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From: alain@elevia.UUCP (W.A.Simon)
Newsgroups: comp.theory.dynamic-sys,sci.crypt,sci.math.stat
Subject: Can Chaos Be Predictable?
Summary: Chaotic Logistic Map has Analytic Solution
Keywords: chaos, predictability
Message-ID: <1991Jun22.133638.3258@elevia.UUCP>
Date: 22 Jun 91 13:36:38 GMT
References: <1991Jun20.194552.15875@cunews.carleton.ca>
Followup-To: comp.theory.dynamic-sys,sci.crypt,sci.math.stat
Organization: The Electronic Path - Global Village
Lines: 36

In <1991Jun20.194552.15875@cunews.carleton.ca> rdb@scs.carleton.ca (Robert D. Black) writes:
> I recently read that the chaotic logistic equation
>     u(t+1) = 4u(t)(1-u(t))      u(0) in 0..1,
> has an ANALYTIC SOLUTION: 
>     u(t) = sin**2 (2**(t-1) arccos(1-2u(0)))
> 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!

	Would it be that your equation has just been proven to be
	non chaotic, or would it be that chaos-order is a continuum,
	and that Laplace was right?  What we perceive as chaos is
	just a weakness in our instrumentation... or computing power.

	Half of a |8-)

	My interest in chaotic sequences is due to the belief that
	Poincare could be right, and therefore I could use such
	a sequence to generate cryptanalytically strong keys.

	Half of a |8-(


	Which brings me back to the Ulam sequences we discussed the
	other day (aka hailstone numbers).  Are the odd/even transitions
	in the sequences known to contain identifiable patterns?  In
	other words, would a string of 0's and 1's matching, respectively,
	even numbers and odd numbers in the sequence, be considered to be
	a random bit stream?

> Robert Black
-- 
William "Alain" Simon
                                                   UUCP: alain@elevia.UUCP