TITLE:
Non-smooth pitchfork bifurcations in a quasi-periodically forced
piecewise-linear map

AUTHORS:
Angel Jorba^(1,2), Joan Carles Tatjer^(1) and Yuan Zhang^(3)

(1) Departament de Matematiques i Informatica, Universitat de
    Barcelona, Spain.
(2) Centre de Recerca Matematica, Edifici C, Campus Bellaterra,
    08193 Bellaterra, Spain
(3) Complex Systems Research Center, Shanxi University, China.

ABSTRACT:
We study a class of one-dimensional family of quasiperiodically forced
maps $F_{a,b}(x,\theta)=(f_{a,b}(x,\theta),\theta+\omega)$, where $x$
is real, $\theta$ is an angle, and $\omega$ is an irrational
frequency, such that $f_{a,b}(x,\theta)$ is a real piecewise linear
map with respect to $x$ of certain kind. The family depends on two
real parameters, $a>0$ and $b>0$. For this family, we prove the
existence of non-smooth pitchfork bifurcations. For $a<1$ and any $b$
there is only a continuous invariant curve. For $a>1$ there exists a
smooth map $b=b_0(a)$ such that:

a) For $b<b_0(a)$, $f_{a,b}$ has two continuous attracting invariant
   curves and one continuous repelling one;

 b) For $b=b_0(a)$ it has one continuous repelling invariant curve and
   two semicontinuous (non-continuous) attracting invariant curves
   that intersect the unstable one in a zero-Lebesgue measure set of
   angles;

 c) For $b>b_0(a)$ it has one continuous attracting invariant curve.

The case $a=1$ is a degenerate case that is also discussed in the
paper. It is interesting to note that this family is a simplified
version of the smooth family
$G_{a,b}(x,\theta)=(\arctan(ax)+b\sin(\theta),\theta+\omega)$ for
which there is numerical evidence of a non-smooth pitchfork
bifurcation. Finally, we also discuss the limit case when
$a\rightarrow\infty$.