Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2017 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 30, no 12, p. 4483-4522Article in journal (Refereed) Published
Abstract [en]

Let Phi be a quasi-periodically forced quadratic map, where the rotation constant omega is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under Phi) attracting graph of a nowhere continuous measurable function psi from the circle T to [0, 1]. This paper investigates how a smooth attractor degenerates into a strange one, as a parameter beta approaches a critical value beta(0), and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attractor collides with a repeller. Our results show that the asymptotic minimum distance between the two colliding invariant curves decreases linearly in the parameter beta, as beta approaches the critical parameter value beta(0) from below. Furthermore, we have been able to show that the asymptotic growth of the supremum of the derivative of the attracting graph is asymptotically bounded from both sides by a constant times the reciprocal of the square root of the minimum distance above.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2017. Vol. 30, no 12, p. 4483-4522
Keywords [en]
dynamical systems, strange attractors, quasi-periodic quadratic family, bifurcations, asymptotics
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-219325DOI: 10.1088/1361-6544/aa8c9eISI: 000415852200001Scopus ID: 2-s2.0-85036637885OAI: oai:DiVA.org:kth-219325DiVA, id: diva2:1162673
Funder
Swedish Research Council, 2012-3090
Note

QC 20171205

Available from: 2017-12-05 Created: 2017-12-05 Last updated: 2018-09-17Bibliographically approved
In thesis
1. On the breakdown of regularity of invariant curves in quasi-periodically forced systems
Open this publication in new window or tab >>On the breakdown of regularity of invariant curves in quasi-periodically forced systems
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study the process of torus collisions in one-parameter families of quasi-periodically forced dynamical systems. Specifically, we study the process whereby two invariant curves (homeomorphic to circles), one attracting and one repelling, bifurcate into a strange non-chaotic attractor. In Paper A, the system is a quasi-periodically forced logistic family, but avoids period-doubling. We give an asymptotic analysis of some geometric properties of the attractor, as it approaches the repeller at the bifurcation point. In Paper B, we study the same type of questions as in Paper A, but instead for a class of quasi-periodic C^2 Schrödinger cocycles. In both papers the results confirm a conjecture that the distance between the curves is asymptotically linear in the parameter, for those classes of systems. In addition, we obtain results about the asymptotic growth of C^1-norms. In Paper C, we study the same class of systems as in Paper B, but instead look at the asymptotics of the (maximal) Lyapunov exponent at the bifurcation point. The results show that it has Hölder exponent exactly 1/2, as the energy parameter approaches the lowest energy of the spectrum. This confirms, for this class and setting, similar conjectures about this asymptotic behaviour.

Abstract [sv]

I denna avhandling studeras toruskollisioner i dynamiska system på skevproduktsform, som beror på en parameter. Mer specifikt studeras processen varvid två invarianta kurvor (homeomorfa med cirkeln), varav en attraherande och en repellerande, närmar sig varandra och resulterar i en attraktor med fraktal geometri. I Artikel A studeras en kvasiperiodiskt störd kvadratisk familj, som i detta fall inte genomgår någon perioddubblering. Resultaten visar asymptotiska lagar för hur geometrin hos de invarianta kurvorna ändras när de närmar sig bifurkationsparametern. I Artikel B studeras samma typ av frågor som i Artikel A, för en klass av kvasi-periodiskt störda Schrödinger-cocycler. I båda artiklarna visar resultaten att avståndet mellan kurvorna är asymptotiskt linjärt, med avseende på parametern, nära bifurkationen. Dessutom fås resultat för hur C^1-normerna växer nära bifurkationen. I Artikel C studeras samma klass av system som i Artikel B, men frågeställningen utgår i stället från det asymptotiska beteendet hos Lyapunovexponenten. Resultaten visar att Lyapunovexponenten har Hölderexponent exakt 1/2 för energier lägre än den lägsta energin i spektrumet. Detta visar, för denna klass, att Lyapunovexponenten uppför sig man förmodat.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2018. p. iii-vii,33
Series
TRITA-SCI-FOU ; 2018:39
Keywords
Dynamical systems, strange attractors, Schrödinger cocycles
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-235078 (URN)978-91-7729-937-0 (ISBN)
Public defence
2018-10-05, F3, Kungl Tekniska högskolan, Lindstedtsvägen 26, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20180917

Available from: 2018-09-17 Created: 2018-09-14 Last updated: 2018-09-17Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full textScopus

Authority records BETA

Timoudas, Thomas Ohlson

Search in DiVA

By author/editor
Timoudas, Thomas Ohlson
By organisation
Mathematics (Div.)
In the same journal
Nonlinearity
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 17 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf