TITLE:
Nonsmooth bifurcations in families of one-dimensional
piecewise-linear quasiperiodically forced maps

AUTHORS:
Rafael Martinez-Vergara and Joan Carles Tatjer^(1),
   
(1) Department de Matematiques i Informatica,
    Universitat de Barcelona (UB), Barcelona, Spain.

ABSTRACT:
We study nonsmooth bifurcations of four types of families of one-dimensional
quasiperiodically forced maps of the form 
   F_i (x, \theta) = (f_i (x, \theta), \theta + \omega) for i = 1,...,4, 
where x is real, \theta \in \mathbb{T} is an angle, \omega is an irrational
frequency, and f_i (x, \theta) is a real piecewise linear map with respect to x. 

The first two types of families f_i have a symmetry with respect to x, and the
other two could be viewed as quasiperiodically forced piecewise-linear versions
of saddle-node and period-doubling bifurcations. The four types of families
depend on two real parameters, a \in \mathbb{R} and b \in \mathbb{R}. Under
certain assumptions for a, we prove the existence of a continuous map b_*(a)
where for b = b_*(a) there exists a nonsmooth bifurcation for these types of
systems. In particular we prove that for b = b_*(a) we have a strange
nonchaotic attractor. It is worth to mention that the four families are
piecewise-linear versions of smooth families which seem to have nonsmooth
bifurcations. Moreover, as far as we know, we give the first example of a
family with a nonsmooth period-doubling bifurcation.