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

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 $i=1,\dots,4$, 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.