Nonsmooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps
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arXiv
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2025
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| author | Martinez-Vergara, Rafael Tatjer, Joan Carles |
| author_facet | Martinez-Vergara, Rafael Tatjer, Joan Carles |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_04234 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonsmooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps Martinez-Vergara, Rafael Tatjer, Joan Carles Dynamical Systems 37D25, 37D45 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. |
| title | Nonsmooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps |
| topic | Dynamical Systems 37D25, 37D45 |
| url | https://arxiv.org/abs/2512.04234 |