Stability and chaos in real polynomial maps

TL;DR

This paper develops a unified theory for the stability and chaos of real polynomial maps of any degree, introducing canonical forms and a product distance function to analyze fixed point stability and bifurcations.

Contribution

It introduces canonical polynomial maps and a product distance function to characterize stability and chaos in real polynomial maps of arbitrary degree.

Findings

Stability of fixed points depends on the product distance function.

Canonical polynomial maps are topologically conjugate to general polynomial maps.

Stability bands determine when chaos arises in polynomial maps.

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 $\lambda$, and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real $n$-th 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 Distance Function for a given fixed point. The values of this product distance 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.