Abstract
The development of efficient methods for detecting quasiperiodic oscillations and computing the corresponding invariant tori is a subject of great importance in dynamical systems and their applications in science and engineering. In this paper, we prove the convergence of a new Newton-like method for computing quasiperiodic normally hyperbolic invariant tori carrying quasiperiodic motion in smooth families of real-analytic dynamical systems. The main result is stated as an a posteriori KAM-like theorem that allows controlling the inner dynamics on the torus with appropriate detuning parameters, in order to obtain a prescribed quasiperiodic motion. The Newton-like method leads to several fast and efficient computational algorithms, which are discussed and tested in a companion paper (Canadell and Haro in J Nonlinear Sci, 2017. doi:10.1007/s00332-017-9388-z), in which new mechanisms of breakdown are presented.
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Acknowledgements
We would like to thank Rafael de la Llave, Alejandro Luque, and Carles Simó for fruitful discussions and comments. We also thank the referees for their insightful comments and suggestions on the paper.
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Communicated by Eusebius Doedel.
Marta Canadell and Àlex Haro acknowledge support from the Spanish Grants MTM2012-32541 and MTM2015-67724-P, and the Catalan Grant 2014-SGR-1145. Marta Canadell also acknowledges support from the FPI Grant BES-2010-039663, and the NSF Grant DMS-1500943.
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Canadell, M., Haro, À. Computation of Quasiperiodic Normally Hyperbolic Invariant Tori: Rigorous Results. J Nonlinear Sci 27, 1869–1904 (2017). https://doi.org/10.1007/s00332-017-9389-y
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DOI: https://doi.org/10.1007/s00332-017-9389-y