Journal ArticleDOI
Rational rotation numbers for maps of the circle
TLDR
In this article, the authors consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, and prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0.Abstract:
We consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms, that is maps like
$$x \to x + t + \frac{c}{{2\pi }}\sin (2\pi x)(\bmod 1)$$
withc=1. We prove that the set of parameter values corresponding to irrational rotation numbers has Lebesgue measure 0. In other words, the intervals on which frequency-locking occurs fill up the set of full measure.read more
Citations
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Journal ArticleDOI
Differentiable circle maps with a flat interval
TL;DR: In this paper, weakly order preserving circle maps with a flat interval were studied and a sharp transition from degenerate geometry to bounded geometry was found, depending on the degree of the singularities at the boundary of the flat interval.
Journal ArticleDOI
Dynamics of one-dimensional spiking neuron models.
Romain Brette,Romain Brette +1 more
TL;DR: It is found that, under conditions satisfied in particular by the periodically and aperiodically driven leaky integrator as well as some of its variants, the spike map is increasing on its range, which leaves no room for chaotic behavior.
Journal ArticleDOI
Singular measures in circle dynamics
Jacek Graczyk,Grzegorz Swiatek +1 more
TL;DR: In this paper, it was shown that critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure, and that the singularity of this measure is of Holder type.
Journal ArticleDOI
Piecewise linear models for the quasiperiodic transition to chaos.
TL;DR: Two families of piecewise linear degree one circle maps offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode locking and the quasiperiodic transition to chaos are formed.
Journal ArticleDOI
A decoding problem in dynamics and in number theory.
TL;DR: It is shown that, except for trivial cases, any code determines the rotation number, up to the orientation, of the homeomorphism which generates it, and simple solutions of the decoding problem are given both in the dynamical context and in the number theoretic context.
References
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Book
An Introduction to the Theory of Numbers
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Journal ArticleDOI
Complete Devil's Staircase, Fractal Dimension, and Universality of Mode- Locking Structure in the Circle Map
TL;DR: In this paper, it was shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos and the complementary set to the stability interval is a Cantor set of fractal dimension $D=0.87.
Journal ArticleDOI
A structure theorem in one dimensional dynamics
W. de Melo,S. van Strien +1 more
TL;DR: On considere la classe #7B-A des applications C ∞ f:[0, 1]→[0,1] telles que f(0)=f(1)=0 et f a unique point critique C ∈(0, 2) as mentioned in this paper.