Computation of Quasiperiodic Normally Hyperbolic Invariant Tori: Rigorous Results

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 fo...

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Bibliographic Details
Published in:Journal of nonlinear science 2017-12, Vol.27 (6), p.1869-1904
Main Authors: Canadell, Marta, Haro, Àlex
Format: Article
Language:English
Subjects:
Analysis
Classical Mechanics
Dynamical systems
Economic Theory/Quantitative Economics/Mathematical Methods
Invariants
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Newton methods
Order parameters
Theoretical
Toruses
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Summary: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.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-017-9389-y