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From: scavo@cie.uoregon.edu (Tom Scavo)
Newsgroups: comp.theory.dynamic-sys
Subject: Re: Can Chaos Be Predictable?
Keywords: chaos, predictability
Message-ID: <1991Jun22.195653.26004@ariel.unm.edu>
Date: 22 Jun 91 19:56:53 GMT
References: <1991Jun20.203628.14343@alchemy.chem.utoronto.ca> <1991Jun21.003436.28578@cunews.carleton.ca> <1991Jun21.064503.8325@netcom.COM>
Distribution: comp.theory.dynamic-sys
Organization: Campus Information Exchange, University of Oregon
Lines: 53

[Sorry if this is a repeat, but I have reason to believe it
didn't get through the first time.]

In article <1991Jun21.064503.8325@netcom.COM> kmc@netcom.COM (Kevin McCarty) writes:

	[much of an excellent article deleted]

>So tell us, what happens to precision when you increment t by 1?  You
>*double* the argument value of the sine.  In order to get an answer
>with one bit of accuracy (e.g., whether the result is closer to 1.0
>than 0.0) you need one additional bit of input precision per unit time
>step.  Another way to look at it is that the absolute output error
>roughly *doubles* with each passing unit of time. In this respect
>it is no different from f(x) = 2x mod 1.  Points initially close
>together diverge exponentially over time.

In fact,  D(x) = 2x mod 1  is (semi-)conjugate to  F(x) = 4x(1-x) 
via the two-to-one map  S(x) = [sin(pi x)]^2 .  I believe this is
where the closed form solution to  F^n(x)  (where  F^n  is the nth
iterate of  F ) comes from since  D^n(x) = 2^n x mod 1 .

	...

>Compare this with your other example, f(x) = sqrt(x) (on the unit
>interval [0,1] say).  Note that you need hardly any precision at all
>in the initial condition in order to predict to a pretty good precision
>the result of iterating f(x) a number of times.  In fact, sqrt(x) has
>an attracting fixed point for almost all initial values.

The fixed point  x = 1  attracts the orbits of ALL points in the
function's domain of definition.

	...

>Consider three different dynamical laws described by
>f(x) = sin**2( 2x )
>g(x) = (x + a) mod 1,    (0 < a < 1)
>h(x) = x/2
>Analyze the precision to which the n'th iterate is known, as a function
>of the precision in the initial value of x and the number of iterates.
>Only f(x) has chaotic behavior.

Can you prove this?  Can you find a conjugacy from  f  to some
other map (like  D , F ,  Q(x) = x^2 - 2 , or the tent map) that
is known to be chaotic?

The map  g  has quite interesting (but atypical) dynamics which
I've yet to completely analyze, and of course  h  has the origin
as a globally attracting fixed point.

-- 
Tom Scavo
scavo@cie.uoregon.edu