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Solving the Babylonian problem of quasiperiodic rotation rates
1. | Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York 10012, USA |
2. | Graduate School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan |
3. | JST PRESTO, 4-1-8 Honcho, Kawaguchi-shi, Saitama 332-0012, Japan |
4. | University of Maryland, College Park, MD 20742, USA |
5. | Department of Mathematical Sciences, George Mason University, USA |
6. | University of Maryland, College Park, USA |
A trajectory $ \theta_n : = F^n(\theta_0), n = 0,1,2, \dots $ is quasiperiodic if the trajectory lies on and is dense in some $ d $-dimensional torus $ {\mathbb{T}^d} $, and there is a choice of coordinates on $ {\mathbb{T}^d} $ for which $ F $ has the form $ F(\theta) = \theta + \rho\bmod1 $ for all $ \theta\in {\mathbb{T}^d} $ and for some $ \rho\in {\mathbb{T}^d} $. (For $ d>1 $ we always interpret $ \bmod1 $ as being applied to each coordinate.) There is an ancient literature on computing the three rotation rates for the Moon. However, for $ d>1 $, the choice of coordinates that yields the form $ F(\theta) = \theta + \rho\bmod1 $ is far from unique and the different choices yield a huge choice of coordinatizations $ (\rho_1,\cdots,\rho_d) $ of $ \rho $, and these coordinations are dense in $ {\mathbb{T}^d} $. Therefore instead one defines the rotation rate $ \rho_\phi $ (also called rotation rate) from the perspective of a map $ \phi:T^d\to S^1 $. This is in effect the approach taken by the Babylonians and we refer to this approach as the "Babylonian Problem": determining the rotation rate $ \rho_\phi $ of the image of a torus trajectory - when the torus trajectory is projected onto a circle, i.e., determining $ \rho_\phi $ from knowledge of $ \phi(F^n(\theta)) $. Of course $ \rho_\phi $ depends on $ \phi $ but does not depend on a choice of coordinates for $ {\mathbb{T}^d} $. However, even in the case $ d = 1 $ there has been no general method for computing $ \rho_\phi $ given only the sequence $ \phi(\theta_n) $, though there is a literature dealing with special cases. Here we present our Embedding continuation method for general $ d $ for computing $ \rho_\phi $ from the image $ \phi(\theta_n) $ of a trajectory, and show examples for $ d = 1 $ and $ 2 $. The method is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem.
References:
[1] |
J. Barrow-Green, Poincaré and the Three Body Problem, Amer. Math. Soc., Providence, London, 1997. |
[2] |
A. Belova,
Rigorous enclosures of rotation numbers by interval methods, J. of Comput. Dyn., 3 (2016), 81-91.
doi: 10.3934/jcd.2016004. |
[3] |
M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511755316.![]() ![]() ![]() |
[4] |
J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Ltd, Chichester, 2016.
doi: 10.1002/9781119121534. |
[5] |
X. Cabré, E. Fontich and R. de la Llave,
The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.
doi: 10.1512/iumj.2003.52.2245. |
[6] |
S. Das, C. B. Dock, Y. Saiki, M. Salgado-Flores, E. Sander, J. Wu and J. A. Yorke,
Measuring quasiperiodicity, Europhys. Lett., 116 (2016), 40005.
doi: 10.1209/0295-5075/114/40005. |
[7] |
S. Das, E. Sander, Y. Saiki and J. A. Yorke,
Quantitative quasiperiodicity, Nonlinearity, 30 (2017), 4111-4140.
doi: 10.1088/1361-6544/aa84c2. |
[8] |
S. Das and J. A. Yorke,
Super convergence of ergodic averages for quasiperiodic orbits, Nonlinearity, 31 (2018), 491-501.
doi: 10.1088/1361-6544/aa99a0. |
[9] |
R. de la Llave, A. González, A. Jorba and J. Villanueva,
KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895.
doi: 10.1088/0951-7715/18/2/020. |
[10] |
B. R. Goldstein,
On the Babylonian discovery of the periods of lunar motion, J. Hist. Astro., 33 (2002), 1-13.
|
[11] |
A. Haro and R. de la Llave,
A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.
doi: 10.1137/050637327. |
[12] |
B. Hasselblatt and A. Katok, Principal structures, in Handbook of Dynamical Systems, North-Holland, 1 (2002), 1-203.
doi: 10.1016/S1874-575X(02)80003-0. |
[13] |
G. Huguet, R. de la Llave and Y. Sire,
Fast iteration of cocycles over rotations and computation of hyperbolic bundles, Discrete Contin. Dyn. Syst., (2013), 323-333.
doi: 10.3934/proc.2013.2013.323. |
[14] |
B. Hunt, T. Sauer and J. A. Yorke,
Prevalence: a translation-invariant "almost every" on infinite dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238.
doi: 10.1090/S0273-0979-1992-00328-2. |
[15] |
J. Laskar, Introduction to frequency map analysis, in Hamiltonian Systems with Three or More Degrees of Freedom(S'Agaró, 1995), vol. 533, Kluwer Acad. Publ., Dordrecht, 1999,134-150. |
[16] |
A. Luque and J. Villanueva,
Numerical computation of rotation numbers of quasi-periodic planar curves, Physica D, 238 (2009), 2025-2044.
doi: 10.1016/j.physd.2009.07.014. |
[17] |
A. Luque and J. Villanueva,
Quasi-periodic frequency analysis using averaging-extrapolation methods, SIAM J. Appl. Dyn. Syst., 13 (2014), 1-46.
doi: 10.1137/130920113. |
[18] |
W. Ott and J. A. Yorke,
Prevalence, Bull. Amer. Math. Soc., 42 (2005), 263-290.
doi: 10.1090/S0273-0979-05-01060-8. |
[19] |
H. Poincaré, New Methods of Celestial Mechanics, Translated from the French. NASA TT F-451 National Aeronautics and Space Administration, Washington, D. C., 1967. |
[20] |
E. Sander and J. A. Yorke,
The many facets of chaos, Int. J. Bifurcat. Chaos, 25 (2015), 1530011, 15pp.
doi: 10.1142/S0218127415300116. |
[21] |
T. Sauer, J. A. Yorke and M. Casdagli,
Embedology, J. Stat. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745. |
[22] |
V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, Cambridge, MA, 1967.
