Golden mean renormalization for a generalized Harper equation: The Ketoja–Satija orchid

Mestel, Ben; Osbaldestin, Andrew

doi:10.1063/1.1797532

Cited by 11 publications

(29 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We show that the two systems in this context are equivalent and that for a typical choice of parameter in Region 2, the correlations at Fibonacci times lie on a renormalization strange set which is the quasi-periodic equivalent to the so-called 'orchid' set which occurs in the study of decaying eigenfunctions of the generalised Harper equation. [17] A generalization of this work to a class of quadratic irrational forcing frequencies should be straightforward. In [12], we extended our work on renormalization of the ACF in symmetric barrier billiards to such a class of frequency, and the renormalization equations for Q n for the Glendinning model will be identical to those in [12].…”

Section: Resultsmentioning

confidence: 99%

The non-smooth pitchfork bifurcation: a renormalization analysis

Adamson

Osbaldestin

2015

Dynamical Systems

View full text Add to dashboard Cite

We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions, self-similar behaviour of the attractor on and near the bifurcation curve can be observed, which corresponds to a periodic orbit of an underlying renormalization operator. We examine the scaling properties for various parameter choices including the so-called pitchfork critical point. Finally, we study the autocorrelation function for the system and show that it is equivalent to that present in symmetric barrier billiards.

show abstract

Section: Resultsmentioning

confidence: 99%

The non-smooth pitchfork bifurcation: a renormalization analysis

Adamson

Osbaldestin

2015

Dynamical Systems

View full text Add to dashboard Cite

show abstract

“…In [8] Ketoja and Satija also discover a universal renormalization strange set-the orchid-associated with a generalised Harper equation. In [13] we show how indeed recurrence (1.1) generates such a set. On the subject of the structure of a strange non-chaotic attractor itself, Kuznetsov et al [10] also utilise (1.1).…”

Section: Introductionmentioning

confidence: 89%

“…On the subject of the structure of a strange non-chaotic attractor itself, Kuznetsov et al [10] also utilise (1.1). We anticipate that our work on the orchid in [13] will throw some light on the this problem.…”

Section: Introductionmentioning

confidence: 99%

Self-similar correlations in a barrier billiard

Chapman¹,

Osbaldestin²

2003

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We give a renormalization analysis of the self-similarity of autocorrelation functions in symmetric barrier billiards for golden mean trajectories. For the special case of a half-barrier we present a rigorous calculation of the asymptotic height of the main peaks in the autocorrelation function. Fundamental to this work is a detailed analysis of a functional recurrence equation which has previously been used in the analysis of fluctuations in the Harper equation and of correlations in strange non-chaotic attractors and in quantum two-level systems.

show abstract

“…10,11,14,21,22 As we have seen, we do not use localization to prove the existence of SNA. Besides, localization may not hold in all the Harper maps considered here, because an energy a in the spectrum with nonzero Lyapunov exponent ͑for which the Harper map has a SNA͒ may not be an eigenvalue of the operator.…”

Section: Sna and Localizationmentioning

confidence: 99%

“…These two hypotheses lie at the heart of many heuristic arguments for establishing the existence of SNA in Harper maps. 10,11,14,21,22 …”

Section: Introductionmentioning

confidence: 99%

Strange nonchaotic attractors in Harper maps

Haro

Puig

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We study the existence of strange nonchaotic attractors ͑SNA͒ in the family of Harper maps. We prove that for a set of parameters of positive measure, the map possesses a SNA. However, the set is nowhere dense. By changing the parameter arbitrarily small amounts, the attractor is a smooth curve and not a SNA. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2259821͔In the last two decades there has been an enormous interest in the so-called strange nonchaotic attractors (SNA). These attracting invariant objects of dynamical systems capture the evolution of a large subset of the phase space and are very relevant for their description. In particular, SNA are geometrically complicated (they are strange) and their dynamics is regular (in most of the examples, quasiperiodic). SNA are typically observed in systems where there is a coupling between different dynamics. In contrast to the vast amount of numerical and experimental work in the area, there are only few rigorous proofs. The goal of this paper is to prove the existence and abundance of SNA in the family of Harper maps. This family arises from a model in mathematical physics and it is a paradigm of a 1D quasiperiodically forced map. Our approach connects the spectral theory of Schrödinger operators and the theory of nonuniformly hyperbolic systems. So, even if the proof is made for a concrete family, many of the arguments apply to other families.

show abstract

Golden mean renormalization for a generalized Harper equation: The Ketoja–Satija orchid

Cited by 11 publications

References 31 publications

The non-smooth pitchfork bifurcation: a renormalization analysis

The non-smooth pitchfork bifurcation: a renormalization analysis

Self-similar correlations in a barrier billiard

Strange nonchaotic attractors in Harper maps

Contact Info

Product

Resources

About