Journal of Nonlinear Mathematical Physics

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Volume 27, Issue 2, January 2020, Pages 295 - 307

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

Authors
Isaac A. García
Departament de Matemàtica. Universitat de Lleida, Avda. Jaume II, 69. 25001 Lleida, Spain.,garcia@matematica.udl.cat
Benito Hernández-Bermejo
Departamento de Biología y Geología, Física y Química Inorgánica., Universidad Rey Juan Carlos., Calle Tulipán S/N. 28933–Móstoles–Madrid, Spain.,benito.hernandez@urjc.es
Received 9 June 2019, Accepted 11 September 2019, Available Online 27 January 2020.
DOI
10.1080/14029251.2020.1700637How to use a DOI?
Keywords
Poisson systems; Casimir invariants; Hamiltonian systems; perturbation theory; limit cycles
Abstract

A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the bifurcation phenomena of periodic orbits as a result of these perturbations in the period annulus associated to the unperturbed harmonic oscillator. This is accomplished via the averaging theory up to an arbitrary order in the perturbation parameter ε. In that theory we shall also use both branching theory and singularity theory of smooth maps to analyze the bifurcation phenomena at points where the implicit function theorem is not applicable. When the perturbation is given by a polynomial family, the associated Melnikov functions are polynomial and tools of computational algebra based on Gröbner basis are employed in order to reduce the generators of some polynomial ideals needed to analyze the bifurcation problem. When the most general perturbation of the harmonic oscillator by a quadratic perturbation field is considered, the complete bifurcation diagram (except at a high codimension subset) in the parameter space is obtained. Examples are given.

Copyright
© 2020 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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<Previous Article In Issue
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 2
Pages
295 - 307
Publication Date
2020/01/27
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1700637How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

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TY  - JOUR
AU  - Isaac A. García
AU  - Benito Hernández-Bermejo
PY  - 2020
DA  - 2020/01/27
TI  - Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds
JO  - Journal of Nonlinear Mathematical Physics
SP  - 295
EP  - 307
VL  - 27
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1700637
DO  - 10.1080/14029251.2020.1700637
ID  - García2020
ER  -