Journal of Nonlinear Science

, Volume 27, Issue 6, pp 1829–1868 | Cite as

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



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.


Normally hyperbolic invariant manifolds KAM theory Computational dynamical systems Breakdown of invariant tori 

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.ICERMBrown UniversityProvidenceUSA
  2. 2.Departament de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain

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