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 $$F_i(x,\theta ) = (f_i(x,\theta ), \theta +\omega )$$ F i ( x , θ ) = ( f i ( x , θ ) , θ + ω ) for $$i=1,\dots ,4$$ i = 1 , ⋯ , 4 , where x is real, $$\theta \in \mathbb {T}$$ θ ∈ T is an angle, $$\omega $$ ω is an irrational frequency, and $$f_i(x,\theta )$$ f i ( x , θ ) is a real piecewise linear map with respect to x . The first two types of families $$f_i$$ 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, $$a\in \mathbb {R}$$ a ∈ R and $$b\in \mathbb {R}$$ b ∈ R . Under certain assumptions for a , we prove the existence of a continuous map $$b^*(a)$$ b ∗ ( a ) where for $$b=b^*(a)$$ b = b ∗ ( a ) there exists a nonsmooth bifurcation for these types of systems. In particular we prove that for $$b=b^*(a)$$ b = b ∗ ( a