Journal of Nonlinear Science

, Volume 27, Issue 6, pp 1869–1904 | Cite as

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 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.


Normally hyperbolic invariant manifolds KAM theory Computational dynamical systems 

Mathematics Subject Classification

37D10 37E45 37M99 65P99 



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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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute for Computational and Experimental Research in Mathematics (ICERM)Brown UniversityProvidenceUSA
  2. 2.Department de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain

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