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Abstract
This survey article is concerned with the study of bifurcations of piecewisesmooth maps. We review the literature in circle maps and quasicontractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and \rotation” numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and its proof relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block and the periods of periodic orbits are incremented by a constant term. It is called the period incrementing bifurcation, in its proof relies on results for maps on the interval.
We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasicontractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.
We also discuss the expanding cases, as some of the partial results found in the literature also hold when these maps lose contractiveness. The higher dimensional case is also discussed by means of quasicontractions. We also provide applied examples in control theory, power electronics and neuroscience where these results can be applied to obtain precise descriptions of their dynamics.
Original language  English 

Journal  S I A M Review 
Volume  59 
Issue number  2 
Pages (fromto)  225–292 
ISSN  00361445 
DOIs  
Publication status  Published  2017 
Keywords
 Piecewisesmooth maps
 discontinuous circle maps
 period adding
 devil's staircase
 period incrementing
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Projects
 1 Finished

COFUNDPostdocDTU: COFUNDPostdocDTU
Præstrud, M. R. & Brodersen, S. W.
01/01/2014 → 31/12/2019
Project: Research