Cycles of the logistic map

TL;DR

This paper analyzes the bifurcation points of the logistic map and related polynomial maps by deriving characteristic equations and minimal polynomials for cycles up to certain periods, enhancing understanding of their dynamical behavior.

Contribution

It introduces a method to locate bifurcation points of cycles in polynomial maps and computes minimal polynomials for key maps up to specific cycle lengths, extending previous analyses.

Findings

Derived characteristic equations for n-cycles.

Obtained minimal polynomials for logistic, Hénon, and cubic maps.

Extended cycle analysis up to n=13 for logistic map.

Abstract

The onset and bifurcation points of the $n$-cycles of a polynomial map are located through a characteristic equation connecting cyclic polynomials formed by periodic orbit points. The minimal polynomials of the critical parameters of the logistic, H\'enon, and cubic maps are obtained for $n$ up to 13, 9, and 8, respectively.