The Fallacy Of Chaos
Mark
antithesis of determinism. The argument underlying this proceeds by
the following chain:
* 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 classic example is the question of stability of an N-body system
under the sole influence of gravity. The slightest change in the
initial positions and motions of the bodies can literally make the
difference between whether a given body is expelled from the system
or not.
As such, this leads to the conclusion:
* Chaotic system possess, for all practical purposes,
a fundamental degree of indeterminacy.
Hence, the conception of chaos as the antithesis of strict determinism.
Unfortunately, there is a serious flaw in this conception, namely
the keyphrase: "IF the initial conditions are not known with infinite
precision..." which stands as the pretext to the whole line of
argumentation.
The fallacy is the premise underlying this supposition, namely that
there is and can never be any such thing as a Theory Of Initial
Conditions, particularly one which predicts precise values for
initial data.
I could take the very same ideas and argue in the exact opposite direction
as follows:
* A chaotic system is a dynamic system in which even the slightest
variation in boundary conditions lead to large deviations in the
system at subsequent points.
* Thereby providing us with a manifying lens with which to observe
and measure initial data with dramatically greater and greater
degrees of accuracy
* Thereby giving us increased opportunity to discover whatever
regularities may exist in the boundary data of dynamic systems to
allow us to arrive at a general theory of initial conditions.
Now the chaotic system, quite the opposite from being the antithesis
to determinism, becomes a tool and means by which determinism could
be practically realised.
Squark
news:<a44qpn$3ke$1...@uwm.edu>...
> * A chaotic system is a dynamic system in which even the slightest
> variation in boundary conditions lead to large deviations in the
> system at subsequent points.
>
> * Thereby providing us with a manifying lens with which to observe
> and measure initial data with dramatically greater and greater
> degrees of accuracy
I'm not sure this is a correct point of view. Information is still
preserved, in a sense (for instance, the Liouville phase space
volume is preserved), so we don't get a magnifying lense here,
only "wild" hard-to-compute behavior.
Best regards,
Squark
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Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and use "com" rather than
"exe")
Brian J Flanagan
> 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.
Louis M. Pecora
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.
Chris Hillman
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
Nicolaas Vroom
news:Pine.OSF.4.33.020210...@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://users.pandora.be/nicvroom/initcond.htm
Nick
Chris Hillman
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.
Kevin Aylward
message news:ySNa8.140976$rt4....@afrodite.telenet-ops.be...
> Systems are often clasified in stable or non stable.
> A temperature control loop is stable if the process
> reaches within a certain time its new set point.
> The transfer function of such a process has no
> null points on the imaginary axis (if I remember well)
Er... Actually, a linear system is stable if its transfer function has
no poles in the right half plane. This can also be expressed by the
number of net encirclements of the plot of magnitude and phase
around the -1 point.
> It is a misnomer to call all non stable processes chaotic.
Obviously. A simple electronic oscillator is not (usually) chaotic.
Kevin Aylward , Warden of the Kings Ale
ke...@anasoft.co.uk
http://www.anasoft.co.uk
SuperSpice, a very affordable Mixed-Mode
Windows Simulator with Schematic Capture,
Waveform Display, FFT's and Filter Design.
Nicolaas Vroom
> "Nicolaas Vroom" <nicolaa...@pandora.be@telenet-ops.be> wrote in
> message news:ySNa8.140976$rt4....@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.
Chris Hillman
> > > 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
Nicolaas Vroom
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).
>
>
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:
dx/dt=-y-z, dy/dt=x+ay and dz/dt=bx-cz+xz
Spiral Chaos exists for a=0.32, b=0.3 and c=4.5
Screw chaos for a=0.38, b=0.3 and c=4.5
(homoclinic orbit c=4.82)
The question is what has this to do with our solar system.
Is this relevant ?
One thing becomes clear: you first need some equations
(Newton's Law or GR) and chaos is more like a behaviour.
However it is not clear that Our solar system also
shows this behaviour and exactly what is the difference
in behaviour between with and without.
