Dynamics of axially symmetric perturbed Hamiltonians in 1:1:1 resonance
Fecha
2018Autor
Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Identificador del proyecto
Impacto
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10.1007/s00332-018-9449-y
Resumen
We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system ...
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We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by
normalisation up to terms of degree 4 in rectangular coordinates; after truncation of
higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system. [--]
Materias
Invariants,
Symplectic reductions,
Axial symmetry,
Relative equilibria,
Periodic solutions,
Parametric bifurcations,
Invariant tori
Editor
Springer
Publicado en
Journal of Nonlinear Science (2018) 28:1293–1359
Notas
This is a post-peer-review, pre-copyedit version of an article published in Journal of Nonlinear Science (2018) 28:1293–1359. The final authenticated version is available online at https://doi.org/10.1007/s00332-018-9449-y
Departamento
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila /
Universidad Pública de Navarra/Nafarroako Unibertsitate Publikoa. Institute for Advanced Materials and Mathematics - INAMAT2
Versión del editor
Entidades Financiadoras
The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and
Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of
Spain, and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of
Spain. D. Carrasco is also partially supported by Project DIUBB 165708 3/R, Universidad del Bío-Bío,
Chile and by FONDECYT Project 1181061, CONICYT (Chile).