Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss

Abstract

Let $g_{\alpha}$ be a one-parameter family of one-dimensional mapswith a cascade of period doubling bifurcations. Between each of thesebifurcations, a superstable periodic orbit is known to exist. Anexample of such a family is the well-known logistic map. In this paperwe deal with the effect of a quasi-periodic perturbation (with onlyone frequency) on this cascade. Let us call $\varepsilon$ theperturbing parameter. It is known that, if $\varepsilon$ is smallenough, the superstable periodic orbits of the unperturbed map becomeattracting invariant curves (depending on $\alpha$ and $\varepsilon$)of the perturbed system. In this article we focus on the reducibilityof these invariant curves.  &nbsp The paper shows that, under generic conditions, there are bothreducible and non-reducible invariant curves depending on the valuesof $\alpha$ and $\varepsilon$. The curves in the space$(\alpha,\varepsilon)$ separating the reducible (or the non-reducible)regions are called reducibility loss bifurcation curves. If the mapsatifies an extra condition (condition satisfied by thequasi-periodically forced logistic map) then we show that, from eachsuperattracting point of the unperturbed map, two reducibility lossbifurcation curves are born. This means that these curves are presentfor all the cascade.