Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds
MetadataShow full item record
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.
Is part ofJournal of Nonlinear Mathematical Physics, 2020, vol. 27, núm. 2, p. 295-307
Showing items related by title, author, creator and subject.
García, I. A. (Isaac A.); Hernández Bermejo, Benito (IOP Publishing, 2017-07-15)Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of ...
Period annulus of the harmonic oscillator with zero cyclicity under perturbations with a homogeneous polynomial field García, I. A. (Isaac A.); Maza Sabido, Susanna (Bolyai Institute. University of SzegedHungarian Academy of Sciences, 2019-01-14)In this work we prove, using averaging theory at any order in the small perturbation parameter, that the period annulus of the harmonic oscillator has cyclicity zero (no limit cycles bifurcate) when it is perturbed by any ...
García, I. A. (Isaac A.); Llibre, Jaume (World Scientific Publishing, 2017-11-15)In this paper we show planar quadratic polynomial differentialsystems exhibiting as solutions some famous planar invariant algebraic curves. Also we put particular attention to the Darboux integrability of these differential ...