Nonsmooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps
Rafael Martinez-Vergara, Joan Carles Tatjer
TL;DR
The paper investigates nonsmooth bifurcations in four families of one-dimensional piecewise-linear quasiperiodically forced maps. It identifies a bifurcation curve $b^*(a)$ in the $(a,b)$-plane where invariant-curve collisions produce nonsmooth bifurcations and Strange Nonchaotic Attractors, including a rare nonsmooth period-doubling case. The authors prove the attracting set closure has positive Lebesgue measure at bifurcation, demonstrate fractalization of invariant curves, and provide a criterion for noncontinuity via uniform convergence of a monotone sequence of curves. They also treat a uniform contraction regime ($|a|<1$) where a unique Lipschitz invariant curve persists, and discuss implications for SNAs in forced maps and potential extensions to higher dimensions and smooth models.
Abstract
We study nonsmooth bifurcations of four types of families of one-dimensional quasiperiodically forced maps of the form $F_i(x,θ) = (f_i(x,θ), θ+ω)$ for $i=1,\dots,4$, where $x$ is real, $θ\in\mathbb{T}$ is an angle, $ω$ is an irrational frequency, and $f_i(x,θ)$ 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.