Numerical evidences of universality and self-similarity in the Forced Logistic Map

TL;DR

This paper numerically investigates the Forced Logistic Map, revealing universality and self-similarity in bifurcation structures related to superattracting periodic orbits under quasi-periodic forcing.

Contribution

It provides numerical evidence of universality and self-similarity in bifurcation diagrams of the Forced Logistic Map, extending known properties from the classical Logistic Map.

Findings

Existence of universality in bifurcation slopes

Self-similarity in the bifurcation diagram

Numerical confirmation of asymptotic behaviors

Abstract

We explore different families of quasi-periodically Forced Logistic Maps for the existence of universality and self-similarity properties. In the bifurcation diagram of the Logistic Map it is well known that there exist parameter values $s_n$ where the $2^n$-periodic orbit is superattracting. Moreover these parameter values lay between one period doubling and the next. Under quasi-periodic forcing, the superattracting periodic orbits give birth to two reducibility-loss bifurcations in the two dimensional parameter space of the Forced Logistic Map, both around the points $s_n$. In the present work we study numerically the asymptotic behavior of the slopes of these bifurcations with respect to $n$. This study evidences the existence of universality properties and self-similarity of the bifurcation diagram in the parameter space.