Fractal boundary for the existence of invariant circles for area-preserving maps: Observations and renormalisation explanation

J. A. Ketoja, R. S. MacKay

Research output: Contribution to journalArticleScientificpeer-review

42 Citations (Scopus)

Abstract

Breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters. The critical line in parameter space turns out to have a Cantor set of cusps which can be organized in a self-similar tree. A theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator. The results of some other researchers on similar systems are shown to fit this explanation as well.

Original languageEnglish
Pages (from-to)318-334
Number of pages17
JournalPhysica D: Nonlinear Phenomena
Volume35
Issue number3
DOIs
Publication statusPublished - 1 Jan 1989
MoE publication typeNot Eligible

Fingerprint

Standard Map
Horseshoe
Cantor set
Breakup
Cusp
Renormalization
Fractals
preserving
Parameter Space
Two Parameters
Fractal
fractals
Circle
Invariant
Line
Operator
cusps
operators
Observation

Cite this

@article{e074fe152adb4b3d8917f5156eadbe77,
title = "Fractal boundary for the existence of invariant circles for area-preserving maps: Observations and renormalisation explanation",
abstract = "Breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters. The critical line in parameter space turns out to have a Cantor set of cusps which can be organized in a self-similar tree. A theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator. The results of some other researchers on similar systems are shown to fit this explanation as well.",
author = "Ketoja, {J. A.} and MacKay, {R. S.}",
year = "1989",
month = "1",
day = "1",
doi = "10.1016/0167-2789(89)90073-0",
language = "English",
volume = "35",
pages = "318--334",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "3",

}

Fractal boundary for the existence of invariant circles for area-preserving maps : Observations and renormalisation explanation. / Ketoja, J. A.; MacKay, R. S.

In: Physica D: Nonlinear Phenomena, Vol. 35, No. 3, 01.01.1989, p. 318-334.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Fractal boundary for the existence of invariant circles for area-preserving maps

T2 - Observations and renormalisation explanation

AU - Ketoja, J. A.

AU - MacKay, R. S.

PY - 1989/1/1

Y1 - 1989/1/1

N2 - Breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters. The critical line in parameter space turns out to have a Cantor set of cusps which can be organized in a self-similar tree. A theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator. The results of some other researchers on similar systems are shown to fit this explanation as well.

AB - Breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters. The critical line in parameter space turns out to have a Cantor set of cusps which can be organized in a self-similar tree. A theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator. The results of some other researchers on similar systems are shown to fit this explanation as well.

UR - http://www.scopus.com/inward/record.url?scp=0009545767&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(89)90073-0

DO - 10.1016/0167-2789(89)90073-0

M3 - Article

AN - SCOPUS:0009545767

VL - 35

SP - 318

EP - 334

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3

ER -