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ABOUT WHAT IS CALLED "GUMOWSKI-MIRA MAP"

Christian Mira

Email : christian.mira@sfr.fr

The term "Gumowski Mira map" was never used by the two authors who have studied the

equations so designated. Such an attitude would have been contrary to scientific ethics. A web

search based on operative words "Gumowski Mira map" shows that several authors have

found here a subject of articles, mainly related to "esthetical" images of chaotic solutions

generated by these Dim2 nonlinear maps. In particular, samples of these images can be found

via the link

https://www.google.fr/search?q=Gumowski+Mira+map&es_sm=93&tbm=isch&tbo=u&sourc

e=univ&sa=X&ei=0uCvU_OcDqKa0QXqk4HYCw&ved=0CC0QsAQ&biw=1164&bih=645

and also the video http://www.youtube.com/watch?v=AdcTAYLgYFI

It is in the framework of the international colloquium "Points Mappings and Applications"

(French title "Transformations Ponctuelles et leurs Applications") [1] that I announced an

exhibition of "stochastic" (i.e. chaotic, at that time this adjective did not exist) images,

quoting the Birkhoff's papers dealing with the laws of aesthetic (vol.3 of [3], p. 320-364), and

a Poincaré's text extracted from a "Notice sur Halphen" [Journal de l'Ecole Polytechnique 60,

137-161 (1890)], also cf. [4]. In this text Poincaré deals with the aesthetic emotion which can

be communicated by mathematics in the following terms:

"Le savant digne de ce nom, le géomètre surtout, éprouve en face de son œuvre la même

impression que l'artiste; sa jouissance est aussi grande et de même nature. Si je n'écrivais

pour un public amoureux de la Science, je n'oserais pas m'exprimer ainsi; je redouterais

l'incrédulité des profanes. Mais ici, je puis dire toute ma pensée. Si nous travaillons, c'est

moins pour obtenir ces résultats positifs auxquels le vulgaire nous croit uniquement attachés,

que pour ressentir cette émotion esthétique et la communiquer à ceux qui sont capables de

l'éprouver".

I took the liberty of saying that exhibited images had begun to manifest such an emotion in

a form opened not only to specialists as Poincaré said, but also to a general public (cf. [2],

page 27), this due to the new possibilities offered by numerical simulations. Such images can

be shown in chapter 8 (I. Gumowski and a Toulouse research group in the "prehistoric" times

of chaotic dynamics) of the book [5], the cover of which represents one of such images.

It appears that an originality of the 1973 Toulouse colloquium was the exhibition of chaotic

images, generated by solutions of nonlinear maps. The same exhibition, entitled

"Morphogénèse et Mathématiques", was organized by Marcel Barthes, Director of the

"Alliance Française Rio de Janeiro-Centre" in the "Centre Culturel de la Maison de France",

from May 8 to 30, 1975.

Equations of the map

As equation-creating of two-dimensional conservative maps, the general map was used

(Jacobian J being J 1) :

(a) x' = y+F(x), y' =-x+F(x'),

Two forms of the nonlinear term F(x) were considered, called "bounded nonlinearities" :

(b1) F(x)= x+ 2(1-)x2 / (1+ x2),

Map (a-b1) with =-0.2

Map (a-b1) with =0.05

Map (a-b1) with =0.3

Map (a-b1) with =0.25

(b2) F(x)= x + (1-)x2 exp[(1-x2)/4].

The following form of dissipative maps, quasi-conservative maps if >0 is small, with a

nonlinear damping term, were used for having given rise to interesting images

(a') x' = y+y(1-y2 )+ F(x), y' =-x+F(x'),

The corresponding results are the object of chapter 3 (conservative maps) and 4 (almost

conservative maps) of the book [6], and chapter 5 of the book [7]. An abbreviated

presentation of the results is given in chapter 8 of [5]. As far as I know these results, with

the considered particular non-linear characteristics, were new at the time of their

publications. Their application was made to two models of the longitudinal motion of

particles in an accelerator

Another variant (quartic form) of the above equations is given by the image below

Chaotic Attractors of Dim2 Noninvertible Maps (1968-1975)

They are not generated by what is called "Gumowski-Mira maps", but they were presented

in the framework of the chaotic images exhibition related to the 1973 colloquium "Points

Mappings and Applications" (French title "Transformations Ponctuelles et leurs

Applications") [1]

More details on the different types of attractors generated by noninvertible maps are given

with references in chapter 4 (1996 book [13], pp. 185-337).

References

1. Colloque International du CNRS n° 229: Transformations Ponctuelles et Applications

(Toulouse Sept. 1973). Proceedings: Editions du CNRS Paris (1976)

2. Mira, C. : "Exposé d'Introduction". Colloque International du CNRS n° 229

Transformations Ponctuelles et Applications (Toulouse Sept. 1973), pp. 19-27. Proceedings:

Editions du CNRS Paris, (1976)

3. Birkhoff, G.D. : Collected Mathematical Papers. Dover Publications, Inc., New York

(1968)

4. Poincaré, H. : Oeuvres, Tome II. Gauthier-Villars, Paris (1916)

5. Abraham, R., Ueda Y. (Editors): The chaos avant-garde. Memories of the early days of

chaos theory. World Scientific Series on Nonlinear Science. Series A 39 (2000)

6. Gumowski I., Mira, C.: Recurrences and discrete dynamic systems - An introduction. 250

pages. Lecture notes in mathematics n° 809, Springer Berlin (1980)

7. Gumowski I., Mira, C.: Dynamique chaotique. Transition ordre-désordre. Ed. Cépadués

Toulouse (1980)

13. Mira, C., Gardini, L., Barugola, A., Cathala, J.C.: Chaotic Dynamics in two-dimensional

noninvertible maps. World Scientific, Series A on Nonlinear Sciences, 20 (1996)