<?xml version="1.0" encoding="UTF-8" standalone="no"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <title>chuap</title> <link rel="stylesheet" type="text/css" href="csound.css" /> <link rel="stylesheet" type="text/css" href="syntax-highlighting.css" /> <meta name="generator" content="DocBook XSL Stylesheets Vsnapshot" /> <link rel="home" href="index.html" title="The Canonical Csound Reference Manual" /> <link rel="up" href="OpcodesTop.html" title="Orchestra Opcodes and Operators" /> <link rel="prev" href="chnset.html" title="chnset" /> <link rel="next" href="cigoto.html" title="cigoto" /> </head> <body> <div class="navheader"> <table width="100%" summary="Navigation header"> <tr> <th colspan="3" align="center">chuap</th> </tr> <tr> <td width="20%" align="left"><a accesskey="p" href="chnset.html">Prev</a> </td> <th width="60%" align="center">Orchestra Opcodes and Operators</th> <td width="20%" align="right"> <a accesskey="n" href="cigoto.html">Next</a></td> </tr> </table> <hr /> </div> <div class="refentry"> <a id="chuap"></a> <div class="titlepage"></div> <a id="Indexchuap" class="indexterm"></a> <div class="refnamediv"> <h2> <span class="refentrytitle">chuap</span> </h2> <p>chuap — Simulates Chua's oscillator, an LRC oscillator with an active resistor, proved capable of bifurcation and chaotic attractors, with k-rate control of circuit elements. </p> </div> <div class="refsect1"> <a id="idm281472946772904"></a> <h2>Description</h2> <p> Simulates Chua's oscillator, an LRC oscillator with an active resistor, proved capable of bifurcation and chaotic attractors, with k-rate control of circuit elements. </p> </div> <div class="refsect1"> <a id="idm281472946731400"></a> <h2>Syntax</h2> <pre class="synopsis">aI3, aV2, aV1 <span class="command"><strong>chuap</strong></span> kL, kR0, kC1, kG, kGa, kGb, kE, kC2, iI3, iV2, iV1, ktime_step</pre> </div> <div class="refsect1"> <a id="idm281472946729224"></a> <h2>Initialization</h2> <p> <span class="emphasis"><em>iI3</em></span> -- Initial current at G </p> <p> <span class="emphasis"><em>iV2</em></span> -- Initial voltage at C2 </p> <p> <span class="emphasis"><em>iV1</em></span> -- Initial voltage at C1 </p> </div> <div class="refsect1"> <a id="idm281472946725928"></a> <h2>Performance</h2> <p> <span class="emphasis"><em>kL</em></span> -- Inductor L </p> <p> <span class="emphasis"><em>kR0</em></span> -- Resistor R0 </p> <p> <span class="emphasis"><em>kC1</em></span> -- Capacitor C1 </p> <p> <span class="emphasis"><em>kG</em></span> -- Resistor G </p> <p> <span class="emphasis"><em>kGa</em></span> -- Resistor V (nonlinearity term) </p> <p> <span class="emphasis"><em>kGb</em></span> -- Resistor V (nonlinearity term) </p> <p> <span class="emphasis"><em>kGb</em></span> -- Resistor V (nonlinearity term) </p> <p> <span class="emphasis"><em>ktime_step</em></span> -- Delta time in the difference equation, can be used to more or less control pitch. </p> <p> <span class="emphasis"><em>Chua's oscillator</em></span> is a simple LRC oscillator with an active resistor. The oscillator can be driven into period bifurcation, and thus to chaos, because of the nonlinear response of the active resistor. </p> <div class="mediaobject"> <img src="images/600px-Chua's_circuit.svg.png" /> <div class="caption"> <p>Diagram of Chua's Oscillator Circuit</p> </div> </div> <p> The circuit is described by a set of three ordinary differential equations called Chua's equations: </p> <div class="literallayout"> <p><br /> dI3 R0 1<br /> --- = - -- I3 - - V2<br /> dt L L<br /> <br /> dV2 1 G<br /> --- = -- I3 - -- (V2 - V1)<br /> dt C2 C2<br /> <br /> dV1 G 1<br /> --- = -- (V2 - V1) - -- f(V1)<br /> dt C1 C1<br /> </p> </div> <p> where f() is a piecewise discontinuity simulating the active resistor: </p> <div class="literallayout"> <p><br /> f(V1) = Gb V1 + - (Ga - Gb)(|V1 + E| - |V1 - E|)<br /> </p> </div> <p>A solution of these equations (I3,V2,V1)(t) starting from an initial state (I3,V2,V1)(0) is called a trajectory of Chua's oscillator. The Csound implementation is a difference