword">double</span> <span class="identifier">cos_sum</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">sin_sum</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">n</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">cos_sum</span> <span class="special">+=</span> <span class="identifier">cos</span><span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">);</span> |
| <span class="identifier">sin_sum</span> <span class="special">+=</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">);</span> |
| <span class="special">}</span> |
| <span class="identifier">cos_sum</span> <span class="special">/=</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">);</span> |
| <span class="identifier">sin_sum</span> <span class="special">/=</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">);</span> |
| |
| <span class="keyword">double</span> <span class="identifier">K</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span> <span class="identifier">cos_sum</span> <span class="special">*</span> <span class="identifier">cos_sum</span> <span class="special">+</span> <span class="identifier">sin_sum</span> <span class="special">*</span> <span class="identifier">sin_sum</span> <span class="special">);</span> |
| <span class="keyword">double</span> <span class="identifier">Theta</span> <span class="special">=</span> <span class="identifier">atan2</span><span class="special">(</span> <span class="identifier">sin_sum</span> <span class="special">,</span> <span class="identifier">cos_sum</span> <span class="special">);</span> |
| |
| <span class="keyword">return</span> <span class="identifier">make_pair</span><span class="special">(</span> <span class="identifier">K</span> <span class="special">,</span> <span class="identifier">Theta</span> <span class="special">);</span> |
| <span class="special">}</span> |
| |
| |
| <span class="keyword">struct</span> <span class="identifier">phase_ensemble</span> |
| <span class="special">{</span> |
| <span class="identifier">container_type</span> <span class="identifier">m_omega</span><span class="special">;</span> |
| <span class="keyword">double</span> <span class="identifier">m_epsilon</span><span class="special">;</span> |
| |
| <span class="identifier">phase_ensemble</span><span class="special">(</span> <span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">g</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">epsilon</span> <span class="special">=</span> <span class="number">1.0</span> <span class="special">)</span> |
| <span class="special">:</span> <span class="identifier">m_omega</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_epsilon</span><span class="special">(</span> <span class="identifier">epsilon</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">create_frequencies</span><span class="special">(</span> <span class="identifier">g</span> <span class="special">);</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">void</span> <span class="identifier">create_frequencies</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">g</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">rng</span><span class="special">;</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="identifier">cauchy</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">,</span> <span class="identifier">g</span> <span class="special">);</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">variate_generator</span><span class="special"><</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span><span class="special">&,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="special">></span> <span class="identifier">gen</span><span class="special">(</span> <span class="identifier">rng</span> <span class="special">,</span> <span class="identifier">cauchy</span> <span class="special">);</span> |
| <span class="identifier">generate</span><span class="special">(</span> <span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">m_omega</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">gen</span> <span class="special">);</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">void</span> <span class="identifier">set_epsilon</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">epsilon</span> <span class="special">)</span> <span class="special">{</span> <span class="identifier">m_epsilon</span> <span class="special">=</span> <span class="identifier">epsilon</span><span class="special">;</span> <span class="special">}</span> |
| |
| <span class="keyword">double</span> <span class="identifier">get_epsilon</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="keyword">const</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">m_epsilon</span><span class="special">;</span> <span class="special">}</span> |
| |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="identifier">container_type</span> <span class="special">&</span><span class="identifier">dxdt</span> <span class="special">,</span> <span class="keyword">double</span> <span class="comment">/* t */</span> <span class="special">)</span> <span class="keyword">const</span> |
| <span class="special">{</span> |
| <span class="identifier">pair</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">mean</span> <span class="special">=</span> <span class="identifier">calc_mean_field</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span> |
| <span class="keyword">for</span><span class="special">(</span> <span class="identifier">size_t</span> <span class="identifier">i</span><span class="special">=</span><span class="number">0</span> <span class="special">;</span> <span class="identifier">i</span><span class="special"><</span><span class="identifier">x</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">;</span> <span class="special">++</span><span class="identifier">i</span> <span class="special">)</span> |
| <span class="identifier">dxdt</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">m_omega</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">+</span> <span class="identifier">m_epsilon</span> <span class="special">*</span> <span class="identifier">mean</span><span class="special">.</span><span class="identifier">first</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span> <span class="identifier">mean</span><span class="special">.</span><span class="identifier">second</span> <span class="special">-</span> <span class="identifier">x</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">);</span> |
| <span class="special">}</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| Note, that we have used <span class="emphasis"><em>Z</em></span> to simplify the equations |
| of motion. Next, we create an observer which computes the value of <span class="emphasis"><em>Z</em></span> |
| and we record <span class="emphasis"><em>Z</em></span> for different values of <span class="emphasis"><em>ε</em></span>. |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">statistics_observer</span> |
| <span class="special">{</span> |
| <span class="keyword">double</span> <span class="identifier">m_K_mean</span><span class="special">;</span> |
| <span class="identifier">size_t</span> <span class="identifier">m_count</span><span class="special">;</span> |
| |
| <span class="identifier">statistics_observer</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> |
| <span class="special">:</span> <span class="identifier">m_K_mean</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">m_count</span><span class="special">(</span> <span class="number">0</span> <span class="special">)</span> <span class="special">{</span> <span class="special">}</span> |
| |
| <span class="keyword">template</span><span class="special"><</span> <span class="keyword">class</span> <span class="identifier">State</span> <span class="special">></span> |
