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Transactions of the Moscow Mathematical Society

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On symplectic dynamics near a homoclinic orbit to 1-elliptic fixed point


Authors: Lev Lerman and Anna Markova
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2015, 271-299
MSC (2010): Primary 37J10, 37J30, 37J45, 70H07
DOI: https://doi.org/10.1090/mosc/245
Published electronically: November 18, 2015
MathSciNet review: 3468068
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Abstract: We study the orbit behavior of a 4-dimensional smooth symplectic diffeomorphism $ f$ near a homoclinic orbit $ \Gamma $ to a 1-elliptic fixed point under some natural genericity assumptions. A 1-elliptic fixed point has two real eigenvalues outside the unit circle and two on the unit circle. Thus there is a smooth 2-dimensional center manifold $ W^c$ where the restriction of the diffeomorphism has the elliptic fixed point supposed to be generic (no strong resonances and first Birkhoff coefficient is non-zero). Then the Moser theorem guarantees the existence of a positive measure set of KAM invariant curves. $ W^c$ itself is a normally hyperbolic manifold in the whole phase space and due to Fenichel results in every point on $ W^c$ having 1-dimensional stable and unstable smooth invariant curves smoothly foliating the related stable and unstable manifolds. In particular, each KAM invariant curve has stable and unstable smooth 2-dimensional invariant manifolds being Lagrangian ones. Stable and unstable manifolds of $ W^c$ are 3-dimensional smooth manifolds which are assumed to be transverse along homoclinic orbit $ \Gamma $. One of our theorems presents conditions under which each KAM invariant curve on $ W^c$ in a sufficiently small neighborhood of $ \Gamma $ has four transverse homoclinic orbits. Another result ensures that under some Moser genericity assumption for the restriction of $ f$ on $ W^c$ saddle periodic orbits in the resonance zone also have homoclinic orbits in the whole phase space, though its transversality or tangency cannot be verified directly. All this implies the complicated dynamics of the diffeomorphism and can serve as a criterion of its non-integrability.


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Additional Information

Lev Lerman
Affiliation: Department of Differential Equations & Mathematical Analysis — and — Research Institute for Applied Mathematics & Cybernetics, Lobachevsky State University of Nizhny Novgorod
Email: lermanl@mm.unn.ru

Anna Markova
Affiliation: Department of Differential Equations & Mathematical Analysis — and — Research Institute for Applied Mathematics & Cybernetics, Lobachevsky State University of Nizhny Novgorod
Email: anijam@yandex.ru

DOI: https://doi.org/10.1090/mosc/245
Keywords: 1-elliptic fixed point, homoclinic, invariant curve, periodic orbits
Published electronically: November 18, 2015
Additional Notes: The authors thank R. de la Llave and S.V.Gonchenko for useful discussions
The authors acknowledge partial support from the Russian Foundation for Basic Research under the grants 13-01-00589a (first author) and 14-01-00344 (second author)
The first author is also grateful for support from the Russian Ministry of Science and Education (project 1.1410.2014/K, target part) and from the Russian Science Foundation (project 14-41-00044).
Article copyright: © Copyright 2015 L. Lerman, A. Markova

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