The Fallacy Of Chaos

1 Mark	The Fallacy Of Chaos	10 Feb 2002 03:52:23 GMT
2 "Chris Hillman"	Re: The Fallacy Of Chaos	maandag 11 februari 2002 6:00
3 "Louis M. Pecora"	Re: The Fallacy Of Chaos	maandag 11 februari 2002 14:23
4 "Brian J Flanagan"	Re: The Fallacy Of Chaos	maandag 11 februari 2002 20:57
5 "Nicolaas Vroom"	Re: The Fallacy Of Chaos	donderdag 14 februari 2002 13:14
6 "Chris Hillman"	Re: The Fallacy Of Chaos	vrijdag 15 februari 2002 4:33
7 "Kevin Aylward"	Re: The Fallacy Of Chaos	vrijdag 15 februari 2002 9:58
8 "Nicolaas Vroom"	Re: The Fallacy Of Chaos	Sat, 16 Feb 2002 20:50:35
9 "Nicolaas Vroom"	Re: The Fallacy Of Chaos	Sun, 17 Feb 2002 12:01:12
10 "Nicolaas Vroom"	Re: The Fallacy Of Chaos	donderdag 21 februari 2002 10:36
11 "Chris Hillman"	Re: The Fallacy Of Chaos	dinsdag 26 februari 2002 6:21
12 "Nicolaas Vroom"	Re: The Fallacy Of Chaos	donderdag 7 maart 2002 0:46
13 "Chris Hillman"	Re: The Fallacy Of Chaos	zondag 10 maart 2002 18:41

On 10 Feb 2002, Mark wrote:

>	The existence of chaos in dynamic systems is normally taken to be the antithesis of determinism.

Where did you read that?!

-Every- book on dynamical systems I have seen which mentions "chaos" at all points out that the concept of "sensitive dependence on initial conditions" is a quite different concept from "deterministic". For example, just note the -title- of this undergraduate textbook!:

author = {J. L. McCauley},
title = {Chaos, Dynamics and Fractals: an Algorithmic Approach to Determinisitic Chaos},
publisher = {Cambridge University Press},
series = {Nonlinear Science},
volume = 2,
year = 1993}

(I mention some other books below which I think might be better for first readings in the area of dynamical systems theory, however.)

Most books also point out that no mathematical definition of "chaos" is standard; indeed, few books attempt to offer a definition! (The one by Devaney cited below is an exception.)

The dynamical systems which are generally agreed to exhibit "chaotic behavior" and which are most likely to be familiar to most readers are simply endomaps on some space, for example x -> x^2+c, c an appropriately chosen negative real constant. Can't get any more "deterministic" than that! Hidden inside this there are generally (one-dimensional) shifts of finite type. Shift spaces are compact metric spaces of sequuences and are the most idealized of all deterministic dynamical systems.
(One-dimensional) shifts of finite type have dense periodic points, so one need only show that all but finitely many of the periodic orbits in the shift space correspond to -repelling- cycles, and to observe that shift spaces exhibit sensitive dependence on initial conditions. Voila!-- you have chaos, according to the definition offered by Devaney (which is a rather strong notion of "chaos"-- many authors demand only SDIC, or even leave the term undefined, but certainly I think everyone would agree that SDIC is a minimal condition).

I might as well tell you what a shift of finite type is. Make infinite the set of sequences x:Z->A, where A is a finite set of "symbols", into a compact space either using the product topology induced from the discrete topology on A, or using a metric in which d(x,y) = 1/2^n where +/-n is the smallest index in absolute value where x(j)=/=y(j). The shift map simply shifts a sequence one place to the left. The result is called the "full shift". Closed shift-invariant subspaces of A^Z are called shift spaces (or sometimes, "subshifts"). A shift space is a shift of finite type if its "language" can be described by giving some finite list of -forbidden- blocks. For example, if A = {0,1}, the shift with the block "11" forbidden consists of all sequences like "..1001000101001...". This shift space is an SFT. SFT's are nice because there are simple methods which enable you to easily compute the topological entropy of an SFT and also to write down a generating function, called the dynamical zeta function, for the number of periodic sequences with period dividing n, for each positive integer n. Entropy and the zeta function are invariant under shift respecting homeomorphisms, and are thus invariants of a shift space. But they are rather weak invariants--- there are much stronger ones known, including a host of interesting guys arising from the K-theory of C-* algebras.

