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

Abstract

We study a family of one-dimensional quasi-periodically forced maps Fa,b(x, θ) = (fa,b(x, θ),θ + ω), where x is real, θ is an angle, and ω is an irrational frequency, such that fa,b(x, θ) 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 = b0(a) such that: (a) For b < b0(a), fa,b has two continuous attracting invariant curves and one continuous repelling curve; (b) For b = b0(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 > b0(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 Ga,b(x, θ) = (arctan(ax) + b sin(θ), θ + ω) for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when a→∞.