scholarly journals ROTATION SETS FOR ORBITS OF DEGREE ONE CIRCLE MAPS

2002 ◽  
Vol 12 (02) ◽  
pp. 429-437
Author(s):  
LLUÍS ALSEDÀ ◽  
FRANCESC MAÑOSAS ◽  
MOIRA CHAS
Keyword(s):  
Circle Map ◽  
Circle Maps ◽  
Rotation Interval ◽  
The One ◽  
Rotation Set ◽  
Dynamical Information

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point [Formula: see text] was given. In this paper we define and study a new notion of a rotation set of point which preserves more of the dynamical information contained in the sequences [Formula: see text] than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.

scholarly journals Twist sets for maps of the circle

1984 ◽  
Vol 4 (3) ◽  
pp. 391-404 ◽  
Author(s):  
Michał Misiurewicz
Keyword(s):  
Rotation Number ◽  
Closed Set ◽  
Twist Map ◽  
Rotation Interval ◽  
Continuous Map ◽  
The One

AbstractLet f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.

OBSERVED ROTATION NUMBERS IN FAMILIES OF CIRCLE MAPS

2001 ◽  
Vol 11 (01) ◽  
pp. 73-89 ◽  
Author(s):  
MICHAEL A. SAUM ◽  
TODD R. YOUNG
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Narrow Band ◽  
Initial Conditions ◽  
Positive Probability ◽  
Circle Maps ◽  
Numerical Evidence ◽  
Rotation Interval ◽  
Large Numbers ◽  
Rotation Numbers

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

scholarly journals The speed interval: a rotation algorithm for endomorphisms of the circle

1988 ◽  
Vol 8 (1) ◽  
pp. 17-33 ◽  
Author(s):  
Jan Barkmeijer
Keyword(s):  
Rotation Interval ◽  
Continuous Map ◽  
Rotation Set

AbstractLetfbe a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates eachz∈S1with an interval, the so-called speed intervalS(z,f), which is contained in the rotation interval ρ(f) off. In contrast with the rotation set ρ(z,f), the intervalS(z,f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn(z) |n≥ 0}. We further show that all choices for ρ(z,f) andS(z,f) occur, for certainz∈S1provided that ρ(z,f) ⊂S(z,f) ⊂ ρ(f).

scholarly journals Quasisymmetric orbit-flexibility of multicritical circle maps

2021 ◽  
pp. 1-40
Author(s):  
EDSON DE FARIA ◽  
PABLO GUARINO
Keyword(s):  
Rotation Number ◽  
A Priori ◽  
Gauss Map ◽  
Circle Map ◽  
A Priori Bounds ◽  
Circle Maps ◽  
Circle Homeomorphism ◽  
Critical Circle ◽  
Full Lebesgue Measure ◽  
Topological Conjugacies

Abstract Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).

scholarly journals BELLS GALORE: OSCILLATIONS AND CIRCLE-MAP DYNAMICS FROM SPACE-FILLING FRACTAL FUNCTIONS

Fractals ◽  
2008 ◽  
Vol 16 (04) ◽  
pp. 367-378 ◽  
Author(s):  
CARLOS E. PUENTE ◽  
ANDREA CORTIS ◽  
BELLIE SIVAKUMAR
Keyword(s):  
Two Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Circle Map ◽  
Space Filling ◽  
Affine Mappings ◽  
Oscillatory Pattern ◽  
Fractal Functions ◽  
The Many ◽  
The One

The construction of a host of interesting patterns over one and two dimensions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings' parameters) in which there is no limit. Specifically, the one-dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, despite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.

scholarly journals Strange non-chaotic attractors in quasi-periodically forced circle maps: Diophantine forcing

2012 ◽  
Vol 33 (5) ◽  
pp. 1477-1501 ◽  
Author(s):  
T. JÄGER
Keyword(s):  
Positive Measure ◽  
Good Control ◽  
Chaotic Attractors ◽  
Circle Map ◽  
Circle Maps ◽  
Arnold Tongue ◽  
Open Set ◽  
Nonlinear Version ◽  
Study Parameter ◽  
Critical Sets

