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

J. A. Ketoja; R. S. MacKay

doi:10.1016/0167-2789(94)90107-4

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

J. A. Ketoja^*
, R. S. MacKay

^*Corresponding author for this work

Not published at VTT

University of Helsinki
University of Warwick (WMG)

Research output: Contribution to journal › Article › Scientific › peer-review

8 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 language	English
Pages (from-to)	388-398
Number of pages	11
Journal	Physica D: Nonlinear Phenomena
Volume	73
Issue number	4
DOIs	https://doi.org/10.1016/0167-2789(94)90107-4
Publication status	Published - 15 Jun 1994
MoE publication type	Not Eligible

Funding

I Nuffield Foundation Science Research Fellow 92-93. We are gratefutlo the FinnishC ulturaFl oun-dationa ndthe MagnusE hrnroothF oundation for travelg rantse nablingK etojato visit Warwick and to the ResearchIn stitutef or Theoretical Physics,H elsinki,for an invitationt o MacKay to visit Helsinki.T his work was also supportedb y the UK SERC and the Nuffield Foundation.

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Cite this

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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.",

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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 - https://www.scopus.com/pages/publications/0012405295

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

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JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 4

ER -

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

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