Conference rusure::math

    You are in luck!
    There is a book on the subject called...
    
    CHAOS
    by Gleick (spelling is not certain)
    Paperback around $9, hardcover appropriately higher!
    
    Krishna
    

>  ...He believes that this theory proves that NOTHING is random
>    and every event may be predicted and effected by every other event,
>    no matter HOW unrelated it may seem.

Well, yes and no. A simple definition (and the only one I know, so it may
not be totally accurate) is:

Chaos theory is the study of iterated continuous functions (i.e x, f(x),
f(f(x)), ...) that don't converge or diverge to infinity but that have
the property that arbitrarily small differences in x can cause arbitrarily
large differences [relative to the range of f] in some f(f(f(...x...))).

If you think of the function f as being a vector-valued function of a vector,
chaos theory can give some insight into real-world systems where f(x) is
"the state of the system at t=delta, given that the state at t=0 is x".

It does not, however, eliminate randomness; it does allow us to distinguish
patterns that may spontaneously arise out of the randomness, which (I guess)
you could say lets you reduce the "degree of randomness" of a system a little.

An example is the Red Spot on Jupiter; it used to be thought that it was some
special feature of the Jovian geography, but applying Chaos theory to the
equations of state of a large swiftly-rotating atmosphere shows that
a feature like the Red Spot is a self-organizing subsystem of a chaotic
system. It was bound to form somewhere in the Jovian atmosphere, and once
formed will maintain itself. It doesn't help you predict what its exact shape
or location will be at any time, however, so randomness still rules...

(My personal impression is that at this time Chaos theory hasn't risen much
above the taxonomy stage - i.e. "this is a chaotic system of type N" - and
that it is primarily an aid to understanding complex systems rather than an
aid in manipulating them).

Anyone want to clarify this muddy explanation?

    Very briefly:
    
    Physics tends to state its laws in terms of differential equations
    over time: that is, in terms of relationships which describe how
    measurable and infered quantities change over infintessimally
    small differences in time.
    
    For example, the behavior with respect to position along one axis
    of an object on which no forces are acting is described by the
    simple differential equation:
                                 
    			dx/dt = C
    
    Which says that the position changes in an infintesimal amount of
    time by a constant amount (the particular differential dx/dt has
    a special name, as you probably realized -- the instantaneous
    velocity, which, since it is constant in this case, can simply
    be called the velocity).
    
    We can solve this differential equation to tell us where that object
    will be at any time that the differential equation is valid, assuming
    that we have a little bit more information referred to as the "boundary
    conditions".  In this case the boundary condition needed is simply
    the position at some particular time (which we might as well call
    time 0), and the differential equation is very easily solved:
    
    		x = x0 + Ct
    
    where x0 is the position at time 0.
    
    Generally, things are a bit more complex, we don't have a single
    differential equation but a whole series of them each relating
    changes in one quantity to changes in others.  This is called a
    system of differential equations.
    
    In the past most physical theory was devoted to systems of differential
    equations with a particular property called "linearity".  Such
    a system of linear equations can get very complex, involving many,
    many variables, but each equation in the system has a relatively
    simple, well defined form.  Methods have been developed for solving
    systems of linear differential equations, and although you cannot
    always find a specific non-differential equation which solves a
    particular system, you can pretty much always crank specific values
    through such a system and get a specific answer to a question about
    what value a particular variable will have at a particular time
    given specific measured values for constants and boundary conditions.
    
    A feeling grew up in the scientific community by the 19th century
    that it was the "normal" state of affairs for physical laws to be
    reducable to systems of linear differential equations.  It was
    known that some real physical phenomena needed non-linear differential
    equations to be described accurately, but the general feeling was
    that the universe was well enough behaved that any such system of
    non-linear equations describing real phenomena could always be
    approximated, to any degree of accuracy desired, by a complex enough
    system of linear differental equations -- one simply had to be
    clever enough to find them.
    
    If this feeling were really conscious it might have been questioned,
    but it was primarily unconscious -- a part of the ill defined but
    important concept of elegance of a physical theory.  If confronted
    with it most physicists would have been willing to predict that
    not *all* systems of non-linear differential equations could be
    approximated by linear systems, but they would have guessed that
    systems which could not would be complex and arbitrary --
    "pathological."
    
    This was the state of affairs until about ten years ago.  For about
    two decades before that isolated exceptions had been found but had
    made little impact.  But about a decade ago the importance and
    reality of these isolated examples in many fields was realized.
    
    A linear system of equations has an important and very useful property.
    If you plug in initial (boundary) values which differ by a very
    little amount, they will result in answers which differ only by
    a small amount (specifically, an amount which is at worst proportional
    to the amount of time which has passed since the initial conditions).
    This means that small errors in your initial measurements results
    in relatively small errors in your predictions and that very small
    influences could be neglected completely.
    
