From: "denis-feldmann" <denis-feldmann@wanadoo.fr>
Subject: Re: chaos question
Date: Mon, 5 Jun 2000 07:44:22 +0200
Newsgroups: sci.math
Summary: [missing]

Rajarshi Ray <rajarshi@home.com> a écrit dans le message :
393AD58C.456477F@home.com...
> denis-feldmann wrote:
> > [snip]
> >
> > No, not dumb question at all, and indeed essential. A very important theorem
> > of chaos theory ("the pursuit lemma", sorry for what is probably a very bad
> > litteral translation from French) says that in the sensitive dependence
> > case, you are indeed never predicting anything right (not only because of
> > potential mistakes in initial conditions, but because of small calculation
> > mistakes at evry step) *but* what you get is a real trajectory (albeit not
> > the one you think you are calculating) In other words, what you see in
> > computers picture is absolutely significant, i.e all of the trajectories
> > pictured are real ones.
>
> This is very interesting. And it seems pretty important too. How come
> Gleick doesn't say anything about this is "Chaos"?? All the computer
> experiments would be worthless if this point isn't considered.
>
>

Ys; my reference is Ekeland , and i found the translation i was looking for:
this is called the "shadowing lemma". A quick Google search gives, for
instance:

"" Here is a version of the Shadowing Lemma, taken from the book "Nonlinear
Oscillations, Dynamical Systems, and Bifurcations of Vector Fields" by J.
Guckenheimer and P. Holmes, page 251:
For every b > 0 there is an a > 0 such that every a-pseudo orbit is
b-shadowed by an exact orbit.
Here, an "a-pseudo orbit" is a computed orbit that may have an error of size
a at each iteration (eg. computer simulation), while "b-shadowed" means
there is an exact orbit that is within a distance b of the computed orbit at
each iteration. Thus, we can be assured that an exact orbit is very close to
the computed one by making the errors at each iteration small enough. The
theorem applies to "hyperbolic invariant sets" so you need to verify this
for your particular dynamical system. ""

[deletia --djr]