Specific can this chaotic behaviour already be demonstrated
over the period that measurements are available ?
If you study Rosslers model it becomes clear how important
parameters (a, b and c) are.
That is exactly my point in one of my previous posts.
If you want to use Newton's Law you must know the masses
of all the objects (planets, asteroids and meteors) involved.
The same for GR.
IMO the answer on question 3 is NO.
Chris Hillman
> 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.
[snip]
Nicolaas Vroom replied:
> > 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).
>
> 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 ?
Thanks for the praise, but I thought I had made it clear that, first, (2)
is really many questions, because (a) there is more than one good notion
of stability, and (b) there is more than one relevant dynamical time
scale, and second, that the theory of dynamical systems not only greatly
clarifies these distinctions, but gives answers to questions of stability
which yield valuable insight.
So, I tried, but I'll have to call it quits here. In fact, let me say that
in future I'll ignore posts by Vroom and Gorgun, so any future
misstatements/misconceptions by them in s.p.r. or s.a.r. will go
unchallenged unless someone else feels like saying something.
Nicolaas Vroom
The most important question is #2.
The correct way to answer that question (and similar questions) is by
performing observations over a long period. How longer the better.
A practical example is by observing a star cluster.
What we should observe is that the positions of the stars will change
and if you are lucky that two stars will collide.
When that is the case we can define that the cluster was not stable.
What is also possible that a star will escape from the cluster.
Also in that case the cluster was not stable. The whole point is the
only way to answer the question is by performing observations.
A different way is to use the theory of dynamical systems.
A good overview is https://en.wikipedia.org/wiki/Dynamical_systems_theory.
"Dynamical systems theory is an area of mathematics used to describe
the behavior of complex dynamical systems, usually by employing differential
equations or difference equations." There are 11 related fields.
This raises a new question: does this theory answer question #2?
To be more specific: can the question be answered by means of mathematics?
Of course there are systems which can be described by means of mathematics
but this are all mathematical systems, not physical systems.
Is it possible to decide by means of mathematics if a roulette is correct?
No. The only way is by performing an experiment 1000 times and based on the
results to define the roulette as 'correct', otherwise certain corrections
have to be performed.
And what about our solar system. Original this was also a cluster of let me
say 1000 small objects. During the ages the stars collided and merged. Slowly
the number of objects became smaller and a large one appeared at the centre
and smaller ones started to revolve around this large one. At least that is
the way we think it physical happened.
To test this we can try to simulate this system on a PC and use for example
newton's law. We start with a system with 1000 identical objects, give each
an initial position and speed and observe how the simulation evolves.
Most probably all the points will collide at one point.
To make the simulation more realistic you have to adapt the possitions and
velocities such that the simulation becomes more stable. That means you
have found a mathematical solution to a physical question. But is that in
agreement which what physical happened, millions of years a go?
Most probably not.
This leads me more or less to my final conclusion.
It does not make sense to study the evolution of our solar system from
a mathematical point of view starting first with 'acurate' observations
roughly 500 years ago, because our solar system is much older.
The point is that starting from that day our solar system, physical evolved
more or less by itself, based on the influence by a group of local objects,
not under the influence of any mathematical law.
>From a physical point of view studying this evolution there is no accuracy issue.
The same can be said that there is no chaotic issue.
>From a mathematical point of view there is an accuracy problem.
They come to light whan you perform a mathematical simulation of your system.
May be most important are the measurements of positions of the objects involved
and the time of these observations. Using these observations the speed and the
accelerations of the objects involved are calculated.
A second accuracy problem happens when objects come near each other. This is also
called the chaos problem. To decrease the step size can be a solution.
The third problem is the mathematics or equations used as part of the simulation,
specific all the parameters involved.
If the equations use the speed of light also that parameter has to be calculated,
based on observations. You can also call that the fourth problem.
For the problems involved, and the impossiblity, select this link:
http://users.pandora.be/nicvroom/wik_Dynamical_systems_theory.htm#ref2