>

* The practical determination of initial conditions is fundamentally beset by uncertainties of some finite, but non-zero size. * In a chaotic system, even the slightest differences in initial conditions can blow up to become differences so large that they actually determine the difference between whether a system comes to posses a given macroscopic attribute or not.

The latter notion is essentially the condition called "sensitive dependence upon initial conditions" or SDIC.

For dynamical systems theory and chaos in general, see for example

author = {E. Atlee Jackson},
title = {Perspectives of Nonlinear Dynamics},
note = {Two Volumes},
publisher = {Cambridge University Press},
year = 1991}

author = {Robert L. Devaney},
title = {An Introduction to Chaotic Dynamical Systems},
edition = {Second},
publisher = {Addison-Wesley},
year = 1989}

author = {Robert C. Hilborn},
title = {Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers},
publisher = {Oxford University Press},
year = 1994}

author = {Anatole Katok and Boris Hasselblatt},
title = {Introduction to the Modern Theory of Dynamical Systems},
publisher = {Cambridge University Press},
year = 1995}

See also volumes 1,3,6,16,39,66 of the Encylopedia of Mathematical Sciences published in zillions of volumes by Springer.

For the complexified version of x -> x^2 + c, see for example

editor = {Brodil Branner and Robert L. Devaney},
title = {Complex dynamical systems : the mathematics behind the Mandelbrot and Julia sets},
address = {Providence, R.I.},
publisher = {American Mathematical Society},
year = 1994}

author = {Carleson, Lennart and Gamelin, Theodore W.},
title = {Complex dynamics},
publisher = {Springer-Verlag},
year = 1993}

For shift spaces, see Devaney and also

title = {Coping with Chaos: Analysis of Chaotic Data and Exploitation of Chaotic Systems},
editor = {Edward Ott and Tim Sauer and James A. Yorke},
series = {Nonlinear Science},
publisher = {Wiley},
year = 1994}

author = {Douglas Lind and Brian Marcus},
title = {Introduction to Symbolic Dynamics and Coding},
publisher = {Cambridge University Press},
year = 1995}

author = {Roy L. Adler},
title = {Symbolic Dynamics and {M}arkov Partitions},
journal = {Bulletin of the American Mathematical Society},
volume = 35,
number = 1,
series = {New},
month = {January},
year = 1998,
pages = {1--56}}

author = {Ya. G. Sinai},
title = {Topics in Ergodic Theory},
series = {Princeton Mathematical Series},
volume = 44,
publisher = {Princeton University Press},
year = 1994}

author = {Manfred Denker and Christian Grillenberger and Karl Sigmund},
title = {Ergodic theory on compact spaces},
series = {Lecture Notes in Mathematics},
volume = 527,
publisher = {Springer-Verlag},
year = 1976}

author = {Bruce P. Kitchens},
title = {Symbolic Dynamics: One-Sided, Two-Sided,
and Countable State {M}arkov Shifts},
publisher = {Springer-Verlag},
date = 1998}

For K-theory of C-* algebras, try

author = {N. E. Wegge-Olsen},
title = {K-theory and {C}-$\ast$ Algebras: a friendly approach},
publisher = {Oxford University Press},
year = 1993}

author = {Rordan, M. and Larsen, F. and Laursen, N. J.},
title = {An Introduction to {K}-theoryfor {$C$}-$\ast$ algebras},
series = {London Mathematical Society Student Texts},
volume = 49,
publisher = {Cambridge University Press},
year = 2000}

author = {Blackadar, Bruce},
title = {K-theory for operator algebras},
publisher = {Springer-Verlag},
series = {MSRI publications},
volume = 5,
year = 1986}

