The "0 to ∞-homoclinic bifurcation" of a class of discrete two-dimensional maps
Abstract
Based on a previous theoretical result of the same authors the presen paper deals with discrete perturbed two-dimensional maps having a semi-hyperbolic fixed point. We give applicable sufficient conditions assuring a particular kind of bifurcation of homoclinic orbits when the perturbative parameter μ varies in a small neighborhood of zero: no homoclinic orbits when μ is on one side of zero, one homoclinic orbit when μ = 0, and infinite homoclinics when μ is on the other side of zero.
References
2. F. Battelli, C. Lazzari, Discrete dichotomies and bifurcations from critical homoclinic orbits, J. Math. Anal. Appl. 219 (1998), 200-228.
3. F. Battelli, C. Lazzari, Homoclinic bifurcation in N-dimensional noninvertible maps having a semi-hyperbolic fixed point, Dynamics of Continuous, Discrete and Impulsive Systems 6 (1999), 1-24.
4. F.R. Marotto, Snap-back repellers imply chaos in ℝ^n, J. Math. Anal. Appl. 63 (1978), 199-223.
5. C. Mira, Complex dynamics in two-dimensional endomorphisms, Nonlin. Analysis, T.M.A. 4 (1980), 1167-1187.
6. C. Mira, L. Gardini, A. Barugola, J.C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps, Series on Nonlin. Sci., vol. 20, World Scientific Pub., London 1996.
Dipartimento di Matematica " V. Volterra", Facoltá di Ingegneria, Universitá di Ancona, Italy Italy
Facoltá di Scienze MM.FF.NN., Universitá di Urbino, Italy Italy
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.