equation simulation of Chua's oscillator with Runge-Kutta integration. </p> <div class="note" style="margin-left: 0.5in; margin-right: 0.5in;"> <table border="0" summary="Note"> <tr> <td rowspan="2" align="center" valign="top" width="25"> <img alt="[Note]" src="images/note.png" /> </td> <th align="left">Note</th> </tr> <tr> <td align="left" valign="top"> <p>This algorithm uses internal non linear feedback loops which causes audio result to depend on the orchestra sampling rate. For example, if you develop a project with <span class="emphasis"><em>sr</em></span>=48000Hz and if you want to produce an audio CD from it, you should record a file with <span class="emphasis"><em>sr</em></span>=48000Hz and then downsample the file to 44100Hz using the <a class="link" href="srconv.html" title="srconv"><em class="citetitle">srconv</em></a> utility.</p> </td> </tr> </table> </div> <div class="warning" style="margin-left: 0.5in; margin-right: 0.5in;"> <table border="0" summary="Warning"> <tr> <td rowspan="2" align="center" valign="top" width="25"> <img alt="[Warning]" src="images/warning.png" /> </td> <th align="left">Warning</th> </tr> <tr> <td align="left" valign="top"> Be careful! Some sets of parameters will produce amplitude spikes or positive feedback that could damage your speakers. </td> </tr> </table> </div> </div> <div class="refsect1"> <a id="idm281472946707320"></a> <h2>Examples</h2> <p> Here is an example of the chuap opcode. It uses the file <a class="ulink" href="examples/chuap.csd" target="_top"><em class="citetitle">chuap.csd</em></a>. </p> <div class="example"> <a id="idm281472946705464"></a> <p class="title"> <strong>Example 126. Example of the chuap opcode.</strong> </p> <div class="example-contents"> <div class="refsect1"> <a id="idm281472780278856"></a> <pre class="programlisting"> <span class="nt"><CsoundSynthesizer></span> <span class="nt"><CsOptions></span> <span class="c1">; Select audio/midi flags here according to platform</span> -odac <span class="c1">;;;RT audio out</span> <span class="c1">;-iadc ;;;uncomment -iadc if RT audio input is needed too</span> <span class="c1">; For Non-realtime ouput leave only the line below:</span> <span class="c1">; -o chuas_oscillator.wav.wav -W ;;; for file output any platform</span> <span class="nt"></CsOptions></span> <span class="nt"><CsInstruments></span> <span class="vg">sr</span> <span class="o">=</span> <span class="mi">44100</span> <span class="vg">ksmps</span> <span class="o">=</span> <span class="mi">32</span> <span class="vg">nchnls</span> <span class="o">=</span> <span class="mi">2</span> <span class="vg">0dbfs</span> <span class="o">=</span> <span class="mi">1</span> gi<span class="n">buzztable</span> <span class="nb">ftgen</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">16384</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1</span> <span class="kd">instr</span> <span class="nf">1</span> <span class="c1">; sys_variables = system_vars(5:12); % L,R0,C2,G,Ga,Gb,E,C1 or p8:p15</span> <span class="c1">; integ_variables = [system_vars(14:16),system_vars(1:2)]; % x0,y0,z0,dataset_size,step_size or p17:p19, p4:p5</span> i<span class="n">step_size</span> <span class="o">=</span> <span class="nb">p5</span> i<span class="n">L</span> <span class="o">=</span> <span class="nb">p8</span> i<span class="n">R0</span> <span class="o">=</span> <span class="nb">p9</span> i<span class="n">C2</span> <span class="o">=</span> <span class="nb">p10</span> i<span class="n">G</span> <span class="o">=</span> <span class="nb">p11</span> i<span class="n">Ga</span> <span class="o">=</span> <span class="nb">p12</span> i<span class="n">Gb</span> <span class="o">=</span> <span class="nb">p13</span> i<span class="n">E</span> <span class="o">=</span> <span class="nb">p14</span> i<span class="n">C1</span> <span class="o">=</span> <span class="nb">p15</span> i<span class="n">I3</span> <span class="o">=</span> <span class="nb">p17</span> i<span class="n">V2</span> <span class="o">=</span> <span class="nb">p18</span> i<span class="n">V1</span> <span