11-83 A. Gonzalez-Enriquez, H. Haro, R. de la Llave

Singularity theory for non-twist KAM tori. (1420K, pdf) May 26, 11

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Abstract. We introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency. This is based on Singularity Theory of critical points of a real-valued function that we call `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. We show that, under general conditions, this classification is robust. Our construction is developed for general Hamiltonian systems and general exact symplectic forms. It is applicable to both the close-to-integrable case and the `far-from-integrable' case where a bifurcation of invariant tori has been detected (e.g. numerically). In the close-to-integrable case, our method applies to any finitely determinate singularity of the frequency map for the integrable system.

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