For the N-body problem in Newtonian gravitation, try

author = {Carl D. Murray and Stanley F. Dermott},
title = {Solar system dynamics},
publisher = {Cambridge University Press},
year = 1999}

author = {Florin Diacu and Philip Holmes},
title = {Celestial Encounters : the Origins of Chaos and Stability},
publisher = {Princeton University Press, 1996}

author = {Yusuke Hagihara},
title = {Celestial Mechanics},
publisher = {MIT Press},
note = {3/2 volumes},
year = 1970}

and this recent reprint of a classic:

author = {J\"urgen Moser},
title = {Stable and Random Motions in Dynamical Systems: with Special Emphasis on Celestial Mechanics},
publisher = {Princeton University Press},
year = 2001}

The above citations are listed within each "section" roughly in order of increasing demands made upon the reader.

Chris Hillman

Home page: http://www.math.washington.edu/~hillman/personal.html

In article , Mark wrote:

>	The existence of chaos in dynamic systems is normally taken to be the antithesis of determinism. The argument underlying this proceeds by the following chain:

[cut]

No. You got the idea about initial conditions sort of right, but in nonlinear dynamics chaotic systems are as deterministic (under the defintion of determinism) as any other systems. There really is no fallacy here. Sorry, but your solution is already similar to how most scientists and mathematicians in nonlinear dynamics think and has been for some time.

-- -- Lou Pecora

- My views are my own.

Mark wrote:

>	The existence of chaos in dynamic systems is normally taken to be the antithesis of determinism.

If memory serves, chaotic systems are usually regarded as deterministic but of limited predictability, owing to the "sensitive dependence on initial conditions" which you mention. Also, there was quite a thriving industry in quantum chaos not too long ago, but I don't know what its status is today.

"Chris Hillman" schreef in bericht news:Pine.OSF.4.33.0202102004010.398916-100000@goedel3.math.washington.edu:

>	On 10 Feb 2002, Mark wrote:

> >	The existence of chaos in dynamic systems is normally taken to be the antithesis of determinism.

>

Where did you read that?! -Every- book on dynamical systems I have seen which mentions "chaos" at all points out that the concept of "sensitive dependence on initial conditions" is a quite different concept from "deterministic". For example, just note the -title- of this undergraduate textbook!: author = {J. L. McCauley}, title = {Chaos, Dynamics and Fractals: an Algorithmic Approach to Deterministic Chaos}, publisher = {Cambridge University Press}, series = {Nonlinear Science}, volume = 2, year = 1993} (I mention some other books below which I think might be better for first readings in the area of dynamical systems theory, however.) Most books also point out that no mathematical definition of "chaos" is standard; indeed, few books attempt to offer a definition! (The one by Devaney cited below is an exception.)

That is also the weakest point of the concept of chaos. Which physical system do you call chaotic and and which not? and why.
You can not say that the behaviour of one animal is chaotic (for example a butterfly) and some other animal not. Except if someone clearly identifies why.

Systems are often clasified in stable or non stable. A temperature control loop is stable if the process reaches within a certain time its new set point.
The transfer function of such a process has no null points on the imaginary axis (if I remember well)

It is a misnomer to call all non stable processes chaotic.

Processes (systems) are described by differential equations. You must solve those equations and then you need the initial conditions at t=0 in order to find the state of that system at t=n.
That is one "direction" of the problem.
The other "direction" is when you start from a process in order to find the differential equations.
To find those equations you have to upset the process and to measure, monitor the state of the process.

In general the more measurements you make the better (more accurate) you can calculate the parameters of your equations. The same is true for initial conditions. And what is more the better you can calculate the future.

To ^define^ chaotic systems as being dependent on initial conditions is a also a misnomer.

