Period doubling and reducibility in the quasi-periodically forced logistic map

TL;DR

This paper investigates the dynamics of the Forced Logistic Map, focusing on period doubling and reducibility of invariant curves, revealing their interconnected bifurcation structures through detailed analysis of the parameter space.

Contribution

It demonstrates the relationship between period doubling and reducibility of invariant curves, providing new insights into bifurcation structures in the Forced Logistic Map.

Findings

Invariant curves double their period within reducible regions.

Reducibility loss occurs as a codimension-one bifurcation.

Reducibility loss and period doubling bifurcation curves meet tangentially.

Abstract

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and the preimages of the critical set (the set where the derivative of the map w.r.t the non-periodic coordinate is equal to zero). Studying these sets we construct constrains in the parameter space for the reducibility of the invariant curve. In second place we consider the reducibility loss of the invariant curve as codimension one bifurcation and we study its interaction with the period doubling bifurcation. This reveals that, if the reducibility loss and the period doubling bifurcation curves meet, they do it in a tangent way.