About rigorous proof of chaos in dynamical system

[John Claud_Michigan]

Hi,
I search for a good referene about rigorous proof of chaos in
dynamical system.
thanks to all.
John Claud, U of Michigan

[Lee Rudolph]

John Claud_Michigan <tarak...@yahoo.fr> writes:

You must first decide on rigorous definitions of "chaos" and
"dynamical system" (though the latter isn't in much dispute
now, the former still is, and there continue to be occasional
nasty spats about whether one person's "chaotic dynamical
system" is another person's "chaotic dynamical system").

Lee Rudolph

[Lou Pecora]

In article
<0d68e51a-a81f-4205...@t2g2000yqe.googlegroups.com>,
John Claud_Michigan <tarak...@yahoo.fr> wrote:

Try Google Scholar. I came up with this:

Rigorous verification of chaos in a molecular model, Thomas Rage, Arnold
Neumaier, Christoph Schlier, Phys. Rev. E 50, 2682­2688 (1994)

I think for certain parameters Sparrow may have proven the Lorenz system
is chaotic. Also search for the double-scroll circuit (also known as
Chua's circuit).

Rigorous proofs of chaos in ODEs are very hard. Low-dimensional maps
are easier (but certainly not trivial).

--
-- Lou Pecora

[richard miller]

A couple of leads, perhaps.

Dynamical Systems, Stability, Symbolic Dynamics and Chaos
Robinson C. (Clark), ISBN 0-8493-8493-1

Chaos and Integrability in Nonlinear Dynamics
An Introduction, Michael Tabor, ISB 0-471-82728-2

If nothing else, the list of references at the back should be of use.

Richard M.

[Lou Pecora]

In article <i39ajm$dra$1...@reader1.panix.com>,
Lee Rudolph <lrud...@panix.com> wrote:

>
> You must first decide on rigorous definitions of "chaos" and
> "dynamical system" (though the latter isn't in much dispute
> now, the former still is, and there continue to be occasional
> nasty spats about whether one person's "chaotic dynamical
> system" is another person's "chaotic dynamical system").
>
> Lee Rudolph

Really? That's news to me. I thought most people were settled on the
definitions of attractor and chaos (ignoring Hamiltonian systems for
now). Can you give me an example of definitions of chaos that differ
and which people are debating?

Thanks.

--
-- Lou Pecora

[Lee Rudolph]

Lou Pecora <pec...@anvil.nrl.navy.mil> writes:

I refer you to Letters to the Editor in the November 2009 issue
of the Notices of the American Mathematical Society, where James
Yorke and David Ruelle appear to have different definitions of
chaos. Not a "nasty spat"; on the contrary, very, very polite.

Lee Rudolph

[Lou Pecora]

In article <i3aip5$fds$1...@reader1.panix.com>,
Lee Rudolph <lrud...@panix.com> wrote:

Thanks. I'll check those out. I know Jim Yorke and he gave me the
impression that this is a closed book. But maybe that's because he has
his own approach that he likes. Not a criticism there. I certainly am
partial to my own views. I could also have missed some subtly that he
did mention.

--
-- Lou Pecora

[Lou Pecora]

In article <pecora-7CBA5F....@ra.nrl.navy.mil>,
Lou Pecora <pec...@anvil.nrl.navy.mil> wrote:

> In article <i3aip5$fds$1...@reader1.panix.com>,
> Lee Rudolph <lrud...@panix.com> wrote:
>
> > Lou Pecora <pec...@anvil.nrl.navy.mil> writes:

[cut]

> > >
> > >Really? That's news to me. I thought most people were settled on the
> > >definitions of attractor and chaos (ignoring Hamiltonian systems for
> > >now). Can you give me an example of definitions of chaos that differ
> > >and which people are debating?
> > >
> > >Thanks.
> >
> > I refer you to Letters to the Editor in the November 2009 issue
> > of the Notices of the American Mathematical Society, where James
> > Yorke and David Ruelle appear to have different definitions of
> > chaos. Not a "nasty spat"; on the contrary, very, very polite.
> >
> > Lee Rudolph
> >
>
> Thanks. I'll check those out. I know Jim Yorke and he gave me the
> impression that this is a closed book. But maybe that's because he has
> his own approach that he likes. Not a criticism there. I certainly am
> partial to my own views. I could also have missed some subtly that he
> did mention.

Just a follow up on the letters and an article in American Mathematical
Society journal. The letters are interesting. They are stimulated by
an article by Freeman Dyson on types of mathematicians throughout
history, especially people he knew. He covers a lot including chaos,
string theory, and number theory. It's a wonderful article and very
readable. I recommend it to people on this news group.

See the Feb. 2009 issue of Notices of the American Mathematical Society
for Dyson's article, the Jun. 2009 issue of Notices of the American
Mathematical Society for the original letters stimulated by Dyson's
article, and Nov. 2009 issue for Yorke's and Ruelle's responses.

Yes, they are polite.

Thanks for the heads up, Mr. Rudolph.

--
-- Lou Pecora