    What was discovered was that it was possible for very simple non-linear
    equations -- arising quite naturally in describing real physical
    phenomena -- did not act this way.  Small errors in measurements
    or infintessimally small neglected factors could result in arbitrarily
    large errors after relatively short amounts of time.  Basically,
    this meant that to make meaningful predictions about such systems
    very far into the future one would have to have effectively infinitely
    precise measurements, which is impossible.
    
    Eventually, what was realized was that this "chaotic" behavior was
    the norm for non-linear equations -- that if you picked a random
    set of non-linear equations then more likely than not, for at least
    some values of its constants and broad regions of its boundary
    conditions, it would be chaotic.
    
    The first really well studied set of chaotic non-linear equations
    was the Lorenz equations, which had been invented to be a simple
    approximation of what happens in atmosphere to produce weather.
    As I remember, they are three rather simple equations in three
    unknowns (pressure, temperature and humidity?).  From their chaotic
    behavior and the unproven but likely proposition that more realistic
    (and complex) equations would be unlikely to be any less chaotic
    comes the frequently cited statement that the beating of the wings
    of a butterfly in Brazil on Monday could have a significant affect
    on the weather in England on Sunday (or words to that effect;
    probably a slight exageration but in principle correct).
    
    The consequence is that there is a severe limit to how far ahead
    we can predict the weather: doubling the accuracy of our measurements
    will only add a small increment in the range of time over which
    we can make accurate predictions.
    
    It is certainly true that if a situation is described by a chaotic
    system of equations then if we want to make accurate predictions
    about something then no influence -- no matter how small or how
    distant -- can be neglected.  But in reality that influence is
    indistinguishable from random -- this is very far from the sense
    of orderly connectedness associated with the so called alpha/omega
    mystical state.
    
    It certainly does not say that events may be "predicted" by every
    other event -- just the contrarary it says that even in a deterministic
    universe there are things which cannot be predicted in any truly
    meaningful sense.
    
    And it says nothing about things not being random.  It says that
    IF the universe were deterministic THEN it would still not be
    predictable.  But that does not mean that it *is* deterministic.
    The arguments which have placed randomness at the core of the
    universe in quantum mechanics are in no way invalidated.  Indeed
    one can plausibly argue that it may lift that randomness from the
    essentially unobservable subnuclear regime to that of everyday
    life -- that real, absolute randomness is part of our macroscopic
    world.  Long before the Brazilian butterflies wings come into the
    calculations the random movements of the electrons around and within
    us right now would have to be taken into account.
    
    The bright side of all this, is that once scientists started looking
    at chaotic systems, some amazing regularities became apparent. 
    Precise predictions are impossible but statistical predictions
    are not.  We cannot predict when and where a whirlpool will form
    in the river but we can predict pretty well how many will form.
    We cannot predict the weather three weeks from now, but we can,
    perhaps, predict the climate three years from now.  And because
    of the regularities which are being discovered, the ability to
    predict the climate implies the ability to predict the liklihood
    under given conditions for a given individual of the chaotic cardiac
    nerve firings believed to be the cause of heart fibrillations and
    the ability to predict what changes would reduce that liklihood.
    
    						Topher

    Re .3:
    
    This is a marvelous explanantion.  May I (we) please have permission
    to copy it?  I want to include it in my file of "great explanations
    of hard to understand topics".

One of my coworkers is very interested in this topic.

What is the relation (if any) of this area of mathematics to
catastrophe theory ?

Catastrophe theory is (bear with me, I'm a neophyte in this) the study
of the qualitative behavior of solutions of multidimensional differential
equations as parameters within those equations vary.  Most of the current
results are confined to the class of differential equations that describe
gradient systems.

Is chaos theory a generalisation of this ?

RE: .4
    
    Thank you -- as far as I'm concerned anything I post in an "open"
    conference is public, so you are welcome to copy it.  I just read
    what I wrote for the first time and it *is* rather more coherent
    than I thought as I wrote it.  It does have some of the characteristics
    of a rough draft, however, (e.g., different terms for essentially
    the same thing) so maybe I'll make the effort to clean it up.  The
    subject has come up in several different conferences that I know
    of.
    
    One thing, though, the last few paragraphs were written as direct
    responses to statements in .0, so it would be a bit clearer if you
    "picked up" that note also as preface.
    
    					Topher

    Catastrophe theory (more legitimately known as singularity theory)
    is the study and classification of singularities of mappings.  Probably
    the best introduction is the little paperback by Vladimir Arnold
    titled "Catastrophe Theory", Springer Verlag.

