Newsgroups: comp.theory.dynamic-sys
Path: utzoo!utgpu!cunews!rdb
From: rdb@scs.carleton.ca (Robert D. Black)
Subject: Re: Can Chaos Be Predictable?
Message-ID: <1991Jun21.003436.28578@cunews.carleton.ca>
Keywords: chaos, predictability
Sender: news@cunews.carleton.ca
Organization: School of Computer Science, Carleton University, Ottawa, Canada
References: <1991Jun20.194552.15875@cunews.carleton.ca> <1991Jun20.203628.14343@alchemy.chem.utoronto.ca>
Distribution: comp.theory.dynamic-sys
Date: Fri, 21 Jun 1991 00:34:36 GMT

In article <1991Jun20.203628.14343@alchemy.chem.utoronto.ca> mroussel@alchemy.chem.utoronto.ca (Marc Roussel) writes:
>
>If you stuck some initial condition u(0) into your analytic solution and
>found (say) u(10000) in single precision and then in double precision,
>I'll wager that the two answers would be substantially different.

True, but differences in single vs double precision would be true
of most any calculation (square root for example).  Chaos enters
into an *iterative* process through the rapid growth of errors in
the initial condition, or from errors resulting from finite
precision arithmetic. 

Because there is a general solution to the difference equation, 
we can simply compute U(10000) without computing the first N-1
values (and suffering accumulated round-off error).  The value of 
U(10000) is a close approximation to the theoretical value at time 
10000.  The value of U(10001) is independent and would not be affected
by any error in U(10000).  

> ... Small differences in u(0) (or differences in the precision
>of the arithmetic used) will be greatly amplified by the 2**(t-1) term.

Agreed, if your initial condition is not exact, then you quickly
lose accuracy as you go to larger times.  However, if we assume
the initial condition is exact (0.75 say), then we can plug this
exact value into the general formula, and get a close approximation
to the theoretical value at that time (even for large t).

--
--
Robert Black                               rdb@scs.carleton.ca
School of Computer Science
Carleton University, Ottawa, Canada