Computation of Quasiperiodic Normally Hyperbolic Invariant Tori: Rigorous Results

Article

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.

Keywords

Normally hyperbolic invariant manifolds KAM theory Computational dynamical systems 

Mathematics Subject Classification

37D10 37E45 37M99 65P99 

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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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