Conjugating the Poincaré-map to a rotation

Pavani, Raffaella; Talamo, Rodolfo

doi:10.1007/bf01765644

Cited by 6 publications

(4 citation statements)

References 4 publications

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“…In particular, Properties 2-4 derive from known results about Farey sequences (e.g., [5]). Actually, Farey sequences can provide an efficient tool to the rational approximation of real numbers (e.g., [7,8]) and we follow this approach. Property 1.…”

Section: Main Results and Algorithmmentioning

confidence: 99%

A Fast Computation of the Best $${k}$$ k -Digit Rational Approximation to a Real Number

Citterio

Pavani

2016

Mediterr. J. Math.

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Given a real number α, we aim at computing the best rational approximation with at most k digits and with exactly k digits at the numerator (denominator). Our approach exploits Farey sequences. Our method turns out to be very fast in the sense that, once the development of α in continued fractions is available, the required operations are just a few and their number remains essentially constant for any k (in double precision finite arithmetic). Estimations of error bounds are also provided.

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Section: Main Results and Algorithmmentioning

confidence: 99%

A Fast Computation of the Best $${k}$$ k -Digit Rational Approximation to a Real Number

Citterio

Pavani

2016

Mediterr. J. Math.

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“…We estimate the error of the rotation number by using (24) with ν = 10, and we validate the rotation number when this error is smaller than 10 −24 . Bottom-right: We plot the error (24) for the points displayed in the previous figure. The horizontal axis is α ∈ [0, 2π] and the vertical one is log 10 of this error.…”

Section: Bottom-leftmentioning

confidence: 99%

“…This problem has been formerly considered by many other authors, and several algorithms have been developed. See for instance [31,4,20,24,23,3,11,12,7]. On the one side, the level of complexity of these algorithms ranges from the definition itself to sophisticated methods of frequency analysis.…”

Section: Introductionmentioning

confidence: 99%

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On the numerical computation of Diophantine rotation numbers of analytic circle maps

Seara

Villanueva

2006

Physica D: Nonlinear Phenomena

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In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that, under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, albeit it is not fully justified by our construction, the method turns to be quite efficient for computing rational rotation numbers. We discuss the method through several numerical examples.

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Numerical Treatment of the Rotation Number for the Forced Pendulum

Pavani

2002

Lecture Notes in Computer Science

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Conjugating the Poincaré-map to a rotation

Cited by 6 publications

References 4 publications

A Fast Computation of the Best $${k}$$ k -Digit Rational Approximation to a Real Number

A Fast Computation of the Best $${k}$$ k -Digit Rational Approximation to a Real Number

On the numerical computation of Diophantine rotation numbers of analytic circle maps

Numerical Treatment of the Rotation Number for the Forced Pendulum

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