Re: Neato chaotic equations for analog computers to display?

From: Lou Pecora (pecora_at_anvil.nrl.navy.mil)
Date: 12/22/04 

Date: Wed, 22 Dec 2004 10:43:31 -0500

In article <e9_xd.7982$yK.5295@newsread3.news.atl.earthlink.net>,
 "Nicholas O. Lindan" <see@sig.com> wrote:

> > The definition of chaos is very rigorous and precise.
>
> What is it? I'd be very interested...

A dynamical system (differential equations, iterated maps, or whatever)
is chaotic if it has at least one positive Lyapunov exponent. A
Lyapunov Exponent is a measure of long-time stability of nearby points
on trajectories of the system. Chaotic Systems have intrinsically
unstable trajectories, i.e. two nearby trajectories diverge from each
other (like two balls rolling down opposite sides of a hill to use a
simple metaphor). The positive sign of the exponent is a mathematical
way of saying this. There are theorems that show that such exponents
(both positive and negative) are well defined and meaningful.

> > It's the definition of 'random' that is the problem.
>
> Can't argue there. Some say random == hand of God; "God
> is the Dice". There may be some truth to it. Certainly
> God is _the_ problem (as in 'unsolvable problem').

Well, the problem is coming up with a mathematically rigorous definition
of random (like is done for chaos). I know of none. But maybe others
can point to something. Algorithmic complexity may be a way to go, i.e.
a process is random if it cannot be reduced to a finite set of rules.
That's rough defintion.

-- Lou Pecora (my views are my own)

Relevant Pages

  • Re: Neato chaotic equations for analog computers to display?
    ... Lyapunov Exponent is a measure of long-time stability of nearby points ... unstable trajectories, i.e. two nearby trajectories diverge from each ... the problem is coming up with a mathematically rigorous definition ...
    (sci.electronics.design)
  • Re: Neato chaotic equations for analog computers to display?
    ... > is chaotic if it has at least one positive Lyapunov exponent. ... > unstable trajectories, i.e. two nearby trajectories diverge from each ... on Info Theory asserting that randomness ... There are certain at least theoretical tests that they cite by Martin-Lof. ...
    (sci.electronics.design)
  • Re: Neato chaotic equations for analog computers to display?
    ... > is chaotic if it has at least one positive Lyapunov exponent. ... > unstable trajectories, i.e. two nearby trajectories diverge from each ... on Info Theory asserting that randomness ... There are certain at least theoretical tests that they cite by Martin-Lof. ...
    (sci.nonlinear)