Newsgroups: comp.theory.dynamic-sys
Path: utzoo!utgpu!cunews!rdb
From: rdb@scs.carleton.ca (Robert D. Black)
Subject: Can Chaos Be Predictable?
Message-ID: <1991Jun20.194552.15875@cunews.carleton.ca>
Summary: Chaotic Logistic Map has Analytic Solution
Keywords: chaos, predictability
Sender: news@cunews.carleton.ca
Organization: School of Computer Science, Carleton University, Ottawa, Canada
Distribution: comp.theory.dynamic-sys
Date: Thu, 20 Jun 1991 19:45:52 GMT

I recently read that the chaotic logistic equation

    u(t+1) = 4u(t)(1-u(t))      u(0) in 0..1,
                                t = 0,1,2,...
has an ANALYTIC SOLUTION: 

    u(t) = sin**2 (2**(t-1) arccos(1-2u(0)))

    Reference "Differential Equations" by Walter G. Kelly and
    Alan C. Peterson, Academic Press 1991, p184.

This is CONFUSING!  Wasn't it the case that solvable systems 
are by definition predictable and hence not chaotic?  Here you
can find the value of the system at any time t without computing
intermediate values.  Yet the logistic equation above is said to be
chaotic!

What's going on here?  

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