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On the numerical approximation of the Perron-Frobenius and Koopman operator
Rigorous enclosures of rotation numbers by interval methods
1. | Box 480, SE-75106, Uppsala, Sweden |
References:
[1] |
CAPD, Computer assisted proofs in dynamics, a package for rigorous numerics., Available from: , (). Google Scholar |
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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. |
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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. |
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S. Das, Y. Saiki, E. Sander and J. A. Yorke, Quantitative quasiperiodicity, preprint,, , (). Google Scholar |
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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).
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[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. 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. |
[13] |
W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations,, Princeton University Press, (2011).
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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: , (). 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. |
[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,, , (). 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. |
[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. 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. |
[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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