Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential
Fecha
2019Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Identificador del proyecto
Impacto
|
10.1088/1361-6544/ab1bc6
Resumen
A family of perturbed Hamiltonians H = 1/2 (x^2 + X^2) − 1/2 (y^2 + Y^2)+1/2
(z^2 + Z^2) + 2[ (x^4 + y^4 + z^4) + (x^2 y^2 + x^2 z^2 + y^2 z^2)] in 1: −1:1 resonance
depending on two real parameters is considered. We show the existence and
stability of periodic solutions using reduction and averaging. In fact, there are
at most thirteen families for every energy level h < 0 and at most twe ...
[++]
A family of perturbed Hamiltonians H = 1/2 (x^2 + X^2) − 1/2 (y^2 + Y^2)+1/2
(z^2 + Z^2) + 2[ (x^4 + y^4 + z^4) + (x^2 y^2 + x^2 z^2 + y^2 z^2)] in 1: −1:1 resonance
depending on two real parameters is considered. We show the existence and
stability of periodic solutions using reduction and averaging. In fact, there are
at most thirteen families for every energy level h < 0 and at most twenty six
families for every h > 0. The different types of periodic solutions for every
nonzero energy level, as well as their bifurcations, are characterised in terms
of the parameters. The linear stability of each family of periodic solutions,
together with the determination of KAM 3-tori encasing some of the linearly
stable periodic solutions is proved. Critical Hamiltonian bifurcations on the
reduced space are characterised. We find important differences with respect
to the dynamics of the 1:1:1 resonance with the same perturbation as the one
given here. We end up with an intuitive interpretation of the results from a
cosmological viewpoint. [--]
Materias
Resonant Hamiltonians,
Friedmann–Lemaître–Robertson–Walker model,
Normalisation and reduction,
Hamiltonian Hopf bifurcation,
KAM tori,
Cosmological Hamiltonian,
Reduced space and invariants
Editor
IOP Publishing
Publicado en
Nonlinearity 32 3406
Notas
This is a peer-reviewed, un-copyedited version of an article published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/ab1bc6.
Departamento
Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas /
Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematika 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 Science, Innovation and Universities of Spain. C Vidal is partially supported by Project Fondecyt 1180288.