Abstract
We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasi-periodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this “linearization” (or conjugacy) from knowledge of a single quasi-periodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian problem: determining the characteristics of a quasi-periodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method “the weighted Birkhoff average”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Goldstein, B. R., On the Babylonian Discovery of the Periods of Lunar Motion, J. Hist. Astron., 2002, vol. 33, pp. 1–13.
Das, S., Saiki, Y., Sander, E., and Yorke, J. A., Solving the Babylonian Problem of Quasiperiodic Rotation Rates, Discrete Contin. Dyn. Syst. Special Topics, to appear, 2019.
de la Llave, R., Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: A Simple Proof and Extensions, Regul. Chaotic Dyn., 2018, vol. 23, no. 1, pp. 1–11.
Siegel, C. and Moser, J., Lectures on Celestial Mechanics, Grundlehren Math. Wiss., vol. 187, New York: Springer, 1971.
Luque, A. and Villanueva, J., Numerical Computation of Rotation Numbers of Quasi-Periodic Planar Curves, Phys. D, 2009, vol. 238, no. 20, pp. 2025–2044.
Siegel, C. L., Iteration of Analytic Functions, Ann. of Math. (2), 1942, vol. 43, pp. 607–612.
de la Llave, R. and Petrov, N. P., Boundaries of Siegel Disks: Numerical Studies of Their Dynamics and Regularity, Chaos, 2008, vol. 18, no. 3, 033135, 11 pp.
Milnor, J.W., Dynamics in One Complex Variable: Introductory Lectures, 3rd ed., Ann. of Math. Stud., vol. 160, Princeton: Princeton Univ. Press, 2006.
Brjuno, A.D., Analytic Form of Differential Equations: 1, Trans. Moscow Math. Soc., 1971, vol. 25, pp. 131–288; see also: Tr. Mosk. Mat. Obs., 1971, vol. 25, pp. 119–262.
Brjuno, A.D., Analytic Form of Differential Equations: 2, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 199–239; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 199–239.
Yoccoz, J.-Ch., Théorème de Siegel, Nombres de Bruno et polynomes quadratiques, in Petits diviseurs en dimension: 1, J.-Ch.Yoccoz (Ed.), Astérisque, vol. 231, Paris: Soc. Math. France, 1995, pp. 3–88.
Ushiki, S., personal communications, 2016.
Gómez, G., Mondelo, J.-M., and Simó, C., A Collocation Method for the Numerical Fourier Analysis of Quasi-Periodic Functions: 1. Numerical Tests and Examples, Discrete Contin. Dyn. Syst. Ser. B, 2010, vol. 14, no. 1, pp. 41–74.
Gómez, G., Mondelo, J.-M., and Simó, C., A Collocation Method for the Numerical Fourier Analysis of Quasi-Periodic Functions: 2. Analytical Error Estimates, Discrete Contin. Dyn. Syst. Ser. B, 2010, vol. 14, no. 1, pp. 75–109.
Cremer, H., Zum Zentrumproblem, Math. Ann., 1928, vol. 98, no. 1, pp. 151–163.
Jorba, A., Numerical Computation of the Normal Behaviour of Invariant Curves of N-Dimensional Maps, Nonlinearity, 2001, vol. 14, no. 5, pp. 943–976.
Das, S., Dock, Ch.B., Saiki, Y., Salgado-Flores, M., Sander, E., Wu, J., and Yorke, J. A., Measuring Quasiperiodicity, Europhys. Lett., 2016, vol. 116, no. 4, 40005, 6 pp.
Das, S., Saiki, Y., Sander, E., and Yorke, J. A., Quantitative Quasiperiodicity, Nonlinearity, 2017, vol. 30, no. 11, pp. 4111–4140.
Das, S. and Yorke, J.A., Super Convergence of Ergodic Averages for Quasiperiodic Orbits, Nonlinearity, 2018, vol. 31, no. 2, pp. 491–501.
Brin, M. and Stuck, G., Introduction to Dynamical Systems, Cambridge: Cambridge Univ. Press, 2002.
Laskar, J., Introduction to Frequency Map Analysis, in Hamiltonian Systems with Three or More Degrees of Freedom (S’Agaró, 1995), C. Simó (Ed.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 533, Dordrecht: Kluwer, 1999, pp. 134–150.
Laskar, J., Frequency Analysis of a Dynamical System, Celestial Mech. Dynam. Astronom., 1993, vol. 56, nos. 1–2, pp. 191–196.
Laskar, J., Frequency Analysis for Multi-Dimensional Systems. Global Dynamics and Diffusion, Phys. D, 1993, vol. 67, nos. 1–3, pp. 257–283.
Laskar, J., Frequency Map Analysis and Particle Accelerators, in Proc. of the 20th IEEE Particle Accelerator Conference (PAC’03, 2–16 May 2003, Portland, Oregon), pp. 378–382.
Seara, T. M. and Villanueva, J., On the Numerical Computation of Diophantine Rotation Numbers of Analytic Circle Maps, Phys. D, 2006, vol. 217, no. 2, pp. 107–120.
Luque, A. and Villanueva, J., Numerical Computation of Rotation Numbers of Quasi-Periodic Planar Curves, Phys. D, 2009, vol. 238, no. 20, pp. 2025–2044.
Luque, A. and Villanueva, J., Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods, SIAM J. Appl. Dyn. Syst., 2014, vol. 13, no. 1, pp. 1–46.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Saiki, Y., Yorke, J.A. Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem. Regul. Chaot. Dyn. 23, 735–750 (2018). https://doi.org/10.1134/S1560354718060084
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354718060084