Towards a renormalization theory for quasi-periodically forced one dimensional maps I. Existence of reducibility loss bifurcations

TL;DR

This paper extends the renormalization theory to quasi-periodically forced one-dimensional maps, proving the existence of fixed points and bifurcation structures, which enhances understanding of complex bifurcation phenomena in such systems.

Contribution

It develops a renormalization framework for quasi-periodic maps, establishing fixed points and bifurcation curves, advancing the theoretical understanding of these complex dynamical systems.

Findings

Fixed point of one-dimensional renormalization extends to quasi-periodic case

Differentiability of the renormalization operator around the fixed point

Existence of reducibility loss bifurcation curves in parameter space

Abstract

We propose an extension of the one dimensional (doubling) renormalization operator to the case of maps on the cylinder. The kind of maps considered are commonly referred as quasi-periodic forced one dimensional maps. We prove that the fixed point of the one dimensional renormalization operator extends to a fixed point of the quasi-periodic forced renormalization operator. We also prove that the operator is differentiable around the fixed point and we study its derivative. Then we consider a two parametric family of quasi-periodically forced maps which is a unimodal one dimensional map with a full cascade of period doubling bifurcations plus a quasi-periodic perturbation. For one dimensional maps it is well known that between one period doubling and the next one there exists a parameter value where the $2^n$-periodic orbit is superatracting. Under appropriate hypotheses, we prove that the two parameter family has two curves of reducibility loss bifurcation around these points.