Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a -smooth Hamiltonian circle action, which is persistent under small integrable perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent...

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Bibliographic Details
Published in:Regular & chaotic dynamics 2021-11, Vol.26 (6), p.732-741
Main Authors: Kudryavtseva, Elena A., Martynchuk, Nikolay N.
Format: Article
Language:English
Subjects:
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
Orbital stability
Orbits
Perturbation
Structural stability
Toruses
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Summary:We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a -smooth Hamiltonian circle action, which is persistent under small integrable perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.
ISSN:1560-3547
1560-3547
1468-4845
DOI:10.1134/S1560354721060101