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

We study a family of one-dimensional quasi-periodically forced maps $F_{a,b}(x,𝜃)=(f_{a,b}(x,𝜃),𝜃+\omega )$, where $x$ is real, $𝜃$ is an angle, and $\omega$ is an irrational frequency, such that $f_{a,b}(x,𝜃)$ 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 $b=b_0(a)$ 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 $G_{a,b}(x,𝜃)=(arctan(ax)+bsin(𝜃),𝜃+\omega )$ for which there is numerical evidence of a nonsmooth pitchfork bifurcation. Finally, we also discuss the limit case when $a\to \infty$.