Abstract
In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.
Highlights
Various bifurcation scenarios have always been in focus of many researchers from different theoretical and applied fields
Bimodal PWL Map: Bifurcation Structures In Panchuk et al [17] we described certain parameter regions corresponding to chaotic attractors and related bifurcation structures
In this work we carry on studying asymptotic dynamics of a 1D continuous bimodal PWL map f with the two outermost slopes being positive
Summary
Various bifurcation scenarios have always been in focus of many researchers from different theoretical and applied fields. In Panchuk et al [16] we disclosed three basic bifurcation structures related to attracting cycles: the skew tent map (STM) structure, the period adding (PA) structure, and the fin structure. In the current paper we summarize the results of previous works and describe new family of periodicity regions observed in the parameter space of the bimodal map. They emerge from the border that separates the parameter region D1/D2 where asymptotic dynamics is associated with two adjacent branches (STM structure) from the region D0 where all three branches can be involved. For such a practical usage see, for instance, [21, 22]
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