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: , ().

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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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

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