Les,
My quick answer is that in your list 1-3 are more
foundational and have little direct applications in
economics. One does find quite a few applications
of points 4-6, although largely in pretty theoretical
and mathematically oriented literature.
I think Ralph Abraham is a genius. He is also
a great coiner of phrases, including the "blue
bagel catastrophe" and "chaostrophe" and
"morphodynamics." He also discovered "chaotic
hysteresis," although I am the one who coined that
term. His book with Marsden is hardly his farthest
out, and he has become rather metaphysical in
more recent years. Anybody who wants to see a
really far out example should check out his _Chaos,
Gaia, Eros_, 1994, New York: Harper Collins.
But I like all his funky figures the best.
Barkley Rosser
----- Original Message -----
From: "Les Schaffer" <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Sent: Wednesday, June 18, 2003 3:14 PM
Subject: Re: [PEN-L] Complexity
> i agree chaos and complexity studies have a fad __component__.
>
> Sabri Oncu writes:
>
> : However, with what I know about chaos, and it is not much, mind
> : you, my subjective judgment is that "chaos" is a fad as
> : "topology" was once to "mathematical analysis" or "game theory"
> : was to "economics".
>
> topology has interesting applications in quantum gravity. as a branch
> of analysis i think it has real beauty in terms of classifications of
> sets.
>
> on the other hand, there is a branch of workers in nonlinear dynamics
> who have gone topology-beserk: see Abraham and Marsden, for example.
>
> i just read an news article this morning that a researcher in Germany
> has devised a unique structural element using what he calls
> "Topological interlocking of protective tiles"..
>
> : He once said this, in exaggeration,
> : maybe: all this chaos theory does is proving the "non-existence"
> : of the solutions of this or that nonlinear dynamical problem.
>
> your friend is missing something.
>
> it is true that rigorous chaos theory proves the non-existence of
> certain kinds of solutions to dynamical systems. that is, the
> non-existence of classical style solutions (and failure of convergence
> of orbit expansions in perturbation parameter) to differential
> equations. i.e., the clockwork solar system which exhibits strong
> (quasi-)periodic behavior.
>
> but look at what we got in return:
>
> 1.) super-convergent expansion techniques (a la
> Kolmogorov-Arnold-Moser) on a set of finite measure in phase space,
> and which also show explictly where the failure to converge
> lies. breakup of invariant tori (KAM et al): geometric descriptions
> of how quasi-periodic behavior tends towards chaotic behavior and
> attendant sensitive dependence on initial condx.
>
> 2.) shadowing lemmas/theorems: reassurances and proofs that the
> classical approximate solutions shadow or tail the real solution in
> some sense, so that they have a limited but rigorous justification.
>
> 3.) advances in geometric thinking on dynamical systems; realization
> of Poincare's program. one i remember from the late 80's turnstile's
> (pieces of previously existing invariant tori) that act to mix up
> phase space and control diffusion across "webs".
>
> 4.) perturbation techniques for analytically calculating onset of
> homoclinic tangles, developed by yet another Russian: Melnikov.
>
> 5.) topological (!) and algebraic techniques for classifying chaotic
> things like the Lorenz attractor. knot holders, horseshoes, baker's
> transformations, symbol sequences.
>
> 6.) last but not least: we got cautionary warnings on what to be
> careful about in our numerical integration schemes.
>
> the fad stuff is simply what people say to their newspapers and their
> funding agencies, and write about in more or less popular books (the
> Sov's a notable exception)
>
> to what degree 1-6 are important in econ i'll leave to Prof. Rosser.
>
> : What kind of mathematics is that? As mathematicians, aren't we
> : supposed to solve some problems?
>
> what does he think of Goedel's work??? to my mind his theorem
> highlights BOTH the strengths and weaknesses of axiomatic systems, as
> he utiliized ingenious techniques to derive said theroems.
>
> les schaffer
>
> p.s. google: "KAM theorem finite measure"
> "Melnikov method | integral | function"
>