In the book Pierre Simon-Laplace By Charles Coulston Gillispie at page 271 you can read:
"More recently, it has been calculated in the light of the chaos theory that the motions of the planets become unpredictable after some 100 million years".

I do not agree with the tendency of that sentence. The more measurements we make, the more accurate, the more objects we include (large or small) in our simulations the better we can predict the positions of the planets over a period of 100 millions years.

In my library I have the book: Exploring Complexity - An Introduction By Gregoure Nicolis Ilya Prigogine.

http://www.nicvroom.be/initcond.htm

Nick

On Thu, 14 Feb 2002, Nicolaas Vroom wrote, apparently addressing me:

>	You can not say that the behaviour of one animal is chaotic (for example a butterfly) and some other animal not.

I didn't!

>	It is a misnomer to call all non stable processes chaotic.

I didn't!

>	Processes (systems) are described by differential equations.

Many dynamical systems are indeed defined in this way, but many are -not-. Indeed, none of the examples in my posts were of this nature.

>	To ^define^ chaotic systems as being dependent on initial conditions is a also a misnomer.

I didn't!

I can't comment further since it seems to me that your followup has nothing to do with what I said in the post to which you are nominally responding.

Chris Hillman

Home page: http://www.math.washington.edu/~hillman/personal.html

"Nicolaas Vroom" wrote in message news:ySNa8.140976$rt4.13055@afrodite.telenet-ops.be...

>	Systems are often clasified in stable or non stable. A temperature control loop is stable if the process reaches within a certain time its new set point. The transfer function of such a process has no null points on the imaginary axis (if I remember well)

Er... Actually, a linear system is stable if its transfer function has no poles in the right half plane. This can also be expressed by the number of net encirclements of the plot of magnitude and phase around the -1 point.

>	It is a misnomer to call all non stable processes chaotic.

Obviously. A simple electronic oscillator is not (usually) chaotic.

Kevin Aylward , Warden of the Kings Ale kevin@anasoft.co.uk
http://www.anasoft.co.uk SuperSpice, a very affordable Mixed-Mode Windows Simulator with Schematic Capture, Waveform Display, FFT's and Filter Design.

Chris Hillman wrote:

>	On Thu, 14 Feb 2002, Nicolaas Vroom wrote, apparently addressing me:

> >	You can not say that the behaviour of one animal is chaotic (for example a butterfly) and some other animal not.

>

I didn't!

Correct. As part of your reply you gave an excellent list of books (with their authors) about chaos theory. With "You" my intention was to express the opinion of these authors. The authors (at least some authors about chaos) use the behaviour of butterflies (the movement of their wings) as an example of a chaotic process. I do not agree with that opinion specific because it seems a too broad definition.

> >	It is a misnomer to call all non stable processes chaotic.

>

I didn't!

Correct. Same reason as above. The authors in general do not define which process are chaotic and which are not.

> >	Processes (systems) are described by differential equations.

>	Many dynamical systems are indeed defined in this way, but many are -not-. Indeed, none of the examples in my posts were of this nature.

> >	To ^define^ chaotic systems as being dependent on initial conditions is a also a misnomer.

>

I didn't!

Correct. Same reason as above. The authors emphasize too much the importance on initial conditions. All parameters of the equations that describe those processes are also important.

>	I can't comment further since it seems to me that your followup has nothing to do with what I said in the post to which you are nominally responding. Chris Hillman Home page: http://www.math.washington.edu/~hillman/personal.html

Chris Hillman wrote:

>	On Thu, 14 Feb 2002, Nicolaas Vroom wrote, apparently addressing me:

> >	You can not say that the behaviour of one animal is chaotic (for example a butterfly) and some other animal not.

>

I didn't!

That is correct. You did not.

Instead of: "You can not say" I should have written: "They can not say" With they I mean the authors of the books about chaos (theory).

The reason why I wrote this comment is because you mentioned: "few books attempt to offer a definition of "chaos""! IMO this is difficult.

> >	It is a misnomer to call all non stable processes chaotic.

>

I didn't!

