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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 680970, 13 pages
http://dx.doi.org/10.1155/2015/680970
Research Article

Stability of Real Parametric Polynomial Discrete Dynamical Systems

1Applied Mathematics, CIMAT, 36240 Guanajuato, GTO, Mexico
2Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Received 23 November 2014; Revised 22 January 2015; Accepted 23 January 2015

Academic Editor: Zhan Zhou

Copyright © 2015 Fermin Franco-Medrano and Francisco J. Solis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter λ and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real mth degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of canonical polynomial maps which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed stability bands. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.