Breaking Points in Quartic Maps

TL;DR

This paper investigates the bifurcation behavior of quartic maps derived from quadratic maps with alternating parameters, revealing abrupt changes called breaking points in their bifurcation diagrams.

Contribution

It introduces a method to analyze quartic maps generated by alternating quadratic maps and identifies the presence of breaking points in their bifurcation diagrams.

Findings

Identification of breaking points in quartic maps bifurcation diagrams

Demonstration that alternating quadratic maps produce complex quartic dynamics

Analysis of how parameter alternation affects system stability

Abstract

Dynamical systems, whether continuous or discrete, are used by physicists in order to study non-linear phenomena. In the case of discrete dynamical systems, one of the most used is the quadratic map depending on a parameter. However, some phenomena can depend alternatively of two values of the same parameter. We use the quadratic map $x_{n+1} =1-ax_{n}^{2} $ when the parameter alternates between two values during the iteration process. In this case, the orbit of the alternate system is the sum of the orbits of two quartic maps. The bifurcation diagrams of these maps present breaking points where abruptly change their evolution.