Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!sunybcs!bingvaxu!vu0112
From: vu0112@bingvaxu.cc.binghamton.edu (Cliff Joslyn)
Newsgroups: comp.ai
Subject: Re: PROBABLE COMPLEXITY QUOTIENT
Message-ID: <2219@bingvaxu.cc.binghamton.edu>
Date: 23 Jun 89 04:05:07 GMT
References: <1591@infmx.UUCP> <361@calmasd.Prime.COM> <1600@infmx.UUCP>
Reply-To: vu0112@bingvaxu.cc.binghamton.edu.cc.binghamton.edu (Cliff Joslyn)
Organization: SUNY Binghamton, NY
Lines: 57

In article <1600@infmx.UUCP> briand@infmx.UUCP (brian donat) writes:
>>> Given that the Human Brain is complex, complex to the point that we regard
>>> it now in the terminology of chaos theory...
>
>>Complexity alone is not a sufficient condition for chaos.  The
>>necessary and sufficient condition for chaotic behavior is that the
>>system be a non-linear dynamic system.  
>
>I have some misgivings that a lot of people really do not know (fully)
>what Chaos Theory implies.   You for example, are certainly correct in
>saying that complexity alone is not a sufficient condition for chaos.

This is all very confused.  Chaos means (generally) that any two
arbitrary trajectories of a system's behavior can become arbitrarily
close, so that they cannot be distinguished, and/or that the margins of
error increase exponentially in time.  Being a dynamical system is not
necessary for chaos, rather chaos is only *defined* on dynamical
systems.  Non-linearity is a necessary, but *not* a sufficient, condition
for chaos. 

While chaos is formally defined, there are many formal and informal
meanings of complexity.  While many chaotic systems appear complex,
complexity and chaos are not mutually necessary.  Complexity means to
have a lot interacting parts, or to take a long time to construct, or a
lot of work. 

>Now this 'non-linear dynamic' thing is more revealing.    However, chaos
>theory sees 'order from chaos' and therefore there is implied a necessary
>requirement that linear systems also be inlcuded as part of a study of 
>chaos.   The textbook definition is misleading.

In a differential system, only three dimensional non-linear systems show
chaotic behavior, where linear means that the state variables have only
constant coefficients.  Chaos is not well defined outside of
differential/difference equations. 

>But what
>really is a 'static' system?   It can be argued that a static system is
>really dynamic, but balanced.   

A static system is one that does not change in time.  No static systems
can be chaotic.  All static systems appear "balanced", but not all
"balances" are static, e.g.  dynamic equilibria (steady states), say in
a chemical reaction or in population dynamics, where there is continual
movement of two opposite kinds which is balanced *on the average* and
appears static only *on the large scale*.

The answers to the original poster's questions about the state of the
art of complexity metrics, complex systems theory, dynamic systems
theory, and chaos theory are all readily available in the literature. 
I'll be happy to post or mail a bibliography. 

-- 
O---------------------------------------------------------------------->
| Cliff Joslyn, Cybernetician at Large
| Systems Science, SUNY Binghamton, vu0112@bingvaxu.cc.binghamton.edu
V All the world is biscuit shaped. . .