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January  2016, 3(1): 81-91. doi: 10.3934/jcd.2016004

Rigorous enclosures of rotation numbers by interval methods

Anna Belova 1,

1. 

Box 480, SE-75106, Uppsala, Sweden

Received  September 2015 Revised  March 2016 Published  September 2016

We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the rotation number. A few numerical experiments are presented to show that the implementation of interval methods produces a good enclosure of the rotation number of a circle map.
Keywords: rigorous computation, circle map, Rotation number, interval arithmetic, set-valued mathematics..
Mathematics Subject Classification: Primary: 37E45, 37E10, 65G3.
Citation: Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004
References:
[1]

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W. de Melo and S. van Strien, One-dimensional Dynamics, 25 Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

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A. Neumaier, Interval Methods for Systems of Equations, 37. Encyclopedia of Mathematics and its Applications, Cambridge university press, 1990.  Google Scholar

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R. Pavani, A numerical approximation of the rotation number, Applied Mathematics and Computation, 73 (1995), 191-201. doi: 10.1016/0096-3003(94)00249-5.  Google Scholar

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H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (1ère partie), Journal de mathématiques pures et appliquées, 7 (1881), 375-422. Google Scholar

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T. M. Seara and J. Villanueva, On the numerical computation of diophantine rotation numbers of analytic circle maps, Physica D: Nonlinear Phenomena, 217 (2006), 107-120. doi: 10.1016/j.physd.2006.03.013.  Google Scholar

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W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011.  Google Scholar

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show all references

References:
[1]

CAPD, Computer assisted proofs in dynamics, a package for rigorous numerics., Available from: , ().   Google Scholar

[2]

H. Bruin, Numerical determination of the continued fraction expansion of the rotation number, Physica D: Nonlinear Phenomena, 59 (1992), 158-168. doi: 10.1016/0167-2789(92)90211-5.  Google Scholar

[3]

M. J. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1997-2026. doi: 10.1088/0951-7715/25/7/1997.  Google Scholar

[4]

S. Das, Y. Saiki, E. Sander and J. A. Yorke, Quantitative quasiperiodicity, preprint,, , ().   Google Scholar

[5]

Z. Galias, Proving the existence of long periodic orbits in 1D maps using interval Newton method and backward shooting, Topology Appl., 124 (2002), 25-37. doi: 10.1016/S0166-8641(01)00227-9.  Google Scholar

[6]

A. Luque and J. Villanueva, Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms, Physica D: Nonlinear Phenomena, 237 (2008), 2599-2615. doi: 10.1016/j.physd.2008.03.047.  Google Scholar

[7]

W. de Melo and S. van Strien, One-dimensional Dynamics, 25 Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[8]

R. Moore, Interval Analysis, Prentice-Hall series in automatic computation, Prentice-Hall, 1966.  Google Scholar

[9]

A. Neumaier, Interval Methods for Systems of Equations, 37. Encyclopedia of Mathematics and its Applications, Cambridge university press, 1990.  Google Scholar

[10]

R. Pavani, A numerical approximation of the rotation number, Applied Mathematics and Computation, 73 (1995), 191-201. doi: 10.1016/0096-3003(94)00249-5.  Google Scholar

[11]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (1ère partie), Journal de mathématiques pures et appliquées, 7 (1881), 375-422. Google Scholar

[12]

T. M. Seara and J. Villanueva, On the numerical computation of diophantine rotation numbers of analytic circle maps, Physica D: Nonlinear Phenomena, 217 (2006), 107-120. doi: 10.1016/j.physd.2006.03.013.  Google Scholar

[13]

W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011.  Google Scholar

[14]

M. Van Veldhuizen, On the numerical approximation of the rotation number, Journal of Computational and Applied Mathematics, 21 (1988), 203-212. doi: 10.1016/0377-0427(88)90268-3.  Google Scholar

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