Nonsmooth Pitchfork Bifurcations in a Quasi-Periodically Forced Piecewise-Linear Map

Abstract

We study a family of one-dimensional quasi-periodically forced maps [Formula: see text], where [Formula: see text] is real, [Formula: see text] is an angle, and [Formula: see text] is an irrational frequency, such that [Formula: see text] is a real piecewise-linear map with respect to [Formula: see text] of certain kind. The family depends on two real parameters, [Formula: see text] and [Formula: see text]. For this family, we prove the existence of nonsmooth pitchfork bifurcations. For [Formula: see text] and any [Formula: see text] there is only one continuous invariant curve. For [Formula: see text] there exists a smooth map [Formula: see text] such that: (a) For [Formula: see text], [Formula: see text] has two continuous attracting invariant curves and one continuous repelling curve; (b) For [Formula: see text] it has one continuous repelling invariant curve and two semi-continuous (noncontinuous) attracting invariant curves that intersect the unstable one in a zero-Lebesgue measure set of angles; (c) For [Formula: see text] it has one continuous attracting invariant curve. The case [Formula: see text] is a degenerate case that is also discussed in the paper. It is interesting to note that this family is a simplified version of the smooth family [Formula: see text] for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when [Formula: see text].