Stability of real parametric polynomial discrete dynamical systems

TL;DR

This paper develops a unified theory for the stability of fixed points in real polynomial discrete dynamical systems of any degree, introducing canonical maps and a new stability criterion based on the Product Position Function.

Contribution

It generalizes stability analysis from quadratic to higher-degree polynomial maps using canonical conjugacy and introduces the Product Position Function as a key stability indicator.

Findings

Stability depends solely on the Product Position Function for canonical polynomial maps.

The concept of stability bands describes regions of parameter space with stable fixed points.

The framework applies to polynomial maps of any degree, extending previous quadratic-only results.

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 $m$-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 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.