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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Poincaré Recurrences in the Circle Map: Fibonacci Stairs

Discontinuity, Nonlinearity, and Complexity 4(2) (2016) 111--119 | DOI:10.5890/DNC.2016.06.001

V.S. Anishchenko; N.I. Semenova; T.E. Vadivasova

Saratov State University - 83 Astrakhanskaya str., Saratov, Russia, 410012

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Abstract

We show that the dependence of the mimimal Poincaré return time on the vicinity size is universal for the golden and silver ratios in the circle map and can be referred to as the “Fibonacci stairs”. The rigorous result for the Afraimovich-Pesin dimension equality αc = 1 is confirmed for irrational rotation numbers with the measure of irrationality μ = 2. It is shown that some transcendental number are Diophantine and have the measure μ = 2. It is also confirmed that the gauge function 1/t cannot be applied for Liouvillian numbers. All the obtained features hold for both the linear and the nonlinear circle map.

Acknowledgments

The reported study was partially supported by RFBR, research project No. 15-02-02288. N.I. Semenova gratefully acknowledges the Dynasty Foundation.

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