Singularity theory for non-twist KAM tori

González-Enríquez, A.; Haro, A.; de la Llave, R.

doi:10.1090/memo/1067

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Singularity theory for non-twist KAM tori

About this Title

A. González-Enríquez, Dept. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona. Gran Via 585 08007 Barcelona Spain, A. Haro, Dept. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona. Gran Via 585, 08007 Barcelona, Spain and R. de la Llave, School of Mathematics, Georgia Institute of Technology. 686 Cherry Street, Atlanta, Georgia 30332-0160

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 227, Number 1067
ISBNs: 978-0-8218-9018-9 (print); 978-1-4704-1428-3 (online)
DOI: https://doi.org/10.1090/memo/1067
Published electronically: June 24, 2013
Keywords: Parametric KAM theory, non-twist tori, quasi-periodic bifurcations, Lagrangian deformations, moment map, Singularity theory for critical points of functions
MSC: Primary 37J20, 37J40

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Abstract

In this monograph we introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameter-dependent Hamiltonian systems. It is based on Singularity Theory of critical points of a real-valued function which we call the potential. The potential is constructed in such a way that: nondegenerate critical points of the potential correspond to twist invariant tori (i.e. with nondegenerate torsion) and degenerate critical points of the potential correspond to non-twist invariant tori. Hence, bifurcating points correspond to non-twist tori.

Invariant tori are classified using the classification of critical points of the potential as provided by Singularity Theory: the degeneracy class of an invariant torus is defined to be the degeneracy class of the corresponding critical point of the potential. Under rather general conditions this classification is robust: given a family of Hamiltonian systems (unperturbed family) for which there is a Hamiltonian with an invariant torus (unperturbed torus) satisfying general conditions, explicitly given in the monograph, we show that for any sufficiently close family of Hamiltonian systems (perturbed family) there is a torus (perturbed torus) that is invariant for a Hamiltonian of the perturbed family and such that both perturbed and unperturbed tori have the same frequency and belong to the same degeneracy class.

Our construction is developed for general Hamiltonian systems and general exact symplectic forms. It is applicable to the case in which a bifurcation of invariant

tori has been detected (e.g. numerically) but the system is not necessarily written as a perturbation of an integrable one. In the case that the system is written as a close-to-integrable one, our method applies to any finitely determinate singularity of the frequency map for the integrable system.

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Singularity theory for non-twist KAM tori

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Singularity theory for non-twist KAM tori