Circle maps with symmetry-breaking perturbations
type: Item
instance of: scholarly article
title: Circle maps with symmetry-breaking perturbations
zbMATH Open document ID: 0759.58038
DOI: 10.1016/0167-2789(92)90088-5
author: Anders Eriksson
author: T. Einarsson
author: Stellan Östlund
published in: Physica D
publication date: 1993-01-16T00:00:00Z
review text: The circle maps under investigation in this paper are of the form \(f_ \Omega(\theta)=\theta+\Omega-(a/2\pi)\sin(2\pi\theta)\) \(- (b/4\pi)\sin(4\pi\theta)\), with \(a,b\geq 0\). For the parameters \(a\) and \(b\), interesting features occur near the critical surface \(a+b=1\), where the map ceases to be a diffeomorphism. The parameter \(\Omega\) is used to control the winding number \(\rho\), which is taken to be either \(\gamma^{-1}\) or \(2\gamma^{-1}-1\), where \(\gamma\) is the golden mean. Universal exponents related to circle maps were first discovered numerically and later explained by renormalization techniques, where they appear as eigenvalues of the linearized renormalization operator around a fixed point. In the present work, four such universal exponents make their appearance. The numerical experiments which produce them rely on symmetry breaking perturbations, whereby it is essential to introduce with \(a=\varepsilon\) and \(b=1-\varepsilon\) a continuous parameter \(\varepsilon\) to control the symmetry violating terms. New crossover properties thus appear, which could not be obtained by discretely changing the universality class. For a rigorous explanation of the four exponents, the authors introduce an extended renormalization transformation, involving a third functional component. The interpretation which emerges is that one of the exponents corresponds to the change in winding number; two others form a degenerate pair corresponding to crossing the critical surface; the fourth and truly new one is somehow associated with antisymmetric perturbations of the circle map.
Mathematics Subject Classification ID: 37C55
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 91638
zbMATH Keywords: circle maps
zbMATH Keywords: renormalization
zbMATH Keywords: universal exponents
reviewed by: W. Sarlet
MaRDI profile type: Publication
full work available at URL: 0167-2789(92)90088-5
OpenAlex ID: W2085332188
cites work: Universal properties of the transition from quasi-periodicity to chaos in dissipative systems
cites work: Fractal boundary for the existence of invariant circles for area- preserving maps: Observations and renormalisation explanation
cites work: The universal metric properties of nonlinear transformations
cites work: Higher-order fixed points of the renormalisation operator for invariant circles
cites work: Q3232347
cites work: Renormalization in a circle map with two inflection points
Recommended article: Symmetry breaking and symmetry locking in equivariant circle maps
Recommended article: Circle expanding maps and symmetric structures
Recommended article: Q3806396
Recommended article: On quasi-symmetry of conjugacies between critical circle maps with break points
Recommended article: Q3598072
Recommended article: The dynamics of circle mapping under a parametric perturbation
Recommended article: Conjugation between circle maps with several break points
Recommended article: Discretizing randomly perturbed circle maps
Recommended article: Symbolic dynamics and skeletons of circle maps
Recommended article: Dynamics of circle maps with flat spots
description: scientific article; zbMATH DE number 91638