ABOUT WHAT IS CALLED "GUMOWSKI-MIRA MAP"

Abstract

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

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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)