Also correct. Same reasoning as above.

>

> >	Processes (systems) are described by differential equations.

>	Many dynamical systems are indeed defined in this way, but many are -not-. Indeed, none of the examples in my posts were of this nature.

You are right that the examples in your post are not described by differential equations.

> >	To ^define^ chaotic systems as being dependent on initial conditions is a also a misnomer.

>

I didn't!

You are correct. You did not write down this definition.

On the other hand the authors of books you mention discuss what is called: "sensitive dependence upon initial conditions" or SDIC.

IMO this is a property of all (dynamic) systems. If a system is described by a differential equation than you need initial conditions in order to solve them. Different initial conditions will give different solutions.

For those systems two things are important: The initial conditions and the parameters. The importance of the last is under estimated by most writers.

>	I can't comment further since it seems to me that your followup has nothing to do with what I said in the post to which you are nominally responding. Chris Hillman Home page: http://www.math.washington.edu/~hillman/personal.html

The following link gives a good overview:

http://www.santafe.edu/~shalizi/notebooks/chaos.html

The following link discusses: Steering Chaos.

http://www.npl.washington.edu/AV/altvw51.html

Nick

Kevin Aylward wrote:

>	"Nicolaas Vroom" wrote in message news:ySNa8.140976$rt4.13055@afrodite.telenet-ops.be...

> >	Systems are often clasified in stable or non stable. A temperature control loop is stable if the process reaches within a certain time its new set point. The transfer function of such a process has no null points on the imaginary axis (if I remember well)

>	Er... Actually, a linear system is stable if its transfer function has no poles in the right half plane. This can also be expressed by the number of net encirclements of the plot of magnitude and phase around the -1 point.

You are right. It is roughly thirty years ago that I studied control theory.
I did a search using Nyquist diagrams and I found these results:

http://www.engin.umich.edu/group/ctm/freq/nyq.html Excellent also select the links "PID" and "Root Locus"

http://www.ame.arizona.edu/ame455/L17n.pdf

http://www.engr.udayton.edu/faculty/rkashani/mee527/nyq_stab/nyq_stab_margins.htm

This article contains the following sentence:
"For these systems, in addition to determining the absolute stability of a system, the Nyquist diagram provides qualitative information as to the degree of stability.
The (-1,0) point plays the same role in the Nyquist diagram as the imaginary axis does in the in the root locus diagram."

The Nyquist diagram makes a clear distinction between which processes are stable and which are not.
You do not need chaos theory for that.

A similar issue is raised for the solar system:
is it stable or not. and which theory describes that.

IMO the best answer is comes from Newton and GR.

Nick.

Nicolaas Vroom wrote:

> > >

A temperature control loop is stable if the process reaches within a certain time its new set point. The transfer function of such a process has no null points on the imaginary axis (if I remember well)

Kevin Aylward commented:

> >	Er... Actually, a linear system is stable if its transfer function has no poles in the right half plane. This can also be expressed by the number of net encirclements of the plot of magnitude and phase around the -1 point.

Vroom replied:

>	The Nyquist diagram makes a clear distinction between which processes are stable and which are not. You do not need chaos theory for that.

Au contraire--- this stuff is -part- of the theory of chaotic dynamical systems!

Transfer operators (infinite dimensional generalizations of "transfer matrices") and zeta functions (generalizations of the Riemann zeta function) were defined in a general dynamical setting by Ruelle, and they have been intensively studied by Mark Pollicott and Viviane Baladi, among others. The motivation is to understand things like the rate at which a dynamical system "mixes" a phase space; more precisely, the rate at which statistical correlations between the orbits of two points decay. This rate turns out to be closely related to the nature of the spectrum of the transfer operator.

