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Wed Nov 21 01:08:37 CET 2007

Chaotic Oscillators

making chaos with a mixer usually involves an EQ, feedback and
gain. the nonlinear element is a saturation. i've never seen this
particular cicuit realized as a special purpose chaotic
oscillator. lots of comparator based stuff, but no saturation..

the simplest i can find is this one:
http://www.ecse.rpi.edu/~khaled/papers/iscas00.pdf

which uses 3 integrators, a comparator and a summing amp. i'd like to
get the parts count down to 4 amps, so i can use a quad. this means
the summing amp needs to be incorporated somewhere else.

i'd like to try the following. on the plane i made a (faulty) circuit
with a svf with positive feedback (timeconstant t = 1 for simplicity)

   x'' =  -x + a x'

if a > 0 this circuit is unstable. for a < 0 this is a standard
biquad.

now, if the integrators can be biased to some voltage, this bias can
be derived from a smith trigger acting on one of the state variables x
or x', switching between two unstable points. the st is stateful, so
chaos is possible on two planes: R^2 x {1,-1}

( what i did wrong is to apply the bias to the (+) input, which can't
be correct since the capacitor voltages are relative to this point, so
changing bias would also change the state variables. )

so.. what about using the saturation of the opamp? by measuring the
voltage at the noninverting inputs, saturation can be detected. in the
phase plane of an svf, there are 4 points where saturation can
occur.

using the general timer principle: 

  * detect a certain condition (one of the state variables saturated)
  * discharge one of the state variables

see:
http://www.scholarpedia.org/article/Chaotic_Spiking_Oscillators

switched 2nd order: a classic unstable biquad with state variables x
and y, where x is decremented with the output of a schmitt trigger
before it's fed into the y integrator and the integrator chain input
summer.

see:
http://www.cs.rmit.edu.au/~jiankun/Sample_Publication/IECON.pdf

which basicly contains the circuit i just (re)invented..


EDIT:
some refs from johan suykens:

M.E. Yalcin, J.A.K. Suykens, J.P.L. Vandewalle, Cellular Neural 
Networks, Multi-Scroll Chaos and Synchronization, World Scientific 
Series on Nonlinear Science, Series A - Vol. 50, Singapore, 2005





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