Reordering of the Logistic Map with a Nonlinear Growth Rate

TL;DR

This paper explores how replacing the linear coefficient in the logistic map with a nonlinear, hyperbolic tangent-based function significantly alters its bifurcation structure, leading to new dynamical behaviors.

Contribution

It introduces a generalized logistic map with a nonlinear coefficient decay, revealing new bifurcation phenomena and chaotic regimes not seen in the classical model.

Findings

Nonlinear coefficient decay modifies asymptotic values.

Bifurcation diagrams show new period doubling and chaos.

Regimes with small periods emerge due to nonlinear effects.

Abstract

In the well known logistic map, the parameter of interest is weighted by a coefficient that decreases linearly when this parameter increases. Since such a linear decrease forms a specific case, we consider the more general case where this coefficient decreases nonlinearly as in a hyperbolic tangent relaxation of a system toward equilibrium. We show that, in this latter case, the asymptotic values obtained via iteration of the logistic map are considerably modified. We demonstrate that both the steepness of the nonlinear decrease as well as its upper and lower boundaries significantly alter the bifurcation diagram. New period doubling features and transitions to chaos appear, possibly leading to regimes with small periods. Computations with a variety of parameter values show that the logistic map can be significantly reordered in the case of a nonlinear growth rate.