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Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps. (English) Zbl 1087.37029

This paper contains the detailed analysis of the following two-parameter family of piecewise smooth unimodal maps with one break point:

f(x)=f 1 (x)=rx,if0xx ¯,f 2 (x)=ax(1-x),ifx ¯x1,

where x ¯=1-r/a and a,r are real parameters varying in the range a>3 and 1<r<a. Primarily, the 2D bifurcation diagram in the (r,a)-parameter plane is investigated. Particular attention is paid to the border-collision on bifurcation of the superstable n-cycle of the map f. The use of the symbolic representation of the superstable cycles is an interesting method to describe the structure of the periodicity regions of the 2D bifurcation diagram.

37E05Maps of the interval (piecewise continuous, continuous, smooth)
37G35Attractors and their bifurcations
37N30Dynamical systems in numerical analysis
37B10Symbolic dynamics