Which theorem proves that every chaotic system exhibits predicable

[Shubee]

I recall hearing of a very interesting theorem ages ago. I believe it
was in Ergodic theory. I believe that the theorem proved that every
chaotic system exhibits predicable behavior. Does that theorem have a
name?

Shubee

[Lou Pecora]

In article
<22ecdc67-87dc-44b4...@h11g2000prf.googlegroups.com>,
Shubee <e.Sh...@gmail.com> wrote:

You may be thinking of determinism. Many dynamical systems, including
chaotic ones, are deterministic. That means knowing the current state
of the system *exactly* will mean the future states are totally
determed, exactly. That's a result of theorems about differential
equations, for example. Of course, there's a lot hidden in the word
"exactly."

--
-- Lou Pecora

[Shubee]

On Jan 7, 9:54 am, Lou Pecora <pec...@anvil.nrl.navy.mil> wrote:
> In article
> <22ecdc67-87dc-44b4-84e8-d1357eb20...@h11g2000prf.googlegroups.com>,

>
> Shubee <e.Shu...@gmail.com> wrote:
> > I recall hearing of a very interesting theorem ages ago. I believe it
> > was in Ergodic theory. I believe that the theorem proved that every
> > chaotic system exhibits predicable behavior. Does that theorem
> > have a name?
>
> > Shubee
>
> You may be thinking of determinism. Many dynamical systems, including
> chaotic ones, are deterministic. That means knowing the current state
> of the system *exactly* will mean the future states are totally
> determed, exactly. That's a result of theorems about differential
> equations, for example. Of course, there's a lot hidden in the word
> "exactly."
>
> --
> -- Lou Pecora

Lou,

I realize that the underlying math of a chaotic system could be
deterministic but the point of the theorem, if I remember correctly,
is that it assumed chaotic behavior and concluded the existence of a
deterministic property.

Shubee

[Igor Khavkine]

Just guessing here, but you may be talking about the Poincare
recurrence theorem. It states that for a volume preserving dynamical
system, if a trajectory is confined to a finite volume, then after a
long enough time the trajectory will come arbitrarily close to its
initial point.

For example, if a chaotic system is Hamiltonian, Liouville's theorem
implies volume conservation. If the system also possesses a conserved
quantity, such as energy, which confines a given trajectory to a
compact hypersurface, the second hypothesis of the theorem is also
satisfied.

Hope this helps.

Igor

[Ian Parker]

What you have described for systems of low dimensionality is indeed
the case. Liovilles theoem does indeed imply volume conservation, but
this volume may be multidimensional. What we have in general is a
Hausdorff set - a topological space, which because of the properties
of the Hamiltonian has a lower dimensionality than the number of
degrees of freedom. In a complex topology return to an initial state
is virtually impossible.

Chaos implies that a small disturbance is amplified "A butterfly in
Japan causes a hurricane in the Gulf". Shubee also talks about the
"ergodic theorem". The ergodic theorem applies when we have a large
number of states. It says (in effect) I don't know what each
individual particle is doing but collectively I can make
generalizations. We say the second law of thermodynamics is obeyed.
There is a second law of thermodynamics because the number of states
is so large that the probability of a group of particles being in a
particlar state is effectively zero.

- Ian Parker

[Ken Stahl]

Could this be relevant:

http://en.wikipedia.org/wiki/Poincare-Bendixson_theorem

"Shubee" <e.Sh...@gmail.com> wrote in message
news:22ecdc67-87dc-44b4...@h11g2000prf.googlegroups.com...

[Oh No]

Thus spake Igor Khavkine <igo...@gmail.com>

You're on the wrong track. This was to do with a discussing between
myself and Shubee on s.p.f. We are thinking in terms of statistical
properties of a chaotic system, e.g. mean behaviours are predictable
while individual behaviours are not. Examples in physics would include
black body radiation, the second law of thermodynamics, the gas laws,
but I think the theorem, if it exists, is much more general and applies
to any statistical system.

Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.teleconnection.info/rqg/MainIndex

[Maarten Bergvelt]

On 2008-01-08, Shubee <e.Sh...@gmail.com> wrote:
> On Jan 7, 9:54 am, Lou Pecora <pec...@anvil.nrl.navy.mil> wrote:
>> In article
>> <22ecdc67-87dc-44b4-84e8-d1357eb20...@h11g2000prf.googlegroups.com>,
>>
>> Shubee <e.Shu...@gmail.com> wrote:
>> > I recall hearing of a very interesting theorem ages ago. I believe it
>> > was in Ergodic theory. I believe that the theorem proved that every
>> > chaotic system exhibits predicable behavior. Does that theorem
>> > have a name?
>>
>> > Shubee
>>
>> You may be thinking of determinism. Many dynamical systems, including
>> chaotic ones, are deterministic. That means knowing the current state
>> of the system *exactly* will mean the future states are totally
>> determed, exactly. That's a result of theorems about differential
>> equations, for example. Of course, there's a lot hidden in the word
>> "exactly."
>>
>

> Lou,
>
> I realize that the underlying math of a chaotic system could be
> deterministic but the point of the theorem, if I remember correctly,
> is that it assumed chaotic behavior and concluded the existence of a
> deterministic property.

Are you thinking of the KAM theorem:
http://en.wikipedia.org/wiki/KAM_theorem
?

--
Maarten Bergvelt

[Lou Pecora]

In article
<404ffddd-e50d-4611...@h11g2000prf.googlegroups.com>,
Ian Parker <ianpa...@gmail.com> wrote:

Virtually impossible is not impossible. Poincare's theorem does not
specific the time between recurrences AFAIR.

--
-- Lou Pecora

[Lou Pecora]

In article <slrnfo72ts...@u19.math.uiuc.edu>,
Maarten Bergvelt <be...@math.uiuc.edu> wrote:

> > Lou,
> >
> > I realize that the underlying math of a chaotic system could be
> > deterministic but the point of the theorem, if I remember correctly,
> > is that it assumed chaotic behavior and concluded the existence of a
> > deterministic property.
>
> Are you thinking of the KAM theorem:
> http://en.wikipedia.org/wiki/KAM_theorem
> ?

Doesn't that show that for a small enough perturbation to a hamiltonian
system regular behavior (tori) will persist even in the presence of
chaotic behavior of some initial conditions? Not sure I got that right.

--
-- Lou Pecora

[Lou Pecora]

In article
<15811df6-4798-44d8...@c23g2000hsa.googlegroups.com>,
Shubee <e.Sh...@gmail.com> wrote:

The only thing I can think of is the shadowing property. Where
(roughly) for any delta > 0 there is a real trajectory that will shadow
a computed one to within delta for some (finite) time interval.

Although I'm not sure I understand what you stated. I don't see how you
can assume a "chaotic" property without using some form of determinism.
What is the "chaotic" property?

--
-- Lou Pecora