### Abstract

The almost periodic eigenvalue problem described by the Harper equation is connected to other classes of quasiperiodic behavior; the dissipative dynamics on critical invariant tori and quasiperiodically driven maps. Firstly, the strong coupling limit of the supercritical Harper equation and the strong dissipation limit of the critical standard map play equivalent role in describing the universal characteristics of these systems. Secondly, a simple transformation is used to relate the Harper equation to a quasiperiodically forced one-dimensional map. In this case, the localized eigenstates of the supercritical Harper equation correspond to strange but nonchaotic attractors of the driven map. Furthermore, the existence of localization in the eigenvalue problem is associated with the appearance of homoclinic points in the corresponding map.

Original language | English |
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Pages (from-to) | 70-80 |

Number of pages | 11 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 109 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 Jan 1997 |

MoE publication type | Not Eligible |

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### Keywords

- Harper equation
- Quasiperiodicity
- Renormalization localization
- Standard map
- Strange nonchaotic attractor

### Cite this

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**Harper equation, the dissipative standard map and strange nonchaotic attractors : Relationship between an eigenvalue problem and iterated maps.** / Ketoja, Jukka A.; Satija, Indubala I.

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

TY - JOUR

T1 - Harper equation, the dissipative standard map and strange nonchaotic attractors

T2 - Relationship between an eigenvalue problem and iterated maps

AU - Ketoja, Jukka A.

AU - Satija, Indubala I.

PY - 1997/1/1

Y1 - 1997/1/1

N2 - The almost periodic eigenvalue problem described by the Harper equation is connected to other classes of quasiperiodic behavior; the dissipative dynamics on critical invariant tori and quasiperiodically driven maps. Firstly, the strong coupling limit of the supercritical Harper equation and the strong dissipation limit of the critical standard map play equivalent role in describing the universal characteristics of these systems. Secondly, a simple transformation is used to relate the Harper equation to a quasiperiodically forced one-dimensional map. In this case, the localized eigenstates of the supercritical Harper equation correspond to strange but nonchaotic attractors of the driven map. Furthermore, the existence of localization in the eigenvalue problem is associated with the appearance of homoclinic points in the corresponding map.

AB - The almost periodic eigenvalue problem described by the Harper equation is connected to other classes of quasiperiodic behavior; the dissipative dynamics on critical invariant tori and quasiperiodically driven maps. Firstly, the strong coupling limit of the supercritical Harper equation and the strong dissipation limit of the critical standard map play equivalent role in describing the universal characteristics of these systems. Secondly, a simple transformation is used to relate the Harper equation to a quasiperiodically forced one-dimensional map. In this case, the localized eigenstates of the supercritical Harper equation correspond to strange but nonchaotic attractors of the driven map. Furthermore, the existence of localization in the eigenvalue problem is associated with the appearance of homoclinic points in the corresponding map.

KW - Harper equation

KW - Quasiperiodicity

KW - Renormalization localization

KW - Standard map

KW - Strange nonchaotic attractor

UR - http://www.scopus.com/inward/record.url?scp=0003061177&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(97)00160-7

DO - 10.1016/S0167-2789(97)00160-7

M3 - Article

AN - SCOPUS:0003061177

VL - 109

SP - 70

EP - 80

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -