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

García, I. A. (Isaac A.); Hernández Bermejo, Benito

doi:https://doi.org/10.1080/14029251.2020.1700637

View/Open

Postprint (241.2Kb)
Sol·licita una còpia

Issue date

2020-02-05

Author

García, I. A. (Isaac A.)

Hernández Bermejo, Benito

Suggested citation

García, I. A. (Isaac A.); Hernández Bermejo, Benito; . (2020) . Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds. Journal of Nonlinear Mathematical Physics, 2020, vol. 27, núm. 2, p. 295-307. https://doi.org/10.1080/14029251.2020.1700637.

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 Grobner 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.

URI

http://hdl.handle.net/10459.1/67918

DOI

https://doi.org/10.1080/14029251.2020.1700637

Is part of

Journal of Nonlinear Mathematical Physics, 2020, vol. 27, núm. 2, p. 295-307