American Institute of Mathematical Sciences

Advanced
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]

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.  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.  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.  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.  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, (1993).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[8]

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

[9]

A. Neumaier, Interval Methods for Systems of Equations, 37., Encyclopedia of Mathematics and its Applications, (1990).   Google Scholar

[10]

R. Pavani, A numerical approximation of the rotation number,, Applied Mathematics and Computation, 73 (1995), 191.  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.   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.  doi: 10.1016/j.physd.2006.03.013.  Google Scholar

[13]

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

[14]

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

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.  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.  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.  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.  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, (1993).  doi: 10.1007/978-3-642-78043-1.  Google Scholar

[8]

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

[9]

A. Neumaier, Interval Methods for Systems of Equations, 37., Encyclopedia of Mathematics and its Applications, (1990).   Google Scholar

[10]

R. Pavani, A numerical approximation of the rotation number,, Applied Mathematics and Computation, 73 (1995), 191.  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.   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.  doi: 10.1016/j.physd.2006.03.013.  Google Scholar

[13]

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

[14]

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

[1]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[2]

Sergio Zamora. Tori can't collapse to an interval. Electronic Research Archive, , () : -. doi: 10.3934/era.2021005
[3]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[4]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310
[5]

Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016
[6]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107
[7]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119
[8]

Editorial Office. Retraction: Jinling Wei, Jinming Zhang, Meishuang Dong, Fan Zhang, Yunmo Chen, Sha Jin and Zhike Han, Applications of mathematics to maritime search. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 957-957. doi: 10.3934/dcdss.2019064
[9]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[10]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031
[11]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[12]

Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071
[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[14]

Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304
[15]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011
[16]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[17]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[18]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162
[19]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[20]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045

 Impact Factor: 

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences