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<div class="refentry" title="chuap">
<a id="chuap"></a>
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<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" title="Description">
<a id="idp15896896"></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" title="Syntax">
<a id="idp15883504"></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" title="Initialization">
<a id="idp15884656"></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" title="Performance">
<a id="idp15934472"></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" title="Note" style="margin-left: 0.5in; margin-right: 0.5in;">
<table border="0" summary="Note">
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<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>
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</div>
<div class="warning" title="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" title="Examples">
<a id="idp15944840"></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="idp15945808"></a>
<p class="title">
<strong>Example 106. Example of the chuap opcode.</strong>
</p>
<div class="example-contents">
<pre class="programlisting">
<span class="csdtag"><CsoundSynthesizer></span>
<span class="csdtag"><CsOptions></span>
<span class="comment">; Select audio/midi flags here according to platform</span>
-odac <span class="comment">;;;RT audio out</span>
<span class="comment">;-iadc ;;;uncomment -iadc if RT audio input is needed too</span>
<span class="comment">; For Non-realtime ouput leave only the line below:</span>
<span class="comment">; -o chuas_oscillator.wav.wav -W ;;; for file output any platform</span>
<span class="csdtag"></CsOptions></span>
<span class="csdtag"><CsInstruments></span>
<span class="ohdr">sr</span> <span class="op">=</span> 44100
<span class="ohdr">ksmps</span> <span class="op">=</span> 32
<span class="ohdr">nchnls</span> <span class="op">=</span> 2
<span class="ohdr">0dbfs</span> <span class="op">=</span> 1
gibuzztable <span class="ohdr">ftgen</span> 1, 0, 16384, 10, 1
<span class="oblock">instr</span> 1
<span class="comment">; sys_variables = system_vars(5:12); % L,R0,C2,G,Ga,Gb,E,C1 or p8:p15</span>
<span class="comment">; integ_variables = [system_vars(14:16),system_vars(1:2)]; % x0,y0,z0,dataset_size,step_size or p17:p19, p4:p5</span>
istep_size <span class="op">=</span> p5
iL <span class="op">=</span> p8
iR0 <span class="op">=</span> p9
iC2 <span class="op">=</span> p10
iG <span class="op">=</span> p11
iGa <span class="op">=</span> p12
iGb <span class="op">=</span> p13
iE <span class="op">=</span> p14
iC1 <span class="op">=</span> p15
iI3 <span class="op">=</span> p17
iV2 <span class="op">=</span> p18
iV1 <span class="op">=</span> p19
iattack <span class="op">=</span> 0.02
isustain <span class="op">=</span> p3
irelease <span class="op">=</span> 0.02
p3 <span class="op">=</span> iattack <span class="op">+</span> isustain <span class="op">+</span> irelease
iscale <span class="op">=</span> 1.0
adamping <span class="opc">linseg</span> 0.0, iattack, iscale, isustain, iscale, irelease, 0.0
aguide <span class="opc">buzz</span> 0.5, 440, <span class="ohdr">sr</span><span class="op">/</span>440, gibuzztable
aI3, aV2, aV1 <span class="opc">chuap</span> iL, iR0, iC2, iG, iGa, iGb, iE, iC1, iI3, iV2, iV1, istep_size
asignal <span class="opc">balance</span> aV2, aguide
<span class="opc">outs</span> adamping <span class="op">*</span> asignal, adamping <span class="op">*</span> asignal
<span class="oblock">endin</span>
<span class="csdtag"></CsInstruments></span>
<span class="csdtag"><CsScore></span>
<span class="comment">; Adapted from ABC++ MATLAB example data.</span>
<span class="stamnt">i</span> 1 0 20 1500 .1 -1 -1 -0.00707925 0.00001647 100 1 -.99955324 -1.00028375 1 -.00222159 204.8 -2.36201596260071 3.08917625807226e-03 3.87075614929199 7 .4 .004 1 86 30<span class="comment">; torus attractor ( gallery of attractors ) </span>
<span class="stamnt">i</span> 1 + 20 1500 .425 0 -1 1.3506168 0 -4.50746268737 -1 2.4924 .93 1 1 0 -22.28662665 .009506608 -22.2861576 32 10 2 20 86 30 <span class="comment">; heteroclinic orbit</span>
<span class="stamnt">i</span> 1 + 20 1024 .05 -1 -1 0.00667 0.000651 10 -1 .856 1.1 1 .06 51.2 -20.200590133667 .172539323568344 -4.07686233520508 2.5 10 .2 1 66 81 <span class="comment">; periodic attractor (torus breakdown route)</span>
<span class="stamnt">i</span> 1 + 20 1024 0.05 -1 -1 0.00667 0.000651 10 -1 0.856 1.1 1 0.1 153.6 21.12496758 0.03001749 0.515828669 2.5 10 0.2 1 66 81 <span class="comment">; torus attractor (torus breakdown route)'</span>
<span class="csdtag"></CsScore></span>
<span class="csdtag"></CsoundSynthesizer></span>
</pre>
</div>
</div>
<p><br class="example-break" />
</p>
</div>
<div class="refsect1" title="Credits">
<a id="idp15946816"></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>
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