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

Albert Granados, Lluis Alseda, Maciej Krupa

Research output: Contribution to journalJournal articleResearchpeer-review

275 Downloads (Pure)

Abstract

This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in 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 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 quasi-contractions. 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 languageEnglish
JournalS I A M Review
Volume59
Issue number2
Pages (from-to)225–292
ISSN0036-1445
DOIs
Publication statusPublished - 2017

Keywords

  • Piecewise-smooth maps
  • discontinuous circle maps
  • period adding
  • devil's staircase
  • period incrementing

Cite this

Granados, Albert ; Alseda, Lluis ; Krupa, Maciej. / The period adding and incrementing bifurcations: from rotation theory to applications. In: S I A M Review. 2017 ; Vol. 59, No. 2. pp. 225–292.
@article{78a0f027b549492dac289035d3809844,
title = "The period adding and incrementing bifurcations: from rotation theory to applications",
abstract = "This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in 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 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 quasi-contractions. 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.",
keywords = "Piecewise-smooth maps, discontinuous circle maps, period adding, devil's staircase, period incrementing",
author = "Albert Granados and Lluis Alseda and Maciej Krupa",
year = "2017",
doi = "10.1137/140996598",
language = "English",
volume = "59",
pages = "225–292",
journal = "S I A M Review",
issn = "0036-1445",
publisher = "Society for Industrial and Applied Mathematics",
number = "2",

}

The period adding and incrementing bifurcations: from rotation theory to applications. / Granados, Albert; Alseda, Lluis ; Krupa, Maciej.

In: S I A M Review, Vol. 59, No. 2, 2017, p. 225–292.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - The period adding and incrementing bifurcations: from rotation theory to applications

AU - Granados, Albert

AU - Alseda, Lluis

AU - Krupa, Maciej

PY - 2017

Y1 - 2017

N2 - This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in 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 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 quasi-contractions. 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.

AB - This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in 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 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 quasi-contractions. 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.

KW - Piecewise-smooth maps

KW - discontinuous circle maps

KW - period adding

KW - devil's staircase

KW - period incrementing

U2 - 10.1137/140996598

DO - 10.1137/140996598

M3 - Journal article

VL - 59

SP - 225

EP - 292

JO - S I A M Review

JF - S I A M Review

SN - 0036-1445

IS - 2

ER -