Differential Galois theory and non-integrability of planar polynomial vector fields
We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational firs...
- Autores:
-
Acosta-Humánez, Primitivo B.
Lázaro, J. Tomás
Morales-Ruiz, Juan J.
Pantazi, Chara
- Tipo de recurso:
- Fecha de publicación:
- 2018
- Institución:
- Universidad Simón Bolívar
- Repositorio:
- Repositorio Digital USB
- Idioma:
- eng
- OAI Identifier:
- oai:bonga.unisimon.edu.co:20.500.12442/2087
- Acceso en línea:
- http://hdl.handle.net/20.500.12442/2087
- Palabra clave:
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- License
- Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional
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Differential Galois theory and non-integrability of planar polynomial vector fields |
| title |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| spellingShingle |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| title_short |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| title_full |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| title_fullStr |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| title_full_unstemmed |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| title_sort |
Differential Galois theory and non-integrability of planar polynomial vector fields |
| dc.creator.fl_str_mv |
Acosta-Humánez, Primitivo B. Lázaro, J. Tomás Morales-Ruiz, Juan J. Pantazi, Chara |
| dc.contributor.author.none.fl_str_mv |
Acosta-Humánez, Primitivo B. Lázaro, J. Tomás Morales-Ruiz, Juan J. Pantazi, Chara |
| description |
We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrabilitywith some higher transcendent functions, like the error function. |
| publishDate |
2018 |
| dc.date.accessioned.none.fl_str_mv |
2018-05-08T14:40:28Z |
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2018-05-08T14:40:28Z |
| dc.date.issued.none.fl_str_mv |
2018-06 |
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article |
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http://purl.org/coar/resource_type/c_6501 |
| dc.identifier.issn.none.fl_str_mv |
00220396 |
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http://hdl.handle.net/20.500.12442/2087 |
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00220396 |
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http://hdl.handle.net/20.500.12442/2087 |
| dc.language.iso.spa.fl_str_mv |
eng |
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eng |
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http://purl.org/coar/access_right/c_abf2 |
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Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional |
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Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional http://purl.org/coar/access_right/c_abf2 |
| dc.publisher.spa.fl_str_mv |
Elsevier |
| dc.source.eng.fl_str_mv |
Journal of Differential Equations |
| dc.source.spa.fl_str_mv |
Vol. 264, No.12 (2018) |
| institution |
Universidad Simón Bolívar |
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Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacionalhttp://purl.org/coar/access_right/c_abf2Acosta-Humánez, Primitivo B.9439add6-3451-4b3f-b187-8b8ba4133fcd-1Lázaro, J. Tomás06c3bab8-f1b2-4099-a27c-5deee08f24d4-1Morales-Ruiz, Juan J.d211877b-2721-44f8-a9a5-c3f9380b8832-1Pantazi, Characaae82ad-d33a-4535-96f2-de87a60447a6-12018-05-08T14:40:28Z2018-05-08T14:40:28Z2018-0600220396http://hdl.handle.net/20.500.12442/2087We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrabilitywith some higher transcendent functions, like the error function.engElsevierJournal of Differential EquationsVol. 264, No.12 (2018)https://reader.elsevier.com/reader/sd/9AC79E5FF1C81427C9C71E1D6F14CCC5D4BC562F3248DB8006AD5A72DEFA8259852833B9E4CBD7D2BCB88DC7F6706567Differential Galois theory and non-integrability of planar polynomial vector fieldsarticlehttp://purl.org/coar/resource_type/c_6501P.B. Acosta-Humánez, J.T. Lázaro, J. Morales-Ruiz, Ch. Pantazi, On the integrability of polynomial vector fields in the plane by means of Picard–Vessiot theory, Discrete Contin. Dyn. Syst. Ser. A 35(5) (2015) 1767–1800, https://doi .org /10 .3934 /dcds .2015 .35 .1767.M. Ayoul, N.T. Zung, Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris 348 (2010) 1323–1326, https://doi .org /10 .1016 /j .crma .2010 .10 .024.G. Casale, Morales–Ramis theorems via Malgrange pseudogroup, Ann. Inst. Fourier 59(7) (2009) 2593–2610.C. Crespo, Z. Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, Rhode Island, 2011.G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. (2) 2 (1878) 60–96; 123–144; 151–200.J.H. Davenport, The Risch differential equation problem, SIAM J. Comput. 15 (1986) 903–918, https://doi .org /10 .1137 /0215063.F. Dumortier, J. Llibre, J.C. Artés, Qualitative Theory of Planar Polynomial Systems, Springer, Berlin, 2006.J.E. Humphreys, Linear Algebraic Groups, Springer Verlag, New York, 1981.Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, American Mathematical Society, Rhode Island, 2008.E. Kaltofen, A note on the Risch differential equation, in: J. Fitch (Ed.), Proceedings of EUROSAM 84, Cambridge, England, July 9–11, 1984, pp.359–366, https://doi .org /10 .1007 /BFb0032858.E.R. Kolchin, Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948) 1–42.E.R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.J. Liouville, Mémoire sur l’intégration d’une classe de fonctions transcendentes, J. Reine Angew. Math. 13 (1835) 93–118.J. Martinet, J.P. Ramis, Théorie de Galois différentielle et resommation, in: E. Tournier (Ed.), Computer Algebra and Differential Equations, Academic Press, London, 1989, pp.117–214.Juan J. Morales-Ruiz, Picard–Vessiot theory and integrability, J. Geom. Phys. 87 (2015) 314–343, https://doi .org /10 .1016 /j .geomphys .2014 .07 .006.J.J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathe-matics, vol.179, Birkhäuser, Basel, 1999.J.J. Morales-Ruiz, J.P. Ramis, C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. Éc. Norm. Supér. 40 (2007) 845–884, https://doi .org /10 .1016 /j .ansens .2007 .09 .002.M.J. Prelle, M.F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983) 215–229.R.H. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969) 167–189.J.F. Ritt, Integration in Finite Terms, Columbia Univ. Press, New York, 1948.M. Rothstein, Aspects of Symbolic Integration and Simplification of Exponential and Primitive Functions, Ph.D. thesis, Univ. Wisconsin-Madison, 1976.M.F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333(2) (1992) 673–688.M. van der Put, M. Singer, Galois Theory of Linear Differential Equations, Springer Verlag, Berlin, 2003.W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover, New York, 1965.E.T. Whittaker, An expression of certain known functions as generalised hypergeometric functions, Bull. Amer. Math. Soc. 10 (1903) 125–134.X. Zhang, Liouvillian integrability of polynomial differential systems, Trans. Amer. Math. Soc. 368 (2016) 607–620.X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics, vol.47, Springer, Singapore, 2017.X. Gómez-Mont, L. Ortíz-Bobadilla, Sistemas dinámicos holomorfos en superficies, Sociedad Matemática Mexi-cana, México D.F., 1989.LICENSElicense.txtlicense.txttext/plain; charset=utf-8368https://bonga.unisimon.edu.co/bitstreams/87665899-fe8f-4276-bd87-cf5f02b44b8b/download3fdc7b41651299350522650338f5754dMD5220.500.12442/2087oai:bonga.unisimon.edu.co:20.500.12442/20872019-04-11 21:51:30.285metadata.onlyhttps://bonga.unisimon.edu.coDSpace UniSimonbibliotecas@biteca.comPGEgcmVsPSJsaWNlbnNlIiBocmVmPSJodHRwOi8vY3JlYXRpdmVjb21tb25zLm9yZy9saWNlbnNlcy9ieS1uYy80LjAvIj48aW1nIGFsdD0iTGljZW5jaWEgQ3JlYXRpdmUgQ29tbW9ucyIgc3R5bGU9ImJvcmRlci13aWR0aDowIiBzcmM9Imh0dHBzOi8vaS5jcmVhdGl2ZWNvbW1vbnMub3JnL2wvYnktbmMvNC4wLzg4eDMxLnBuZyIgLz48L2E+PGJyLz5Fc3RhIG9icmEgZXN0w6EgYmFqbyB1bmEgPGEgcmVsPSJsaWNlbnNlIiBocmVmPSJodHRwOi8vY3JlYXRpdmVjb21tb25zLm9yZy9saWNlbnNlcy9ieS1uYy80LjAvIj5MaWNlbmNpYSBDcmVhdGl2ZSBDb21tb25zIEF0cmlidWNpw7NuLU5vQ29tZXJjaWFsIDQuMCBJbnRlcm5hY2lvbmFsPC9hPi4= |