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

We study a family of one-dimensional quasi-periodically forced maps F a , b ( x , ) = ( f a , b ( x , ) , + ω ) , where x is real, is an angle, and ω is an irrational frequency, such that f a , b ( x , ) is a real piecewise-linear map with respect to x of certain kind. The family depends on two real parameters, a > 0 and b > 0 . For this family, we prove the existence of nonsmooth pitchfork bifurcations. For a < 1 and any b , there is only one continuous invariant curve. For a > 1 , there exists a smooth map b = b 0 ( a ) such that: (a) For b < b 0 ( a ) , f a , b has two continuous attracting invariant curves and one continuous repelling curve; (b) For b = b 0 ( a ) , 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 b > b 0 ( a ) , it has one continuous attracting invariant curve. The case a = 1 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 G a , b ( x , ) = ( arctan ( a x ) + b sin ( ) , + ω ) for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when a → ∞ .