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Computation of Quasi-Periodic Normally Hyperbolic Invariant Tori: Algorithms, Numerical Explorations and Mechanisms of Breakdown

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Abstract

We present several algorithms for computing normally hyperbolic invariant tori carrying quasi-periodic motion of a fixed frequency in families of dynamical systems. The algorithms are based on a KAM scheme presented in Canadell and Haro (J Nonlinear Sci, 2016. doi:10.1007/s00332-017-9389-y), to find the parameterization of the torus with prescribed dynamics by detuning parameters of the model. The algorithms use different hyperbolicity and reducibility properties and, in particular, compute also the invariant bundles and Floquet transformations. We implement these methods in several 2-parameter families of dynamical systems, to compute quasi-periodic arcs, that is, the parameters for which 1D normally hyperbolic invariant tori with a given fixed frequency do exist. The implementation lets us to perform the continuations up to the tip of the quasi-periodic arcs, for which the invariant curves break down. Three different mechanisms of breakdown are analyzed, using several observables, leading to several conjectures.

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Authors and Affiliations

  1. ICERM, Brown University, 121 South Main street, Providence, RI, 02903, USA

    Marta Canadell

  2. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Àlex Haro

Authors
  1. Marta Canadell
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  2. Àlex Haro
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Correspondence to Àlex Haro.

Additional information

Communicated by Eusebius Doedel.

M.C. and À.H. acknowledge support from the Spanish Grants MTM2012-32541 and MTM2015-67724-P, and the Catalan Grant 2014-SGR-1145. M.C. 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 Quasi-Periodic Normally Hyperbolic Invariant Tori: Algorithms, Numerical Explorations and Mechanisms of Breakdown. J Nonlinear Sci 27, 1829–1868 (2017). https://doi.org/10.1007/s00332-017-9388-z

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  • DOI: https://doi.org/10.1007/s00332-017-9388-z

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