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    Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential

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    NonlinearityVersiOn2019May8.pdf (616.7Kb)
    Date
    2019
    Author
    Palacián Subiela, Jesús Francisco Upna
    Vidal Díaz, Claudio 
    Vidarte, Jhon 
    Yanguas Sayas, Patricia Upna
    Version
    Acceso abierto / Sarbide irekia
    xmlui.dri2xhtml.METS-1.0.item-type
    Artículo / Artikulua
    Version
    Versión aceptada / Onetsi den bertsioa
    Project Identifier
    ES/6PN/MTM2011-28227 
    ES/1PE/MTM2014-59433 
    ES/2PE/MTM2017-88137 
    Impact
     
     
     
    10.1088/1361-6544/ab1bc6
     
     
     
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    Abstract
    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. [--]
    Subject
    Resonant Hamiltonians, Friedmann–Lemaître–Robertson–Walker model, Normalisation and reduction, Hamiltonian Hopf bifurcation, KAM tori, Cosmological Hamiltonian, Reduced space and invariants
     
    Publisher
    IOP Publishing
    Published in
    Nonlinearity 32 3406
    Description
    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.
    Departament
    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. InaMat - Institute for Advanced Materials
     
    Publisher version
    https://doi.org/10.1088/1361-6544/ab1bc6
    URI
    https://hdl.handle.net/2454/34657
    Sponsorship
    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.
    Appears in Collections
    • Artículos de revista DEIM - EIMS Aldizkari artikuluak [108]
    • Artículos de revista - Aldizkari artikuluak [2947]
    • Artículos de revista InaMat - InaMat aldizkari artikuluak [147]
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