This subject is part of ergodic theory, which evolved directly from the studies of Poincare on solar system dynamics and the proposals of Boltzmann concerning statistical dynamics (and the objections of Zermelo and others to Boltzmann's proposals). The point here is that the properties called "mixing" (of various orders or rates) which a given measure-theoretic dynamical system may or may not have are usually regarded as belonging to "chaotic dynamics". Note too that Hamiltonian systems are very special cases of measure-theoretic dynamical systems (the phase space is typically an "energy surface" and the measure is Liouville measure; "Liouville's theorem" says this measure is preserved by a Hamiltonian flow on the phase space.)

References:

For background on ergodic theory, see the undergraduate textbook:

author = {M. Pollicott and M. Yuri},
title = {Ergodic Theory and Dynamical Systems},
publisher = {London Mathematical Society},
series = {Student Texts},
number = 40,
year = 1998}

For expository papers on "decay of correlations" by Baladi and Pollicott, see

http://www.ihes.fr/~baladi/publi.html

http://www.ma.man.ac.uk/~mp/preprints.html

For an expository paper on transfer operators and zeta functions applied to continued fractions (the "Gaussian shift" is an ergodic theoretic model of the simple continued fraction algorithm), see

author = {Mayer, Dieter H.},
title = {Continued fractions and related transformations},
booktitle = {Ergodic theory, symbolic dynamics, and hyperbolic spaces},
editor = {Tim Bedford and Michael Keane and Caroline Series},
publisher = {Oxford University Press},
year = 1991,
pages = {175--222}}

For a monograph on the decay of correlations, see:

author = {Viviane Baladi},
title = {Positive Transfer Operators and Decay of Correlations},
series = {Advanced Series in Nonlinear Dynamics},
volume = 16,
publisher = {World Scientific},
year = 2000}

Note that dynamical zeta functions really are a meaningful generalization of the Riemann zeta function, and the relationship between the spectrum of transfer operator and the corresponding zeta function really is analogous to the notions introduced by Riemann in connection with his prime counting formula. In particular, there is a beautiful dynamical analogue of the famous Prime Number Theorem; see the expository paper by Pollicott on the page cited above and see also:

author = {Pollicott, Mark},
title = {Closed geodesics and zeta functions},
booktitle = {Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces},
editor = {Tim Bedford and Michael Keane and Caroline Series},
publisher = {Oxford University Press},
year = 1991,
pages = {153--174}}

author = {Michel Lapidus and Machiel van Frankenhuysen},
title = {A Prime Orbit Theorem for Self-Similar Flows and {D}iophantine Approximation},
note = {math.SP/0111067}}

(Caveat: dynamical systems theorists do not generally expect to arrive at a dynamical systems theoretic proof of the famous Riemann Hypothesis; rather, just as a proof of RH would give incredibly detailed information about prime numbers, so knowledge of the spectrum of a dynamical transfer operator gives remarkably detailed information about certain dynamical properties of the dynamical system under study. OTH, see http://xxx.lanl.gov/abs/math.GM/0111262.)

For background about geodesics on compact Riemannian manifolds (symbolic dynamics, the most abstract branch of ergodic theory, which is closely related to automata theory, developed in part from Morse's seminal work on this phenomenon, which was in turn motivated by a chess problem!), see

author = {Series, Caroline},
title = {Geometric methods of symbolic coding},
booktitle = {Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces},
editor = {Tim Bedford and Michael Keane and Caroline Series},
publisher = {Oxford University Press},
year = 1991,
pages = {125--152}}

which features a delightful connection between geodesics on the upper half plane model of H^2, the simple continued fraction algorithm, and the Conway knot theoretic game once explained in this forum by John Baez, which arises via the "Farey-Series tiling".

