Given the plane triangle with vertices (0,0), (0,4) and (4,0) and the transformation F: (x,y) ↦ (x(4-x-y),xy) introduced by A. N. Sharkovskiĭ, we prove the existence of the following objects: a unique invariant curve of spiral type, a periodic trajectory of period 4 (given explicitly) and a periodic trajectory of period 5 (described approximately). Also, we give a decomposition of the triangle which helps to understand the global dynamics of this discrete system which is linked with the behavior of the Schrödinger equation.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-5,
author = {Francisco Balibrea and Juan Luis Garc\'\i a Guirao and Marek Lampart and Jaume Llibre},
title = {Dynamics of a Lotka-Volterra map},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {265-279},
zbl = {1107.37032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-5}
}
Francisco Balibrea; Juan Luis García Guirao; Marek Lampart; Jaume Llibre. Dynamics of a Lotka-Volterra map. Fundamenta Mathematicae, Tome 189 (2006) pp. 265-279. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm191-3-5/