Existence and Stability of 3-Cycles in Quadratic Maps

TL;DR

This paper investigates quadratic maps on the real line, demonstrating that the existence, quantity, and stability of 3-cycles depend solely on a specific parameter derived from the map's coefficients.

Contribution

It provides an elementary method to determine 3-cycle properties in quadratic maps based on a single parameter, simplifying analysis compared to previous approaches.

Findings

Existence of 3-cycles depends on a parameter

Number of 3-cycles is determined by this parameter

Stability of 3-cycles is characterized by the same parameter

Abstract

For a discrete dynamical system on $\R$ generated by a quadratic function, we show, using elementary computations, that the existence, number, and stability of 3-cycles are determined by a single parameter depending on the coefficients of the function.