2025/12/03 by Martinez-Vergara, Rafael, Tatjer, Joan Carles
Mathematics · Physics and Astronomy · #37D25 #37D45 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · doi:10.48550/arxiv.2512.04234
openalex publication_date 2025/12/03 · openalex created_date 2025/12/06 · openalex updated_date 2026/07/28
We study nonsmooth bifurcations of four types of families of one-dimensional quasiperiodically forced maps of the form Fi(x,θ) = (fi(x,θ), θ+ω) for i=1,…,4, where x is real, θ∈\mathbbT is an angle, ω is an irrational frequency, and fi(x,θ) is a real piecewise linear map with respect to x. The first two types of families fi 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∈ℝ and b∈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.