    These singularities are the states where a qualitative change occurs in
    the nature of a mapping.  For example a conic results from the intersection
    of a plane with a cone.  As you change the angle of the plane with
    respect to the cone, the intersection will suddenly change from an
    ellipse shape to a hyperbola.  The singular position yields a parabola.
    What distinguishes the singular position is that any infinitesimal
    perturbation (except for a set of measure zero) will qualitatively
    change the situation - in this case the parabola goes over to either
    an ellipse or a hyperbola.

    A lot of absurd claims have been made for the applicability of catastrophe
    theory, but singularity theory is legitimate and worthwhile.

    Catastrophe and chaos theory are related.  Nonlinear dynamic
    systems can go over from a structurally stable condition (where a
    small perturbation of the initial conditions yields a small
    perturbation of the later states of the system) to an unstable
    condition via a small perturbation of the system parameters.

    One of the earliest chaos-theoretic questions to be carefully studied
    was the stability of the solar system (or other n-body Newtonian system.)
    Poincare did early work on the behavior of systems of nonlinear
    differential equations.  He considered multivariate Taylor series
    expansions beyond first order in the solution of such equations, and
    his attempts to qualitativly classify such systems lead to some of the
    earliest fundamental ideas of topology - homotopy and homology.

    - Jim

    thanks for all the information. I just stumbled over, and signed
    up for a M.A.E.T. 1/2 day seminar on CHAOS!
    
    (MAET is Maynard Area ENgineering Training)
    
    The course costs $300.00 internal "cost center" dollars, and requires
    managerial approval. Cant wait!
    
    its sometime in the 1st half of january

Extracted from COURSES account on MILRAT:
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-
Title: CHAOS: An Emerging Science

Course Number: SAT89-45

Date: 1/19/88		Time: 11:00 to 17:00

Location: Lent Satellite Training Room ML04-5

Class Size: 18

Instructor: Celso Grebogi et al

Enrollment Closing: 1/12/89		Cost: $300

Description: These lectures will present the fundamentals of chaotic dynamics,
	including many examples using a variety of illustrations and
	computer-generated figures and movies.  Topics covered will include:
		
		o Introduction and Basic Concepts
		o Strange Attractors
		o Bifurcations to Chaos
		o Fractal Basin Boundaries
		o Sudden Changed in Chaos

Synopsis: Even relatively simple systems can behave in a surprisingly
	complex and erratic manner.  When viewing the time history of such
	systems, one often has the feeling that a statistical description
	is called for.  This type of behavior is one of the attributes of 
	chaotic dynamics.  Within the last decade, there has been an explosion
	interest, as well as many major developments, in the field.

Speakers: Celso Grebogi - Research scientist in the Lab. for Plasma Research
			  at the U of Md, College Park.
	  Edward Ott - Prof. in the Dept. of Physics and Astronomy at the
		       U of Md.
	  James A. Yorke - Director of the Inst. for Physical Science and
		       Technology at the U of Md.

    The current (Jan 6, 1989) issue of Science contains (starting on
    page 25) a fairly good article on a possible practical application
    of chaos to epidemiology.
    
    Traditionally irregularities in epidemiological data has been
    attributed to gross random fluctuations outside the infection models
    -- things like bouts of bad weather at critical times.  Some
    researchers are now claiming that much of the apparent randomness
    is actually due to deterministic chaos within the models.  This
    has implications for prediction and decision making about corrective
    actions such as mass immunization.
    
    A good article, not particularly deep technically, but presenting
    a good picture of both the difficulties and potentials of applying
    chaos theory to "real" systems.
    
    					Topher

    I have read "Chaos," and it seems that this book is an interesting
    mathematical twist of the classical ontological argument (i.e.:
     Fate vs. Freewill) which numerous noted authors have posed.
    Descartes' argument, "I think, therefore I am," sort of blows away
    the theory that the Universe is preordained.  Now, some would argue
    that our own thoughts are subject to these complex mathematical
    laws, but that is hard to prove.  St. Anselm and St. Thomas Aquinas
    have both posed (seemingly) valid arguments for and against freewill
    (and God).  There is also a treatise in Aristotle's Metaphysics
    which pertains to the nature of our existence.  The answer I have
    for anyone who cares is:  "The truth is what you believe."  There
    very well may be "absolute" truths, but in the light of the quantum
    world, these truths are vague.  We live in a probabilistic universe,
    the more we study it, the more complex it appears.  If you don't
    believe that, read "Chaos," and you will see (perhaps) how complex
    the universe is.  Sorry if I digress.
    
    					-Superclam
    

    Reference:  Proceedings of IEEE, August 1987, Special Issue on Chaotic
    Systems.
    
    Barry

    I have been trying to figure out what is such a big deal about this
    chaos thing.  I have always thought that it was a lot of
    hot air.  I had read lots of pop magazines (including Scientific
    America) on chaos, read lots of chaos books full of neat pictures.  I
    even carefully watched the NOVA program on chaos.  What had I learned so
    far?  Not much.  There were talks about small initial disturbance can
    creat divergent results in nonlinear system, but we know that since the
    day of Poincare.  There was the Lorentz attractor which one can't do
    much about.  What else?  Not much, except a bunch of pretty
    pictures.  Nothing fantastic. So why the recent big fuss about this thing?
    