class="o">=</span> <span class="nb">p19</span> i<span class="n">gnore</span> <span class="o">=</span> <span class="nb">p21</span> i<span class="n">gnore</span> <span class="o">=</span> <span class="nb">p22</span> i<span class="n">gnore</span> <span class="o">=</span> <span class="nb">p23</span> i<span class="n">gnore</span> <span class="o">=</span> <span class="nb">p24</span> i<span class="n">gnore</span> <span class="o">=</span> <span class="nb">p25</span> i<span class="n">attack</span> <span class="o">=</span> <span class="mf">0.02</span> i<span class="n">sustain</span> <span class="o">=</span> <span class="nb">p3</span> i<span class="n">release</span> <span class="o">=</span> <span class="mf">0.02</span> <span class="nb">p3</span> <span class="o">=</span> i<span class="n">attack</span> <span class="o">+</span> i<span class="n">sustain</span> <span class="o">+</span> i<span class="n">release</span> i<span class="n">scale</span> <span class="o">=</span> <span class="mf">1.0</span> a<span class="n">damping</span> <span class="nb">linseg</span> <span class="mf">0.0</span><span class="p">,</span> i<span class="n">attack</span><span class="p">,</span> i<span class="n">scale</span><span class="p">,</span> i<span class="n">sustain</span><span class="p">,</span> i<span class="n">scale</span><span class="p">,</span> i<span class="n">release</span><span class="p">,</span> <span class="mf">0.0</span> a<span class="n">guide</span> <span class="nb">buzz</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">440</span><span class="p">,</span> <span class="vg">sr</span><span class="o">/</span><span class="mi">440</span><span class="p">,</span> gi<span class="n">buzztable</span> a<span class="n">I3</span><span class="p">,</span> a<span class="n">V2</span><span class="p">,</span> a<span class="n">V1</span> <span class="nb">chuap</span> i<span class="n">L</span><span class="p">,</span> i<span class="n">R0</span><span class="p">,</span> i<span class="n">C2</span><span class="p">,</span> i<span class="n">G</span><span class="p">,</span> i<span class="n">Ga</span><span class="p">,</span> i<span class="n">Gb</span><span class="p">,</span> i<span class="n">E</span><span class="p">,</span> i<span class="n">C1</span><span class="p">,</span> i<span class="n">I3</span><span class="p">,</span> i<span class="n">V2</span><span class="p">,</span> i<span class="n">V1</span><span class="p">,</span> i<span class="n">step_size</span> a<span class="n">signal</span> <span class="nb">balance</span> a<span class="n">V2</span><span class="p">,</span> a<span class="n">guide</span> <span class="nb">outs</span> a<span class="n">damping</span> <span class="o">*</span> a<span class="n">signal</span><span class="p">,</span> a<span class="n">damping</span> <span class="o">*</span> a<span class="n">signal</span> <span class="kd">endin</span> <span class="nt"></CsInstruments></span> <span class="nt"><CsScore></span> <span class="c1">; Adapted from ABC++ MATLAB example data.</span> <span class="nb">i</span> <span class="mi">1</span> <span class="mi">0</span> <span class="mi">20</span> <span class="mi">1500</span> <span class="mf">.1</span> <span class="o">-</span><span class="mi">1</span> <span class="o">-</span><span class="mi">1</span> <span class="o">-</span><span class="mf">0.00707925</span> <span class="mf">0.00001647</span> <span class="mi">100</span> <span class="mi">1</span> <span class="o">-</span><span class="mf">.99955324</span> <span class="o">-</span><span class="mf">1.00028375</span> <span class="mi">1</span> <span class="o">-</span><span class="mf">.00222159</span> <span class="mf">204.8</span> <span class="o">-</span><span class="mf">2.36201596260071</span> <span class="mf">3.08917625807226e-03</span> <span class="mf">3.87075614929199</span> <span class="mi">7</span> <span class="mf">.4</span> <span class="mf">.004</span> <span class="mi">1</span> <span class="mi">86</span> <span class="mi">30</span><span class="c1">; torus attractor ( gallery of attractors ) </span> <span class="nb">i</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">20</span> <span