![]() |
[23] |
F. Takens, Detecting strange attractors in turbulence, in Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), vol. 898 of Lecture Notes in Math., Springer, Berlin-New York, 1981,366-381. |
[24] |
H. Whitney,
Differentiable manifolds, Annals of Math., 37 (1936), 645-680.
doi: 10.2307/1968482. |
show all references
References:
[1] |
J. Barrow-Green, Poincaré and the Three Body Problem, Amer. Math. Soc., Providence, London, 1997. |
[2] |
A. Belova,
Rigorous enclosures of rotation numbers by interval methods, J. of Comput. Dyn., 3 (2016), 81-91.
doi: 10.3934/jcd.2016004. |
[3] |
M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.
doi: 10.1017/CBO9780511755316.![]() ![]() ![]() |
[4] |
J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, Ltd, Chichester, 2016.
doi: 10.1002/9781119121534. |
[5] |
X. Cabré, E. Fontich and R. de la Llave,
The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.
doi: 10.1512/iumj.2003.52.2245. |
[6] |
S. Das, C. B. Dock, Y. Saiki, M. Salgado-Flores, E. Sander, J. Wu and J. A. Yorke,
Measuring quasiperiodicity, Europhys. Lett., 116 (2016), 40005.
doi: 10.1209/0295-5075/114/40005. |
[7] |
S. Das, E. Sander, Y. Saiki and J. A. Yorke,
Quantitative quasiperiodicity, Nonlinearity, 30 (2017), 4111-4140.
doi: 10.1088/1361-6544/aa84c2. |
[8] |
S. Das and J. A. Yorke,
Super convergence of ergodic averages for quasiperiodic orbits, Nonlinearity, 31 (2018), 491-501.
doi: 10.1088/1361-6544/aa99a0. |
[9] |
R. de la Llave, A. González, A. Jorba and J. Villanueva,
KAM theory without action-angle variables, Nonlinearity, 18 (2005), 855-895.
doi: 10.1088/0951-7715/18/2/020. |
[10] |
B. R. Goldstein,
On the Babylonian discovery of the periods of lunar motion, J. Hist. Astro., 33 (2002), 1-13.
|
[11] |
A. Haro and R. de la Llave,
A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM J. Appl. Dyn. Syst., 6 (2007), 142-207.
doi: 10.1137/050637327. |
[12] |
B. Hasselblatt and A. Katok, Principal structures, in Handbook of Dynamical Systems, North-Holland, 1 (2002), 1-203.
doi: 10.1016/S1874-575X(02)80003-0. |
[13] |
G. Huguet, R. de la Llave and Y. Sire,
Fast iteration of cocycles over rotations and computation of hyperbolic bundles, Discrete Contin. Dyn. Syst., (2013), 323-333.
doi: 10.3934/proc.2013.2013.323. |
[14] |
B. Hunt, T. Sauer and J. A. Yorke,
Prevalence: a translation-invariant "almost every" on infinite dimensional spaces, Bull. Amer. Math. Soc., 27 (1992), 217-238.
doi: 10.1090/S0273-0979-1992-00328-2. |
[15] |
J. Laskar, Introduction to frequency map analysis, in Hamiltonian Systems with Three or More Degrees of Freedom(S'Agaró, 1995), vol. 533, Kluwer Acad. Publ., Dordrecht, 1999,134-150. |
[16] |
A. Luque and J. Villanueva,
Numerical computation of rotation numbers of quasi-periodic planar curves, Physica D, 238 (2009), 2025-2044.
doi: 10.1016/j.physd.2009.07.014. |
[17] |
A. Luque and J. Villanueva,
Quasi-periodic frequency analysis using averaging-extrapolation methods, SIAM J. Appl. Dyn. Syst., 13 (2014), 1-46.
doi: 10.1137/130920113. |
[18] |
W. Ott and J. A. Yorke,
Prevalence, Bull. Amer. Math. Soc., 42 (2005), 263-290.
doi: 10.1090/S0273-0979-05-01060-8. |
[19] |
H. Poincaré, New Methods of Celestial Mechanics, Translated from the French. NASA TT F-451 National Aeronautics and Space Administration, Washington, D. C., 1967. |
[20] |
E. Sander and J. A. Yorke,
The many facets of chaos, Int. J. Bifurcat. Chaos, 25 (2015), 1530011, 15pp.
doi: 10.1142/S0218127415300116. |
[21] |
T. Sauer, J. A. Yorke and M. Casdagli,
Embedology, J. Stat. Phys., 65 (1991), 579-616.
doi: 10.1007/BF01053745. |
[22] |
V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies, Academic Press, Cambridge, MA, 1967.
![]() |
[23] |
F. Takens, Detecting strange attractors in turbulence, in Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), vol. 898 of Lecture Notes in Math., Springer, Berlin-New York, 1981,366-381. |
[24] |
H. Whitney,
Differentiable manifolds, Annals of Math., 37 (1936), 645-680.
doi: 10.2307/1968482. |













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