    Well, I checked out a real physics book on chaos this afternoon, and
    within an hour found the reason.  The real point is a guy named
    Mitchell Feigenbaum discovered a universal equation that describes all
    chaotic systems (well almost all of them).  Yes, it was a true break
    through because for the first time, we discovered that some predictions
    can be made about the chaotic systems that were inaccessable till now.
    This experience tells me that pop science is really for entertainment
    and contains very little substantive information.  Sigh, I almost went to
    study chaotic theory under Wolfram, but didn't partly because I thought
    there weren't much substance in it.  On second thought, I should have
    known better.  It was stupid and presumptious for me to think that
    Wolfram was going after fads.  Well, too late now. 
    
    Eugene

Eugene,

I'll admit to reading only pop books on chaos, but James Gliek (sp?) did
talk about Feigenbaum's work.

I still have trouble understanding what is the big deal.

From his description, physical systems seem to fall into three categories:

	linear systems which we can understand very well

	borderline systems which Feigenbaum and related math helps us with

	fully chaotic systems, which are still unpredictable

From descriptions I have read of the fluid motion between rotating cylinders or
the interdrop interval for a faucet, you must set up a very careful experiment
to put a system into the borderline.  If the system is fully chaotic, then you 
cannot predict the results of your experiment.

You can of course make a non-linear model for a fully chaotic system, like 
fluid flow in a cloud or a river, and you can even get some agreement between 
the behavior of your model and a real system.  But a large number of non-linear 
models may give the same level of agreement.  So you have lost the one-to-one 
correspondence between model and prediction which makes linear physics so 
interesting.

I agree with you that the physics of chaos is interesting and not a fad.  It is 
interesting to see where the borderline is and how far we can push it.  

What is disagree with is the presumption of some of the chaos folks that theirs 
is now the only interesting field in math, physics or whatever.

    re .14,
    
    I will write a review of what I learnt as soon as I figure out a few
    details...
    
    Eugene

The crucial development in the theory of chaos is the discovery of the
universal equations and the universal constant delta=4.669... and 
alpha=2.5029...  The discovery is really about the transition from
predictable behavior to chaotic behavior which itself is still beyond
our reach.

Most systems that eventually become chaotic go through a transition
stage of the so call "period doubling".  For example, consider the 
following simple function:
            2
f(x) = a - x

Step 1: Let a be small (say 1/10)
                                        n
Now if you choose x0=0 and look at lim f (x0), you
will discover the limit exists.

Step 2: Now we increase a and at certain point, you will discover that
             n
        lim f (x0) no longer exists, but rather the number oscillates
        between two distinct values.  Hence, if we consider
         2n
        f  (x0), we will discover that it exists.

Step 3: Now if we increase a still further, the same phenomenon happens
         2n                         n
        f  (x0), and well see that f (x0) oscillates between four values.
                          4n
Step 4:  Now we consider f   and so on and so forth.

Observation:  

1.  This simple example exhibits a classic behavior of a system going
    from a predictable one to a chaotic one--The phenomenon of period
    doubling.
                                                   k
2.  If you do a detail study of all the functions f , you will discover
     k
    f converges functionwise to a function that satisfies the equation:

    g(x) = -alpha * g(g(x/alpha)

This is the so call Universal Equation.



Now here is something big:

Any system that exhibits the phenomenon of period doubling eventually
converges to the behavior dictated by the function g (directly or
indirectly).

If we normalize g with g(0) = 1, then there exists a unique infinitely
differentiable solution to the Universal Equation.

Two immediate concrete results:

1. Suppose we have a system with a nonlinear parameter d, and
   if the system goes through period double at d(1), d(2), d(3),.. d(n)
              d(n) - d(n-1)
   Then lim   ------------- exists and equals to delta (as defined above)
        n->oo d(n+1) - d(n)

2. lim d(n) exists and the system becomes chaotic at the limit.
   n->oo

3. Suppose in the same system the distance of the bifurcation is e(i)
   where i denotes the i'th stage of bifurcation.  In other words,
   e(i) is the difference between the two new oscillation points.

               e(n) - e(n-1)
   Then lim    ------------   exists and equals to alpha (as defined above) 
        n->oo  e(n+1) - e(n)

Now we really have some substantive results (big results, not in the
same scale as F=ma or the law of gravitation, but still big).  Now since
the universal equation is nonlinear, lots of work is needed to figure
out the behavior of g and the operator T that takes g(x) to 
-alpha * g(g(x/a)), eigenvalues and etc...  Heavy duty work.

Eugene