class="mi">1500</span> <span class="mf">.425</span> <span class="mi">0</span> <span class="o">-</span><span class="mi">1</span> <span class="mf">1.3506168</span> <span class="mi">0</span> <span class="o">-</span><span class="mf">4.50746268737</span> <span class="o">-</span><span class="mi">1</span> <span class="mf">2.4924</span> <span class="mf">.93</span> <span class="mi">1</span> <span class="mi">1</span> <span class="mi">0</span> <span class="o">-</span><span class="mf">22.28662665</span> <span class="mf">.009506608</span> <span class="o">-</span><span class="mf">22.2861576</span> <span class="mi">32</span> <span class="mi">10</span> <span class="mi">2</span> <span class="mi">20</span> <span class="mi">86</span> <span class="mi">30</span> <span class="c1">; heteroclinic orbit</span> <span class="nb">i</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">20</span> <span class="mi">1024</span> <span class="mf">.05</span> <span class="o">-</span><span class="mi">1</span> <span class="o">-</span><span class="mi">1</span> <span class="mf">0.00667</span> <span class="mf">0.000651</span> <span class="mi">10</span> <span class="o">-</span><span class="mi">1</span> <span class="mf">.856</span> <span class="mf">1.1</span> <span class="mi">1</span> <span class="mf">.06</span> <span class="mf">51.2</span> <span class="o">-</span><span class="mf">20.200590133667</span> <span class="mf">.172539323568344</span> <span class="o">-</span><span class="mf">4.07686233520508</span> <span class="mf">2.5</span> <span class="mi">10</span> <span class="mf">.2</span> <span class="mi">1</span> <span class="mi">66</span> <span class="mi">81</span> <span class="c1">; periodic attractor (torus breakdown route)</span> <span class="nb">i</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">20</span> <span class="mi">1024</span> <span class="mf">0.05</span> <span class="o">-</span><span class="mi">1</span> <span class="o">-</span><span class="mi">1</span> <span class="mf">0.00667</span> <span class="mf">0.000651</span> <span class="mi">10</span> <span class="o">-</span><span class="mi">1</span> <span class="mf">0.856</span> <span class="mf">1.1</span> <span class="mi">1</span> <span class="mf">0.1</span> <span class="mf">153.6</span> <span class="mf">21.12496758</span> <span class="mf">0.03001749</span> <span class="mf">0.515828669</span> <span class="mf">2.5</span> <span class="mi">10</span> <span class="mf">0.2</span> <span class="mi">1</span> <span class="mi">66</span> <span class="mi">81</span> <span class="c1">; torus attractor (torus breakdown route)'</span> <span class="nt"></CsScore></span> <span class="nt"></CsoundSynthesizer></span> </pre> </div> </div> </div> <p><br class="example-break" /> </p> </div> <div class="refsect1"> <a id="idm281472946703640"></a> <h2>Credits</h2> <p> </p> <table border="0" summary="Simple list" class="simplelist"> <tr> <td>Inventor of Chua's oscillator: <a class="ulink" href="http://www.eecs.berkeley.edu/~chua" target="_top"><em class="citetitle">Leon O. Chua</em></a></td> </tr> <tr> <td>Author of MATLAB simulation: James Patrick McEvoy <a class="ulink" href="http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3541" target="_top"><em class="citetitle">MATLAB Adventures in Bifurcations and Chaos (ABC++)</em></a></td> </tr> <tr> <td>Author of Csound port: Michael Gogins</td> </tr> </table> <p> </p> <p>New in Csound version 5.09</p> <p>Note added by François Pinot, August 2009</p> </div> </div> <div class="navfooter"> <hr /> <table width="100%" summary="Navigation footer"> <tr> <td width="40%" align="left"><a accesskey="p" href="chnset.html">Prev</a> </td> <td width="20%" align="center"> <a accesskey="u" href="OpcodesTop.html">Up</a> </td> <td width="40%" align="right"> <a accesskey="n" href="cigoto.html">Next</a></td> </tr> <tr> <td width="40%" align="left" valign="top">chnset </td> <td width="20%" align="center"> <a accesskey="h" href="index.html">Home</a> </td> <td width="40%" align="right" valign="top"> cigoto</td> </tr> </table> </div> </body> </html>