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May  2015, 35(5): 1767-1800. doi: 10.3934/dcds.2015.35.1767

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

Primitivo B. Acosta-Humánez 1, , J. Tomás Lázaro 2, , Juan J. Morales-Ruiz 3, and  Chara Pantazi 2,

1. 

Department of Mathematics, Universidad del Atlántico and Intelectual.Co, Barranquilla, Colombia
2. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, Spain, Spain
3. 

Department of Applied Mathematics, Technical University of Madrid, Madrid, Spain

Received  September 2013 Revised  September 2014 Published  December 2014

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.
Keywords: Poincaré problem, rational first integral, Riccati equation, Differential Galois theory, integrating factor, Liouvillian solution., Liénard equation, Darboux theory of integrability.
Mathematics Subject Classification: Primary: 12H05; Secondary: 32S6.
Citation: Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767
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show all references

References:
[1]

PhD. Thesis, Universitat Politecnica de Catalunya, 2009, arXiv:0906.3532. Google Scholar

[2]

VDM Verlag Dr. Müller, Berlin, 2010. Google Scholar

[3]

Lecturas Matemáticas, 27 (2006), 21-56.  Google Scholar

[4]

SIAM Journal on Applied Dynamical Systems, 8 (2009), 279-297. doi: 10.1137/080730329.  Google Scholar

[5]

Discrete Continuous Dynam. Systems - B, 10 (2008), 265-293. doi: 10.3934/dcdsb.2008.10.265.  Google Scholar

[6]

Report on Mathematical Physics, 67 (2011), 305-374. doi: 10.1016/S0034-4877(11)60019-0.  Google Scholar

[7]

SIGMA Symmetry Integrability Geom. Methods Appl., 8 (2012), Paper 043, 26 pp. doi: 10.3842/SIGMA.2012.043.  Google Scholar

[8]

preprint 2014. Google Scholar

[9]

J. Diff. Equat., 41 (1981), 44-58. doi: 10.1016/0022-0396(81)90052-8.  Google Scholar

[10]

VDM Verlag Dr. Müller, Berlin, 2008. Google Scholar

[11]

in Differential algebra, complex analysis and orthogonal polynomials, (eds P.B. Acosta-Humánez and F. Marcellán), Contemp. Math., Amer. Math. Soc., Providence, 509 (2010), 1-58. doi: 10.1090/conm/509.  Google Scholar

[12]

Nonlinearity, 25 (2012), 2615-2624. doi: 10.1088/0951-7715/25/9/2615.  Google Scholar

[13]

Ann. Math., 140 (1994), 289-294. doi: 10.2307/2118601.  Google Scholar

[14]

Ann. Inst. Fourier, 56 (2006), 735-779. doi: 10.5802/aif.2198.  Google Scholar

[15]

Ann. Inst. Fourier, 41 (1991), 883-903. doi: 10.5802/aif.1278.  Google Scholar

[16]

J. Phys. A, 37 (2004), 9923-9949. doi: 10.1088/0305-4470/37/42/007.  Google Scholar

[17]

European J. Appl. Math., 14 (2003), 217-229. doi: 10.1017/S0956792503005114.  Google Scholar

[18]

Gordon and Breach Science Publishers, New York-London-Paris, 1978.  Google Scholar

[19]

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J. Diff. Equat., 250 (2011), 1-25. doi: 10.1016/j.jde.2010.10.013.  Google Scholar

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J. Phys. A, 35 (2002), 2457-2476. doi: 10.1088/0305-4470/35/10/310.  Google Scholar

[24]

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[25]

Bull. Sci. math. 2ème série, 2 (1878), 60-96; 123-144; 151-200. Google Scholar

[26]

Historia Math., 25 (1998), 245-264. doi: 10.1006/hmat.1998.2194.  Google Scholar

[27]

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[28]

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[29]

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[30]

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[31]

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[32]

Lectures Notes in Mathematics, 708, Springer-Verlag, New York/Berlin, 1979. doi: 10.1007/BFb0063393.  Google Scholar

[33]

Hermann, Paris, 1957.  Google Scholar

[34]

Funkcialaj Ekvacioj, 12 (1969), 269-281.  Google Scholar

[35]

J. Symb. Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.  Google Scholar

[36]

Academic Press, Pure and Applied Mathematics, 54, New York - London, 1973.  Google Scholar

[37]

J. Differential Geometry, 26 (1987), 1-31.  Google Scholar

[38]

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[39]

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[44]

in Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., 38 (2001), 465-501.  Google Scholar

[45]

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in Computer Algebra and Differential Equations, (ed. E. Tournier), Comput. Math. Appl., Academic Press, London, (1990), 117-214.  Google Scholar

[47]

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Chapman and Hall, Boca Raton, 2003.  Google Scholar

[52]

Dover Publications, Inc., New York, 1960.  Google Scholar

[53]

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[54]

Grundlehren der Mathematischen Wissenschaften, 328, Springer Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7.  Google Scholar

[55]

Trans. Amer. Math. Soc., 333 (1992), 673-688. doi: 10.1090/S0002-9947-1992-1062869-X.  Google Scholar

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Amer. J. Math., {103} (1981), 661-682. doi: 10.2307/2374045.  Google Scholar

[57]

J. Symb. Comp., 22 (1996), 179-200. doi: 10.1006/jsco.1996.0047.  Google Scholar

[58]

J. Symb. Comp., 28 (1999), 495-520. doi: 10.1006/jsco.1999.0312.  Google Scholar

[59]

PhD. Thesis, École Politechnique, 1995. Google Scholar

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in Algorithms Seminar 2001-2002, (ed. F. Chyzak), INRIA, (2003), 43-46. Google Scholar

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Cambridge Univ. Press, New York, 1962.  Google Scholar

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in Differential Galois theory (Bedlewo, 2001), Banach Center Publ., 58, