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Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps. (English) Zbl 1402.37061

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.

MSC:

37G35 Dynamical aspects of attractors and their bifurcations
37E05 Dynamical systems involving maps of the interval
37N30 Dynamical systems in numerical analysis
39A28 Bifurcation theory for difference equations
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