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

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.

Syntax

aI3, aV2, aV1 chuap kL, kR0, kC1, kG, kGa, kGb, kE, kC2, iI3, iV2, iV1, ktime_step

Initialization

iI3 -- Initial current at G

iV2 -- Initial voltage at C2

iV1 -- Initial voltage at C1

Performance

kL -- Inductor L

kR0 -- Resistor R0

kC1 -- Capacitor C1

kG -- Resistor G

kGa -- Resistor V (nonlinearity term)

kGb -- Resistor V (nonlinearity term)

kGb -- Resistor V (nonlinearity term)

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] Note

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 srconv utility.

[Warning] Warning
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 109. 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 chuas_oscillator.wav.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
; sys_variables = system_vars(5:12); % L,R0,C2,G,Ga,Gb,E,C1 or p8:p15
; integ_variables = [system_vars(14:16),system_vars(1:2)]; % x0,y0,z0,dataset_size,step_size or p17:p19, p4:p5
istep_size    =       p5
iL            =       p8
iR0           =       p9
iC2           =       p10
iG            =       p11
iGa           =       p12
iGb           =       p13
iE            =       p14
iC1           =       p15
iI3           =       p17
iV2           =       p18
iV1           =       p19
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    adamping * asignal, adamping * asignal
endin
</CsInstruments>
<CsScore>
;        Adapted from ABC++ MATLAB example data.
i 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; torus attractor ( gallery of attractors ) 
i 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 ; heteroclinic orbit
i 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 ; periodic attractor (torus breakdown route)
i 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  ; torus attractor (torus breakdown route)'
</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