For more about the many remarkable and deep relationships between number theory and dynamical systems, see:

author = {J. C. Lagarias},
title = {Number Theory and Dynamical Systems},
booktitle = {The Unreasonable Effectiveness of Number Theory},
editor = {Burr, Stefan A.},
series = {Proceedings of Symposia in Applied Mathematics},
volume = 46,
publisher = {American Mathematical Society},
address = {Providence, Rhode Island},
year = 1991}

author = {M. M. Dodson and J. A. G. Vickers},
title = {Number Theory and Dynamical Systems},
series = {London Mathematical Society Lecture Notes},
volume = 114,
publisher = {Cambridge University Press},
year = 1989}

author = {P. Cvitovanic},
title = {Circle Maps: Irrationally Winding},
booktitle = {From Number Theory to Physics},
editor = {M. Waldschmidt and P. Moussa and J.-M. Luck and C. Itzkyson},
note = {Lectures given at the meeting ``Number Theory and Physics'',
held at the ``Centre de Physique'', Les Houches,
France, March 7--16, 1989},
publisher = {Springer-Verlag},
year = 1992}

In particular, note that the classic "problem of small divisors", which was stressed by Poincare in his work on solar system dynamics, is closely related to "KAM theory" (on perturbations of Hamiltonian systems) and also to simple continued fractions, as well as to the Fields' Medal winning work of Yoccoz:

author = {S. Marmi},
title = {An Introduction To Small Divisors},
note = {math.DS/0009232}}

author = {M. M. Dodson},
title = {Exceptional sets in dynamical systems and {D}iophantine approximation},
note = {math.NT/0108210}

For connections between dynamical systems, homogeneous spaces, affine forms (c.f. classical invariant theory), and Diophantine approximation, see

author = {Dmitry Kleinbock},
title = {Badly approximable systems of affine forms},
note = {math.NT/9808057}}

author = {Dmitry Kleinbock and Gregory Margulis},
title = {Flows on homogeneous spaces and {D}iophantine approximation on manifolds},
journal = {Ann. Math.},
volume = 148,
year = 1998,
pages = {339--360},
note = {math.NT/9810036}}

For a recent example of a connection between the theory of partitions of natural numbers (c.f. also Young diagrams!) and dynamical systems, see:

http://xxx.lanl.gov/abs/math.CO/0110075

BTW, here is a putative example of a theory of algorithms not limited by the CT thesis:

http://xxx.lanl.gov/abs/physics/0106045

(The work of Traub and Werschulz has been challenged by Parlett and others; check out Math Reviews.) More interesting, I think, is this paper, which introduces yet another nice Galois duality:

http://xxx.lanl.gov/abs/math.DS/0112216

Chris Hillman

Home page: http://www.math.washington.edu/~hillman/personal.html

Chris Hillman wrote:

>	This subject is part of ergodic theory, which evolved directly from the studies of Poincare on solar system dynamics and the proposals of Boltzmann concerning statistical dynamics (and the objections of Zermelo and others to Boltzmann's proposals). References:

This is an excellent post

However I am still trying to find an answer on the following three questions: 1a What is the state of our Solar system over 100 million years ?
1b Can we predict this state ? (Accurate ?)
2. Is our solar system stable ? 3. Can we use the chaos theory to answer the first two questions ?

The first two questions are closely related.

In order two answer the second question you must first have a good difinition of what means stable.
If our definition of stable means that our Solar system will exist forever the answer clearly is No.
In fact our solar system goes through a number of phases: is born, the planets form, evolves, matures and dies (explodes).
All stars behaves in similar ways (with many exceptions).
However the time period involved can be quite different.

To answer the first question two things are important: Measurements and a Model (Law or theory) Measuremnts gives us the state of the past.
The more measurements, the more accurate, the better. Those measurements are impotant to calculate the parameters of your model. Two models can be used: Newton's law and GR. GR is the most accurate if you want to include the behaviour of Mercury.

To answer the third question we must first have a good definition of the chaos theory.
Besides the chaos theory there is also a range of other theories:
1. Modern Theory of Dynamical Systems.
2. Theory of Chaotic Dynamical Systems.
3. Theory of Complex Dynamical Systems.
4. Ergodic Theory
5. Theory of Dynamical Systems which show chaotic behaviour.

At page 124 of the book Exploring Complexity by Gregoire Nicolis chaotic behavior is studied as suggested by Otto Rossler. Starting point are three equations: