Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory

We show the nonintegrability of the three-parameter Armburster--Guckenheimer--Kim quartic Hamiltonian using Morales--Ramis theory, with the exception of the three already known integrable cases. We use Poincaré sections to illustrate the breakdown of regular motion for some parameter values.

Autores:
Acosta-Humánez, P.
Alvarez-Ramírez, M.
Stuchi, T.J.
Tipo de recurso:
Fecha de publicación:
2018
Institución:
Universidad Simón Bolívar
Repositorio:
Repositorio Digital USB
Idioma:
eng
OAI Identifier:
oai:bonga.unisimon.edu.co:20.500.12442/1725
Acceso en línea:
http://hdl.handle.net/20.500.12442/1725
Palabra clave:
Hamiltonian
Integrability of dynamical systems
Diferential Galois theory
Legendre equation
Schrodinger equation
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License
Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional
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dc.title.eng.fl_str_mv Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
title Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
spellingShingle Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
Hamiltonian
Integrability of dynamical systems
Diferential Galois theory
Legendre equation
Schrodinger equation
title_short Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
title_full Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
title_fullStr Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
title_full_unstemmed Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
title_sort Nonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theory
dc.creator.fl_str_mv Acosta-Humánez, P.
Alvarez-Ramírez, M.
Stuchi, T.J.
dc.contributor.author.none.fl_str_mv Acosta-Humánez, P.
Alvarez-Ramírez, M.
Stuchi, T.J.
dc.subject.eng.fl_str_mv Hamiltonian
Integrability of dynamical systems
Diferential Galois theory
Legendre equation
Schrodinger equation
topic Hamiltonian
Integrability of dynamical systems
Diferential Galois theory
Legendre equation
Schrodinger equation
description We show the nonintegrability of the three-parameter Armburster--Guckenheimer--Kim quartic Hamiltonian using Morales--Ramis theory, with the exception of the three already known integrable cases. We use Poincaré sections to illustrate the breakdown of regular motion for some parameter values.
publishDate 2018
dc.date.accessioned.none.fl_str_mv 2018-02-08T14:59:21Z
dc.date.available.none.fl_str_mv 2018-02-08T14:59:21Z
dc.date.issued.none.fl_str_mv 2018-01-09
dc.type.spa.fl_str_mv article
dc.type.coar.fl_str_mv http://purl.org/coar/resource_type/c_6501
dc.identifier.issn.none.fl_str_mv 15360040
dc.identifier.uri.none.fl_str_mv http://hdl.handle.net/20.500.12442/1725
identifier_str_mv 15360040
url http://hdl.handle.net/20.500.12442/1725
dc.language.iso.spa.fl_str_mv eng
language eng
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_abf2
dc.rights.license.spa.fl_str_mv Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional
rights_invalid_str_mv Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional
http://purl.org/coar/access_right/c_abf2
dc.publisher.spa.fl_str_mv Sociedad de Matemática Industrial y Aplicada
dc.source.eng.fl_str_mv Vol. 17, No.1 (2018)
dc.source.spa.fl_str_mv Revista SIAM
institution Universidad Simón Bolívar
dc.source.uri.none.fl_str_mv https://doi.org/10.1137/16M1080689
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spelling Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacionalhttp://purl.org/coar/access_right/c_abf2Acosta-Humánez, P.4fa7634d-7d29-4f96-a6cb-627a48cb7570-1Alvarez-Ramírez, M.6bdd1617-7534-4fbe-bd41-3221d8f3638c-1Stuchi, T.J.90a5cc24-03e9-46b2-983b-1baad492f0ca-12018-02-08T14:59:21Z2018-02-08T14:59:21Z2018-01-0915360040http://hdl.handle.net/20.500.12442/1725We show the nonintegrability of the three-parameter Armburster--Guckenheimer--Kim quartic Hamiltonian using Morales--Ramis theory, with the exception of the three already known integrable cases. We use Poincaré sections to illustrate the breakdown of regular motion for some parameter values.engSociedad de Matemática Industrial y AplicadaVol. 17, No.1 (2018)Revista SIAMhttps://doi.org/10.1137/16M1080689HamiltonianIntegrability of dynamical systemsDiferential Galois theoryLegendre equationSchrodinger equationNonintegrability of the Armbruster-Guckenheimer-Kim Quartic Hamiltonian Through Morales-Ramis Theoryarticlehttp://purl.org/coar/resource_type/c_6501R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin Cummings, San Francisco, 1978.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Wiley, New York, 1984.P. Acosta-Hum anez and D. Bl azquez-Sanz, Non-integrability of some Hamiltonians with rational potential, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), pp. 265{293, https://doi.org/10.3934/dcdsb. 2008.10.265.P. B. Acosta-Hum anez, Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schr odinger Equation by Means of Di erential Galois Theory, VDM Verlag Dr. M uller, Berlin, 2010.P. B. Acosta-Hum anez, J. L azaro, J. Morales-Ruiz, and C. Pantazi, On the integrability of polynomial vector elds in the plane by means of Picard-Vessiot theory, Discrete Contin. Dyn. Syst., 35 (2015), pp. 1767{1800, https://doi.org/10.3934/dcds.2015.35.1767.P. B. Acosta-Hum anez, J. Morales-Ruiz, and J. A. Weil, Galoisian approach to integrability of Schr odinger equation, Rep. Math. Phys., 67 (2011), pp. 305{374, https://doi.org/10.1016/ S0034-4877(11)60019-0.P. Andrle, A third integral of motion in a system with a potential of the fourth degree, Phys. Lett. A, 17 (1966), pp. 169{175.D. Armbruster, J. Guckenheimer, and S. Kim, Chaotic dynamics in systems with square symmetry, Phys. Lett. A, 140 (1989), pp. 416{420, https://doi.org/10.1016/0375-9601(89)90078-9.A. Bostan, T. Combot, and M. E. Din, Computing necessary integrability conditions for planar parametrized homogeneous potentials, in Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, 2014, pp. 67{74, https://doi.org/10.1145/2608628.2608662.G. Contopoulos, Galactic Dynamics, Princeton University Press, Princeton, NJ, 1988.A. A. Elmandouh, On the dynamics of Armbruster-Guckenheimer-Kim galactic potential in a rotating reference frame, Astrophys. Space Sci., 361 (2016), pp. 182{194, https://doi.org/10.1007/ s10509-016-2770-8.J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), pp. 87{154.I. Kaplansky, An Introduction to Di erential Algebra, Hermann, Paris, 1957.E. Kolchin, Di erential Algebra and Algebraic Groups, Pure and Appl. Math. 59, Academic Press, New York, 1973.J. Llibre and L. Roberto, Periodic orbits and non-integrability of Armbruster-Guckenheimer-Kim potential, Astrophys. Space Sci., 343 (2013), pp. 69{74, https://doi.org/10.1007/s10509-012-1210-7.A. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901, https://doi.org/10.1063/1. 1917311.J. Morales-Ruiz, Di erential Galois Theory and Non-Integrability of Hamiltonian Systems, Progr. Math. 178, Birkhauser, Basel, 1999.J. Morales-Ruiz and J. P. Ramis, Integrability of dynamical systems through di erential galois theory: A practical guide, in Di erential Algebra, Complex Analysis and Orthogonal Polynomials, Contemp. Math. 509, AMS, Providence, RI, 2010, pp. 143{220, http://dx.doi.org/10.1090/conm/509.J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods Appl. Anal., 8 (2001), pp. 33{96, https://doi.org/10.4310/MAA.2001.v8.n1.a3.J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems II, Methods Appl. Anal., 8 (2001), pp. 97{112, https://doi.org/10.4310/MAA.2001.v8.n1.a4.K. Nakagawa, Direct Construction of Polynomial First Integrals for Hamiltonian Systems with a Two- Dimensional Homogeneous Polynomial Potential, Ph.D. thesis, Department of Astronomical Science, Graduate University for Advanced Study and the National Astronomical Observatory of Japan, 2002.G. P oschl and E. Teller, Bemerkungen zur quantenmechanik des anharmonischen oszillators, Z. Phys., 83 (1933), pp. 143{151.F. Simonelli and J. P. Gollub, Surface wave mode interactions: E ects of symmetry and degeneracy, J. Fluid Mech., 199 (1989), pp. 471{494, https://dx.doi.org/10.1017/S0022112089000443.E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Di erential Equations, Part I, 2nd ed., Oxford University Press, Oxford, UK, 1962.M. van der Put and M. Singer, Galois Theory of Linear Di erential Equations, Grundlehren Math. Wiss. 328, Springer, Berlin, 2003.S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990.H. Yoshida, Nonintegrability of the truncated Toda lattice Hamiltonian at any order, Comm. Math. Phys., 116 (1988), pp. 529{538.ORIGINALPDF.pdfPDF.pdfFormato Pdf texto completoapplication/pdf20587266https://bonga.unisimon.edu.co/bitstreams/1e121994-6347-4724-97ea-e395cf3cb274/download0d48430995cd71d1ee3b845491398e55MD51LICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://bonga.unisimon.edu.co/bitstreams/b18fa754-54f3-4295-afc2-bc07f40f9fa2/download8a4605be74aa9ea9d79846c1fba20a33MD52TEXTNonintegrability of the Armbruster.pdf.txtNonintegrability of the Armbruster.pdf.txtExtracted texttext/plain45866https://bonga.unisimon.edu.co/bitstreams/48fee0fd-f469-46ef-93a1-9d79dacf02ec/download70e5791ffdf158fde54613e7236e56ecMD53PDF.pdf.txtPDF.pdf.txtExtracted texttext/plain47732https://bonga.unisimon.edu.co/bitstreams/c02f1235-a50e-46b3-a700-be87a1b9f660/downloadb81f4e05353fcec3747c7a36ffb6f36aMD55THUMBNAILNonintegrability of the Armbruster.pdf.jpgNonintegrability of the Armbruster.pdf.jpgGenerated Thumbnailimage/jpeg1543https://bonga.unisimon.edu.co/bitstreams/c1e19d60-6e40-4623-a026-c5f6cc1a4fa8/download165bcba2629cac682805e0a138ab70d0MD54PDF.pdf.jpgPDF.pdf.jpgGenerated Thumbnailimage/jpeg4802https://bonga.unisimon.edu.co/bitstreams/c8a5dfa9-e130-4a27-9b62-6c6cbd4e1021/download10b8d9e5c8c648a5d6dabaaeb45eed9eMD5620.500.12442/1725oai:bonga.unisimon.edu.co:20.500.12442/17252024-07-25 03:02:59.371open.accesshttps://bonga.unisimon.edu.coRepositorio Digital Universidad Simón Bolívarrepositorio.digital@unisimon.edu.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