Advanced Search
Article Contents
Article Contents

Rigorous enclosures of rotation numbers by interval methods

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 37E45, 37E10, 65G30.


    \begin{equation} \\ \end{equation}
  • [1]

    CAPD, Computer assisted proofs in dynamics, a package for rigorous numerics. Available from: http://capd.ii.uj.edu.pl/.


    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.


    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.


    S. Das, Y. Saiki, E. Sander and J. A. Yorke, Quantitative quasiperiodicity, preprint, arXiv:1601.06051.


    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.


    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.


    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.


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


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


    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.


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


    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.


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


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(114) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint