Route to chaos in generalized logistic map

TL;DR

This paper introduces a generalized logistic map with two additional parameters, exploring how these affect the transition from regularity to chaos and analyzing bifurcation sequences and attractor properties.

Contribution

It extends the classical logistic map by adding parameters p and q, systematically studying their impact on chaos transition and bifurcation behavior.

Findings

Transition to chaos varies with p and q parameters.

Bifurcation sequences exhibit different characteristics depending on parameters.

Feigenbaum constant analyzed for the generalized map.

Abstract

Motivated by a possibility to optimize modelling of the population evolution we postulate a generalization of the well-know logistic map. Generalized difference equation reads: \begin{equation} x_{n+1}=rx^p_n(1-x^q_n), \end{equation} $x\in[0,1],\;(p,q)>0,\;n=0,1,2,...$, where the two new parameters $p$ and $q$ may assume any positive values. The standard logistic map thus corresponds to the case $p=q=1$. For such a generalized equation we illustrate the character of the transition from regularity to chaos as a function of $r$ for the whole spectrum of $p$ and $q$ parameters. As an example we consider the case for $p=1$ and $q=2$ both in the periodic and chaotic regime. We focus on the character of the corresponding bifurcation sequence and on the quantitative nature of the resulting attractor as well as its universal attribute (Feigenbaum constant).