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<- [[:chaosbook]]

====== Appendix: A brief history of chaos  ======

(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#appendHist|A brief history of chaos]])  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2010-01-06//

Enter latest posts at the top, blog-style. 

===== Prune Danish =====

**Amy 2010-01-05** I remember you told a story about going on Danish tv with a pastry chef and actually doing the kneading. 

**Predrag 2009-12-01** 
I have the video, but it's some Oy-ro-peon format. Will check.

**Predrag 2012-09-17**
Transferred all this to ChaosBook.org history appendix (not public yet)
===== Cyclist remarks =====

**Predrag 2009-12-01** I derived "cycle expansions" sometimes in 1986 or earlier (?). They arose from trying to figure out the scalings in period-doubling renormalization, and topology of the Hénon attractor (my pruning front theory) developed at Chalmers, Gothenburg 1985-1986 and Cornell Jan-May 1985.

I do not publish until I have worked a topic to death, but I did give lots of conference and seminar talks about cycle expansions. //Inter alia// I talked about the periodic-orbit topology of Lozi and Hénon attractors, and the cycle expansions at the  "Chaos and Related Nonlinear Phenomena: Where Do We Go From Here?", the Fritz Haber Symposium organized by Itamar Procaccia  in 1986(?) and held at a kibbutz in Israel. A great meeting, and Celso Grebogi was in the audience. My first publication on this was

[[http://www.cns.gatech.edu/~predrag/papers/preprints.html#Cycling|Invariant measurement of strange sets in terms of cycles]], Phys. Rev. Lett. 61, 2729 (1988), submitted March 1988.

Celso and the Maryland colleagues had written written many articles about UPOs since, starting with 

C. Grebogi, E. Ott, J. A. Yorke, Unstable periodic orbits and the dimensions
of multifractal chaotic attractors, //Phys Rev A// **37**,  1711–1724 (1988), submitted Sept. 1987.

in which they presented a formula for the natural measure in terms of the unstable periodic orbits (UPOs) with large period embedded in a chaotic attractor.

For some reason it's been the habit ever since for the Maryland school to cite Maryland papers only and some of the mathematicians of the 1970's, and not Copenhagen papers. 

Ott in particular is an exquisite physicist, but the Maryland approach was and remained heuristic, and there is no need for that. These heuristic formulas are approximations to the exact trace formulas (that are derived in ChaosBook with no more effort than the heuristic approximations), but they are not smart for computations; much faster convergence is obtained by using the cycle expansions of dynamical zeta functions and Fredholm determinants. The cycle expansions //are not heuristic//, they are exact expansion in  the unstable periodic orbits for classical dynamics, and semi-classically exact for quantum mechanics. The first paper that derives them is, I believe, the above 1988 Phys. Rev. Lett.. Some other early papers are

Topological and metric properties of Hénon-type attractors (with G.H. Gunaratne and I. Procaccia), Phys. Rev. A 38, 1503 (1988)

Exploring chaotic motion through periodic orbits (with D. Auerbach, J.-P. Eckmann, G. Gunaratne and I. Procaccia), Phys. Rev. Lett. 58, 2387-2389 (1987) 

but the two long papers (combining the Artuso and Aurell PhD theses with my own work) were better:

Recycling of strange sets: I. Cycle expansions (with R. Artuso and E. Aurell), Nonlinearity 3, 325 (1990) 

Recycling of strange sets: II. Applications (with R. Artuso and E. Aurell), Nonlinearity 3, 361 (1990)


The most recent in the Maryland school series is 

[[http://arxiv.org/abs/0912.0596|Comment on "Time-averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems"]]
by Michael A. Zaks and Denis S. Goldobin. It would not hurt to recycle these kinds of computations in terms of spectral determinants, instead of sticking to evaluations of approximate trace formulas. There is no extra work involved, except reading relevant bits of ChaosBook, and inserting the same set of UPOs of the Lorenz equations into cycle expansions. They should converge faster - symbolic dynamics is a problem here, so perhaps one would need to use the stability cutoffs.

If one is reluctant to refer to people with long, weird surnames, one can always refer to Ruelle (not Bowen, he did not do that) for deriving the dynamical (or Ruelle) zeta function. Sinai-Bowen-Ruelle work was much smarter and more profound than 95% of physicists publications from 1980s on. However, continuous time flow traces weighted by cycle periods were introduced by Bowen who treated them as Poincaré section suspensions weighted by the “time ceiling” function. But I do not see cycle expansions that we use in Bowen's work (see, for example, the description in Scholarpedia.org/article/Bowen-Margulis_measure).

BTW, if you are interesting in periodic orbit theory, try reading ChaosBook.org. It's not half bad. And if you ignore this bit of history, [[http://oldpoetry.com/opoem/show/2906-Dylan-Thomas-Do-Not-Go-Gentle-Into-That-Good-Night|I will not mind at all]], but please do master the best tools available to deal with chaos, instead of just running computers. It does not really matter who did it first, as long as we all have decent working conditions to enjoy our research. I certainly have no rights to complain about that.
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