The period adding and incrementing bifurcations: From rotation theory to applications

Albert Granados, Lluís Alsedà, Maciej Krupa

Research output: Contribution to journalReview articleResearchpeer-review

12 Citations (Scopus)

Abstract

© 2017 Society for Industrial and Applied Mathematics. This survey article is concerned with the study of bifurcations of discontinuous piecewisesmooth maps, with a special focus on the one-dimensional case. We review the literature on circle maps and quasi-contractions 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 the proof of its existence 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. This is called the period incrementing bifurcation, and 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 quasi-contractions. We provide applied examples in control theory, power electronics, and neuroscience, where these results can be used to obtain precise descriptions of their dynamics.
Original languageEnglish
Pages (from-to)225-292
JournalSIAM Review
Volume59
Issue number2
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

  • Devil's staircase
  • Discontinuous circle maps
  • Farey tree
  • Period adding
  • Period incrementing
  • Piecewise-smooth maps

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