AbstractWe study parameter families of quasi-periodically forced (qpf) circle maps with Diophantine frequency. Under certain $\mathcal {C}^1$-open conditions concerning their geometry, we prove that these families exhibit non-uniformly hyperbolic behaviour, often referred to as the existence of strange non-chaotic attractors, on parameter sets of positive measure. This provides a nonlinear version of results by Young on quasi-periodic $\mathrm {SL}(2,\mathbb {R})$-cocycles and complements previous results in this direction which hold for sets of frequencies of positive measure, but did not allow for an explicit characterization of these frequencies. As an application, we study a qpf version of the Arnold circle map and show that the Arnold tongue corresponding to rotation number $1/2$collapses on an open set of parameters. The proof requires to perform a parameter exclusion with respect to some twist parameter and is based on the multiscale analysis of the dynamics on certain dynamically defined critical sets. A crucial ingredient is to obtain good control on the parameter dependence of the critical sets. Apart from the presented results, we believe that this step will be important for obtaining further information on the behaviour of parameter families like the qpf Arnold circle map.

scholarly journals Indicator studies: a critical review and a new data­presentation method

1993 ◽  
Vol 40 ◽  
pp. 332-340
Author(s):  
Per Smed
Keyword(s):  
Critical Review ◽  
Lateral Displacement ◽  
Wide Spectrum ◽  
Point Of View ◽  
Circle Map ◽  
Ice Flow ◽  
Circle Maps ◽  
Rock Types ◽  
Made In

This paper deals with general problems of indicator collection and introduces a new datapresentation method, named the circle-map method. The advantages of this method is demonstrated by collecting a wide spectrum of Scandinavian and Baltic rock types in two Danish localities. It is concluded that an indicator count should be made in situ and comprise at least 50 stones, and that the rock types used should represent parent areas as evenly distributed as possible throughout the Scandinavian and Baltic regions. The circle maps indicate the routeway of ice flow from the Scandinavian mountains to the locality dealt with. It emphasizes variations in indicator content and may, therefore, show the lateral displacement of ice-flow routes which should be expected from a glaciological point of view. In addition, the method makes it possible to distinguish between directly brought stones and contaminating stones.

Rotation intervals for a family of degree one circle maps

1988 ◽  
Vol 8 (3) ◽  
pp. 395-409
Author(s):  
Leo B. Jonker
Keyword(s):  
Local Minimum ◽  
Local Maximum ◽  
Circle Map ◽  
Circle Maps ◽  
Rotation Numbers

AbstractLet f be a C0 circle map of degree one with precisely one local minimum and one local maximum, and let [ρ−(f), ρ+(f)] be the interval of rotation numbers of f. We study the structure of the function ρ(λ) = ρ+(Rλ∘f), where Rλ is the rotation through the angle λ.

Non-monotone periodic orbits of a rotational horseshoe

2017 ◽  
Vol 39 (7) ◽  
pp. 1870-1903
Author(s):  
BRÁULIO A. GARCIA ◽  
VALENTÍN MENDOZA
Keyword(s):  
Periodic Orbits ◽  
Rational Numbers ◽  
Markov Partition ◽  
Circle Map ◽  
Rotation Interval

In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair $(r,s)$ of rational numbers with $r<s$ in $(0,1)$, there exists a non-monotone orbit $B_{r,s}$ in this family which has pseudo-Anosov type and rotation interval $[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits $H_{p/q}$, which are the largest for the forcing order among all the $(p,q)$-orbits of the rotational horseshoe.

BIFURCATIONS IN A DISCONTINUOUS CIRCLE MAP: A THEORY FOR A CHAOTIC CARDIAC ARRHYTHMIA

1995 ◽  
Vol 05 (02) ◽  
pp. 359-371 ◽  
Author(s):  
GIL BUB ◽  
LEON GLASS
Keyword(s):  
Lyapunov Exponent ◽  
Cardiac Arrhythmia ◽  
Piecewise Linear ◽  
Circle Map ◽  
Circle Maps ◽  
Normal Sinus ◽  
Coupling Interval ◽  
Wide Range ◽  
Interval Width ◽  
Ectopic Beats

The dynamics of discontinuous circle maps are investigated in the context of modulated parasystole, a cardiac arrhythmia in which there is an interaction between normal (sinus) and abnormal (ectopic) pacemaking sites in the heart. A class of noninvertible discontinuous circle maps with slope greater than 1 displays banded chaos under certain conditions. Banded chaos in these maps is characterized by a zero rotation interval width in the presence of a positive Lyapunov exponent. The bifurcations of a simple piecewise linear circle map are investigated. Parameters that result in banded chaos are organized into discrete, nonoverlapping zones in the parameter space. We apply these results to analyze a circle map that models modulated parasystole. Analysis of the model is complicated by the fact that the map has slope less than 1 for part of its domain. However, numerical simulations indicate that the modulated parasystole map displays banded chaos over a wide range of parameters. Banded chaos in this map produces rhythms with a relatively constant sinus-ectopic coupling interval, long trains of uninterrupted sinus beats, and patterns of successive sinus beats between ectopic beats characteristic of those found clinically.

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