chuap
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.
Description
Plugin opcode in chua. This opcode is part of the plugin repository and has to be installed separately. The plugin repository can be found here: https://github.com/csound/plugins
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.
Initialization
iI3 -- Initial current at G
iV2 -- Initial voltage at C2
iV1 -- Initial voltage at C1
Performance
kL -- Inductor L (L1 in the diagram)
kR0 -- Resistor R0 (R0 in the diagram)
kC1 -- Capacitor C1
kC2 -- Capacitor C2
kG -- Resistor G (part of the active resistor, R1 in the diagram). The G parameter is the time step, which is needed to get the same slope of the piecewise discontinuity from Ga and Gb for all sampling rates.
kGa -- Resistor V (nonlinearity term of the active resistor, one of the R2's in the diagram)
kGb -- Resistor V (nonlinearity term of the active resistor, one of the R2's in the diagram)
kE -- Size of the piecewise discontinuity simulating the active resistor
ktime_step -- Delta time in the difference equation, can be used to more or less control pitch.
Chua's oscillator 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.
Diagram of Chua's Oscillator Circuit
The circuit is described by a set of three ordinary differential equations called Chua's equations:
dI3 R0 1
--- = - -- I3 - - V2
dt L L
dV2 1 G
--- = -- I3 - -- (V2 - V1)
dt C2 C2
dV1 G 1
--- = -- (V2 - V1) - -- f(V1)
dt C1 C1
where f() is a piecewise discontinuity simulating the active resistor:
f(V1) = Gb V1 + - (Ga - Gb)(|V1 + E| - |V1 - E|)
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.
![[Note]](images/note.png)
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Note |
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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 sr=48000Hz and if you want to produce an audio CD from it, you should record a file with sr=48000Hz and then downsample the file to 44100Hz using the src_conv utility. |
![[Warning]](images/warning.png)
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Warning |
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| Be careful! Some sets of parameters will produce amplitude spikes or positive feedback that could damage your speakers. |
Examples
Here is an example of the chuap opcode. It uses the file chuap.csd.
Example 140. Example of the chuap opcode.
<CsoundSynthesizer> <CsOptions> ; Select audio/midi flags here according to platform -odac ;;;RT audio out ;-iadc ;;;uncomment -iadc if RT audio input is needed too ; For Non-realtime ouput leave only the line below: ; -o chuap.wav -W ;;; for file output any platform </CsOptions> <CsInstruments> sr = 44100 ksmps = 32 nchnls = 2 0dbfs = 1 gibuzztable ftgen 1, 0, 16384, 10, 1 instr 1 istep_size = p4 iL = p5 iR0 = p6 iC2 = p7 iG = p8 iGa = p9 iGb = p10 iE = p11 iC1 = p12 iI3 = p13 iV2 = p14 iV1 = p15 iattack = 0.02 isustain = p3 irelease = 0.02 p3 = iattack + isustain + irelease iscale = 1.0 adamping linseg 0.0, iattack, iscale, isustain, iscale, irelease, 0.0 aguide buzz 0.5, 440, sr/440, gibuzztable aI3, aV2, aV1 chuap iL, iR0, iC2, iG, iGa, iGb, iE, iC1, iI3, iV2, iV1, istep_size asignal balance aV2, aguide outs asignal*adamping, asignal*adamping endin </CsInstruments> <CsScore> ; Adapted from ABC++ MATLAB example data. // time_step kL kR0 kC2 kG kGa kGb kE kC1 iI3 iV2 iV1 ; torus attractor ( gallery of attractors ) i 1 0 20 .1 -0.00707925 0.00001647 100 1 -.99955324 -1.00028375 1 -.00222159 -2.36201596260071 3.08917625807226e-03 3.87075614929199 ; heteroclinic orbit i 1 + 20 .425 1.3506168 0 -4.50746268737 -1 2.4924 .93 1 1 -22.28662665 .009506608 -22.2861576 ; periodic attractor (torus breakdown route) i 1 + 20 .05 0.00667 0.000651 10 -1 .856 1.1 1 .06 -20.200590133667 .172539323568344 -4.07686233520508 ; torus attractor (torus breakdown route)' i 1 + 20 0.05 0.00667 0.000651 10 -1 0.856 1.1 1 0.1 21.12496758 0.03001749 0.515828669 </CsScore> </CsoundSynthesizer>
Credits
| Inventor of Chua's oscillator: Leon O. Chua |
| Author of MATLAB simulation: James Patrick McEvoy MATLAB Adventures in Bifurcations and Chaos (ABC++) |
| Author of Csound port: Michael Gogins |
New in Csound version 5.09
Note added by François Pinot, August 2009