| <span class="keyword">void</span> <span class="keyword">operator</span><span class="special">()(</span> <span class="keyword">const</span> <span class="identifier">State</span> <span class="special">&</span><span class="identifier">x</span> <span class="special">,</span> <span class="keyword">double</span> <span class="identifier">t</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">pair</span><span class="special"><</span> <span class="keyword">double</span> <span class="special">,</span> <span class="keyword">double</span> <span class="special">></span> <span class="identifier">mean</span> <span class="special">=</span> <span class="identifier">calc_mean_field</span><span class="special">(</span> <span class="identifier">x</span> <span class="special">);</span> |
| <span class="identifier">m_K_mean</span> <span class="special">+=</span> <span class="identifier">mean</span><span class="special">.</span><span class="identifier">first</span><span class="special">;</span> |
| <span class="special">++</span><span class="identifier">m_count</span><span class="special">;</span> |
| <span class="special">}</span> |
| |
| <span class="keyword">double</span> <span class="identifier">get_K_mean</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="keyword">const</span> <span class="special">{</span> <span class="keyword">return</span> <span class="special">(</span> <span class="identifier">m_count</span> <span class="special">!=</span> <span class="number">0</span> <span class="special">)</span> <span class="special">?</span> <span class="identifier">m_K_mean</span> <span class="special">/</span> <span class="keyword">double</span><span class="special">(</span> <span class="identifier">m_count</span> <span class="special">)</span> <span class="special">:</span> <span class="number">0.0</span> <span class="special">;</span> <span class="special">}</span> |
| |
| <span class="keyword">void</span> <span class="identifier">reset</span><span class="special">(</span> <span class="keyword">void</span> <span class="special">)</span> <span class="special">{</span> <span class="identifier">m_K_mean</span> <span class="special">=</span> <span class="number">0.0</span><span class="special">;</span> <span class="identifier">m_count</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="special">}</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| Now, we do several integrations for different values of <span class="emphasis"><em>ε</em></span> |
| and record <span class="emphasis"><em>Z</em></span>. The result nicely confirms the analytical |
| result of the phase transition, i.e. in our example the standard deviation |
| of the Lorentzian is 1 such that the transition will be observed at <span class="emphasis"><em>ε = |
| 2</em></span>. |
| </p> |
| <p> |
| </p> |
| <pre class="programlisting"><span class="keyword">const</span> <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">16384</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">dt</span> <span class="special">=</span> <span class="number">0.1</span><span class="special">;</span> |
| |
| <span class="identifier">container_type</span> <span class="identifier">x</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">);</span> |
| |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">rng</span><span class="special">;</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uniform_real</span><span class="special"><></span> <span class="identifier">unif</span><span class="special">(</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">2.0</span> <span class="special">*</span> <span class="identifier">M_PI</span> <span class="special">);</span> |
| <span class="identifier">boost</span><span class="special">::</span><span class="identifier">variate_generator</span><span class="special"><</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">mt19937</span><span class="special">&,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uniform_real</span><span class="special"><></span> <span class="special">></span> <span class="identifier">gen</span><span class="special">(</span> <span class="identifier">rng</span> <span class="special">,</span> <span class="identifier">unif</span> <span class="special">);</span> |
| |
| <span class="comment">// gamma = 1, the phase transition occurs at epsilon = 2</span> |
| <span class="identifier">phase_ensemble</span> <span class="identifier">ensemble</span><span class="special">(</span> <span class="identifier">n</span> <span class="special">,</span> <span class="number">1.0</span> <span class="special">);</span> |
| <span class="identifier">statistics_observer</span> <span class="identifier">obs</span><span class="special">;</span> |
| |
| <span class="keyword">for</span><span class="special">(</span> <span class="keyword">double</span> <span class="identifier">epsilon</span> <span class="special">=</span> <span class="number">0.0</span> <span class="special">;</span> <span class="identifier">epsilon</span> <span class="special"><</span> <span class="number">5.0</span> <span class="special">;</span> <span class="identifier">epsilon</span> <span class="special">+=</span> <span class="number">0.1</span> <span class="special">)</span> |
| <span class="special">{</span> |
| <span class="identifier">ensemble</span><span class="special">.</span><span class="identifier">set_epsilon</span><span class="special">(</span> <span class="identifier">epsilon</span> <span class="special">);</span> |
| <span class="identifier">obs</span><span class="special">.</span><span class="identifier">reset</span><span class="special">();</span> |
| |
| <span class="comment">// start with random initial conditions</span> |
| <span class="identifier">generate</span><span class="special">(</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">begin</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">x</span><span class="special">.</span><span class="identifier">end</span><span class="special">()</span> <span class="special">,</span> <span class="identifier">gen</span> <span class="special">);</span> |
| |
| <span class="comment">// calculate some transients steps</span> |
| <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">>()</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">10.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">);</span> |
| |
| <span class="comment">// integrate and compute the statistics</span> |
| <span class="identifier">integrate_const</span><span class="special">(</span> <span class="identifier">runge_kutta4</span><span class="special"><</span> <span class="identifier">container_type</span> <span class="special">>()</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">ensemble</span> <span class="special">)</span> <span class="special">,</span> <span class="identifier">x</span> <span class="special">,</span> <span class="number">0.0</span> <span class="special">,</span> <span class="number">100.0</span> <span class="special">,</span> <span class="identifier">dt</span> <span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">ref</span><span class="special">(</span> <span class="identifier">obs</span> <span class="special">)</span> <span class="special">);</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">epsilon</span> <span class="special"><<</span> <span class="string">"\t"</span> <span class="special"><<</span> <span class="identifier">obs</span><span class="special">.</span><span class="identifier">get_K_mean</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| The full cpp file for this example can be found here <a href="https://github.com/headmyshoulder/odeint-v2/blob/master/examples/phase_oscillator_ensemble.cpp" target="_top">phase_oscillator_ensemble.cpp</a> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2009-2012 Karsten |
| Ahnert and Mario Mulansky<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
| </div></td> |
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