Research Article Stability of Real Parametric Polynomial Discrete Dynamical Systems Fermin Franco-Medrano 1,2 and Francisco J. Solis 1 1 Applied Mathematics, CIMAT, 36240 Guanajuato, GTO, Mexico 2 Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan Correspondence should be addressed to Francisco J. Solis; solis@cimat.mx Received 23 November 2014; Revised 22 January 2015; Accepted 23 January 2015 Academic Editor: Zhan Zhou Copyright © 2015 F. Franco-Medrano and F. J. Solis. his 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. We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coeicients that depend on a single parameter  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 ixed points of any real polynomial map with real ixed 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 ixed points. he stability of the ixed points of canonical polynomial maps has been found to depend solely on a special function termed Product Position Function for a given ixed point. he values of this product position determine the stability of the ixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed stability bands. he 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. 1. Introduction he theory of discrete dynamical systems with iteration func- tions given by polynomials is an intensive research subject where a wide variety of discrete models have been proposed to describe and to analyze diferent mechanisms in various areas of science. For example, in Biology and more speciically in Population Dynamics there are many simple models that are used to study the asymptotic behavior of some species that live in isolated generations; see, for instance, [1–7]. Although the dynamics of parametric polynomial dis- crete systems are very complex their bifurcation diagrams have proved to be a very useful visual tool. A new method for constructing a rich class of bifurcation diagrams for unimodal maps was presented in [8], where the behavior of quadratic maps was analyzed when the dependence of their coeicients was given by continuous functions of a parameter. Conditions on the coeicients of the quadratic maps were given in order to obtain regular reversal maps. Our irst goal is to restate the results for more complex systems (cubic) than the quadratic systems analyzed in [8] and to state the results in the frame of a new formulation that would allow for generalization. Our second goal is to generalize the existing results on real quadratic maps for arbitrary real polynomial maps within a framework that allows us to understand the dynamics for a larger set of discrete systems. It is important to remark that our results are analytical and depend only on the parametric derivative of the system evaluated at equilibrium points. here are diverse results based on other approaches such as the linearized stability due to Lyapunov; see, for instance, [9, 10]. In our opinion, our approach is natural for polynomial iteration functions whereas the linearized stability can be used for more complex discrete systems with iteration functions such as piecewise functions. It is also important to notice that there is diverse numerical sotware specialized in the numerical continuation and bifurcation study of continuous and discrete parameter- ized dynamical systems, such as Auto [11] and MatCont [12]. Before attempting to obtain general results for polyno- mial discrete systems, we want to motivate them with those for a nontrivial system. To do this, we propose in Section 2 analyzing the stability of a general cubic discrete dynamical Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2015, Article ID 680970, 13 pages http://dx.doi.org/10.1155/2015/680970