New phenomena related to the presence of focal points in two-dimensional maps
Abstract
In this paper we consider two-dimensional maps, defined in the whole plane, with the property of mapping a whole curve δ into a single point Q. We relate such property to the fact that at least one inverse map exists with a denominator which can vanish, and assumes the form 0/0 in Q. This allows us to apply the properties of focal points and prefocal curves in order to explain the dynamic phenomena observed. By an example we show that the iteration of such maps may generate discrete dynamical systems with peculiar attracting sets, characterized by the presence of "knots", where infinitely many phase curves shrink into a single point.
References
2. G.I. Bischi, L. Gardini, Basin fractalization due to focal points in a class of triangular maps, International Journal of Bifurcation & Chaos 7 (1997), 1555-1577.
3. G.I. Bischi, L. Gardini, Focal points and basin fractalization in some rational maps, Grazer Math. Ber. (to appear).
4. G.I. Bischi, L. Gardini, C. Mira, Maps with denominator. Part 1: some generic properties, International Journal of Bifurcation & Chaos 9 (1999), 119-153.
5. J.C. Cathala, A. Barugola, On the basin boundary of a quadratic degenerate two-dimensional endomorphism, Grazer Math. Ber. (to appear).
6. L. Gardini, G.I. Bischi, Focal points and focal values in rational maps, Grazer Math. Ber. (to appear).
7. I. Gumowski, C. Mira, Dynamique Chaotique, Cepadues Editions, Toulouse 1980.
8. G. Julia, Remarques sur les transformations ponctuelles planes au visinage d'un point ou le jacobien est nul, Bull. des Sciences mathém. 2 series 53 (1929).
9. C. Mira, Some properties of two-dimensional (Z_0-Z_2) maps not defined in the whole plane, Grazer Math. Ber. (to appear).
10. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic dynamics in two-dimensional noninvertible maps, World Scientific, Singapore 1996.
11. C. Mira, Singularités non classiques d'une recurrence et d'une équation differentielle d'ordre deux, Comptes Rendus Acad. Sci. Paris, Série I 292 (1981), 146-151.
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
Instituto di Scienze Economiche, Universitá di Urbino, Italy Italy
GESNLA-DGE-INSA, Complexe Scientifique de Rangueil, France France
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
