# American Institute of Mathematical Sciences

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

## Rigorous enclosures of rotation numbers by interval methods

 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.
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] CAPD, Computer assisted proofs in dynamics, a package for rigorous numerics. Available from: http://capd.ii.uj.edu.pl/. [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. [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. [4] S. Das, Y. Saiki, E. Sander and J. A. Yorke, Quantitative quasiperiodicity, preprint, arXiv:1601.06051. [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. [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. [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. [8] R. Moore, Interval Analysis, Prentice-Hall series in automatic computation, Prentice-Hall, 1966. [9] A. Neumaier, Interval Methods for Systems of Equations, 37. Encyclopedia of Mathematics and its Applications, Cambridge university press, 1990. [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. [11] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (1ère partie), Journal de mathématiques pures et appliquées, 7 (1881), 375-422. [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. [13] W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. [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.

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##### References:
 [1] CAPD, Computer assisted proofs in dynamics, a package for rigorous numerics. Available from: http://capd.ii.uj.edu.pl/. [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. [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. [4] S. Das, Y. Saiki, E. Sander and J. A. Yorke, Quantitative quasiperiodicity, preprint, arXiv:1601.06051. [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. [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. [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. [8] R. Moore, Interval Analysis, Prentice-Hall series in automatic computation, Prentice-Hall, 1966. [9] A. Neumaier, Interval Methods for Systems of Equations, 37. Encyclopedia of Mathematics and its Applications, Cambridge university press, 1990. [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. [11] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (1ère partie), Journal de mathématiques pures et appliquées, 7 (1881), 375-422. [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. [13] W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011. [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.
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