Stability of Real Parametric Polynomial Discrete Dynamical Systems

Fermin Franco-Medrano,Francisco J Solis

doi:10.1155/2015/680970

Abstract

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameterλand generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to realmth 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 ofcanonical polynomial mapswhich 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 termedProduct Position Functionfor 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 termedstability 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.

Highlights

The theory of discrete dynamical systems with iteration functions given by polynomials is an intensive research subject where a wide variety of discrete models have been proposed to describe and to analyze different mechanisms in various areas of science
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
This turns out to be very useful, since we know that topological conjugacy is an equivalence relation that preserves the property of chaos. This means that the analysis of stability and chaos of real polynomial maps with real fixed points is reduced to the study of the canonical polynomial maps defined above, since we can always take any polynomial in PC[x] to its Canonical Form by means of T, determine the stability properties, and go back to the original polynomial

Summary

Introduction

The theory of discrete dynamical systems with iteration functions given by polynomials is an intensive research subject where a wide variety of discrete models have been proposed to describe and to analyze different mechanisms in various areas of science. 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 coefficients was given by continuous functions of a parameter. 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. 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. We use a general framework in order to analyze the stability for real polynomial discrete dynamical systems by using some of the ideas introduced in the previous section.

Cubic Discrete Dynamical Systems

Polynomial Discrete Dynamical Systems

Examples

Conclusions