On a parametrized difference equation connecting chaotic and integrable mappings

TL;DR

This paper introduces a parametrized difference equation that smoothly connects chaotic logistic mappings with integrable Morishita mappings, exploring their dynamics through bifurcation analysis.

Contribution

It proposes a new two-parameter difference equation unifying chaotic and integrable mappings and analyzes their bifurcation structure.

Findings

The equation interpolates between logistic and Morishita mappings.

Bifurcation diagrams reveal transition from chaos to integrability.

Parameter variations show continuous dynamical behavior changes.

Abstract

We present a new difference equation with two parameters c in [0,1] and A in [1,4]. This equation is equivalent to the logistic mapping if c=1 and the Morishita mapping if c=0, which are the well-known chaotic and integrable mappings, respectively. We first consider the case A=4 and investigate the time evolution by changing the parameter c in [0,1]. We next change both two parameters A in [3,4] and c in [0,1] and present the corresponding 3D bifurcation diagram.