Rotationally-ordered periodic orbits for multiharmonic area-preserving twist maps

J. A. Ketoja, R. S. MacKay

Research output: Contribution to journalArticleScientificpeer-review

7 Citations (Scopus)

Abstract

The Poincaré-Birkhoff theorem guarantees existence of at least two rotationally-ordered periodic orbits of each rational rotation number for each area-preserving twist map. For many maps, however, there are more than two. We prove this for maps near a non-degenerate multi-well anti-integrable limit, and deduce an intricate bifurcation diagram for rotationally-ordered periodic orbits in so-called "multiharmonic" families. Our results are motivated and supported by numerical investigations of the reversible 2-harmonic family. We believe the results will be helpful for understanding the breakup boundary for invariant circles in multiharmonic families, which numerically exhibits a Cantor set of cusps.

Original languageEnglish
Pages (from-to)388-398
Number of pages11
JournalPhysica D: Nonlinear Phenomena
Volume73
Issue number4
DOIs
Publication statusPublished - 15 Jun 1994
MoE publication typeNot Eligible

Fingerprint

Twist Map
Periodic Orbits
preserving
Orbits
orbits
existence theorems
Rotation number
Cantor set
Breakup
Bifurcation Diagram
Cusp
cusps
Numerical Investigation
Deduce
Circle
Harmonic
diagrams
harmonics
Invariant
Theorem

Cite this

@article{041051cd769e4ace87ae378ee657307c,
title = "Rotationally-ordered periodic orbits for multiharmonic area-preserving twist maps",
abstract = "The Poincar{\'e}-Birkhoff theorem guarantees existence of at least two rotationally-ordered periodic orbits of each rational rotation number for each area-preserving twist map. For many maps, however, there are more than two. We prove this for maps near a non-degenerate multi-well anti-integrable limit, and deduce an intricate bifurcation diagram for rotationally-ordered periodic orbits in so-called {"}multiharmonic{"} families. Our results are motivated and supported by numerical investigations of the reversible 2-harmonic family. We believe the results will be helpful for understanding the breakup boundary for invariant circles in multiharmonic families, which numerically exhibits a Cantor set of cusps.",
author = "Ketoja, {J. A.} and MacKay, {R. S.}",
year = "1994",
month = "6",
day = "15",
doi = "10.1016/0167-2789(94)90107-4",
language = "English",
volume = "73",
pages = "388--398",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "4",

}

Rotationally-ordered periodic orbits for multiharmonic area-preserving twist maps. / Ketoja, J. A.; MacKay, R. S.

In: Physica D: Nonlinear Phenomena, Vol. 73, No. 4, 15.06.1994, p. 388-398.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Rotationally-ordered periodic orbits for multiharmonic area-preserving twist maps

AU - Ketoja, J. A.

AU - MacKay, R. S.

PY - 1994/6/15

Y1 - 1994/6/15

N2 - The Poincaré-Birkhoff theorem guarantees existence of at least two rotationally-ordered periodic orbits of each rational rotation number for each area-preserving twist map. For many maps, however, there are more than two. We prove this for maps near a non-degenerate multi-well anti-integrable limit, and deduce an intricate bifurcation diagram for rotationally-ordered periodic orbits in so-called "multiharmonic" families. Our results are motivated and supported by numerical investigations of the reversible 2-harmonic family. We believe the results will be helpful for understanding the breakup boundary for invariant circles in multiharmonic families, which numerically exhibits a Cantor set of cusps.

AB - The Poincaré-Birkhoff theorem guarantees existence of at least two rotationally-ordered periodic orbits of each rational rotation number for each area-preserving twist map. For many maps, however, there are more than two. We prove this for maps near a non-degenerate multi-well anti-integrable limit, and deduce an intricate bifurcation diagram for rotationally-ordered periodic orbits in so-called "multiharmonic" families. Our results are motivated and supported by numerical investigations of the reversible 2-harmonic family. We believe the results will be helpful for understanding the breakup boundary for invariant circles in multiharmonic families, which numerically exhibits a Cantor set of cusps.

UR - http://www.scopus.com/inward/record.url?scp=0012405295&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(94)90107-4

DO - 10.1016/0167-2789(94)90107-4

M3 - Article

AN - SCOPUS:0012405295

VL - 73

SP - 388

EP - 398

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 4

ER -