Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!ut-emx!ccwf.cc.utexas.edu
From: psilsbee@ccwf.cc.utexas.edu (Peter Silsbee)
Newsgroups: comp.theory.dynamic-sys
Subject: Re: Chaotic Pseudo-random Number Generators
Summary: FM synthesis uses one
Keywords: chaos, randomness
Message-ID: <50977@ut-emx.uucp>
Date: 21 Jun 91 20:28:55 GMT
References: <1991Jun20.181619.11029@cunews.carleton.ca>
Sender: news@ut-emx.uucp
Reply-To: psilsbee@ccwf.cc.utexas.edu (Peter Silsbee)
Distribution: comp.theory.dynamic-sys
Organization: The University of Texas at Austin
Lines: 23

In article <1991Jun20.181619.11029@cunews.carleton.ca> rdb@scs.carleton.ca (Robert D. Black) writes:
>Does anyone know of chaotic dynamical systems being used to generate
>random/pseudo-random numbers?  For example, it seems the chaotic 
>logistic map 4x(1-x) could be used to generate pseudo-random numbers

There exists at least one such application, in FM (frequency modulation)
sound synthesis. In the basic FM algorithm, there are two sinewave oscillators,
and the output of one controls the frequency of the other:
        f(t) = sin(2*pi*w1*t + A*sin(2*pi*w2*t))
where f(t) is the final output, w1 is the carrier frequency in radians/sec,
w2 is the modulator frequency, and A is related to the index of modulation.
If, instead of using two oscillators, the output f(t) is fed back in place
of the modulator's output, the result has characteristics of a chaotic
system. In discrete time:
        f(t+1) = sin(2*pi*w1*(t+1) + A*f(t))
As the parameter A is increased from zero, the spectrum becomes increasingly
complex, and one can show that bifurcations take place (frequency-doubling?)
(it's been a while since I looked at some of this). In any case, this is
the technique used by Yamaha to create pseudo-white noise (with the parameter
A giving pretty good control over the color of the noise) in their
commercial FM synthesizers.
        Peter
        (silsbee@vision.ee.utexas.edu)