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

AUTHORS:
Angel Jorba, Pau Rabassa and Joan Carles Tatjer
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
E-mails: angel@maia.ub.es, jcarles@maia.ub.es, prs@maia.ub.es

ABSTRACT:
Let $g_{\alpha}$ be a one-parameter family of one-dimensional maps
with a cascade of period doubling bifurcations. Between each of these
bifurcations, a superstable periodic orbit is known to exist. An
example of such a family is the well-known logistic map. In this paper
we deal with the effect of a quasi-periodic perturbation (with only
one frequency) on this cascade. Let us call $\varepsilon$ the
perturbing parameter. It is known that, if $\varepsilon$ is small
enough, the superstable periodic orbits of the unperturbed map become
attracting invariant curves (depending on $\alpha$ and $\varepsilon$)
of the perturbed system. In this article we focus on the reducibility
of these invariant curves.

The paper shows that, under generic conditions, there are both
reducible and non-reducible invariant curves depending on the values
of $\alpha$ 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 map
satifies an extra condition (condition satisfied by the
quasi-periodically forced logistic map) then we show that, from each
superattracting point of the unperturbed map, two reducibility loss
bifurcation curves are born. This means that these curves are present
for all the cascade.