Towards a bifurcation theory for perturbed monomial dynamical systems modulo a prime

TL;DR

This paper explores the distribution of periodic points in perturbed monomial dynamical systems over finite fields, introducing a visual tool and conjectures to understand their bifurcation behavior.

Contribution

It develops a unified approach to study all such systems simultaneously and introduces the Periodic Point Diagram as a new visualization method.

Findings

Visualization of periodic points distribution via Periodic Point Diagram

Proven results on the distribution patterns of periodic points

Conjecture on the total number of periodic points in the system

Abstract

We investigate perturbed monomial dynamical system over $\mathbb{F}_p$ given by iterations of $x\mapsto x^n+c\bmod{p}$, where $c\in \mathbb{F}_p$. Instead of study the systems one at a time we study all of them at the same time. The complex distibution of periodic points is visualized in the so called Periodic Point Diagram, which can be seen as a discrete version of the classical Bifurcation Diagram. We also prove some general results about the distribution of periodic points. We end the article with a conjecture about the total number of periodic points.