Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps
The authors consider the map \(x_{n+1}=f(x_n)\) where \[ f(x)=\begin{cases} ax-1, & x\leq 0, \\ bx^{-\gamma}-1, & x>0. \end{cases} \] Here \(\gamma>0\) so that the map is discontinuous. If \(a<0,b<0\) the map is invertible and if \(a<0, b>0\) it is noninvertible. A thorough, although not complete, description of the dynamics is given. Bifurcations of all fixed points are described. In the invertible case certain period-2 and period-4 orbits are shown to be involved in both smooth (i.e., occurring in smooth dynamical systems) and border collision bifurcations. These orbits are attracting for some parameter values. In the noninvertible case, for \(\gamma>1\) there is shown to be a sequence of alternating smooth period-doubling bifurcations and border collision bifurcations leading to attracting orbits of period \(n,2n,4n-1,2(4n-1),4(4n-1)-1,\dots\). For \(0<\gamma<1\) there may be unbounded chaotic attractors, or an attracting periodic orbit, or coexistence of a stable fixed point and an unbounded chaotic attractor.
scientific article; zbMATH DE number 6932759
scientific article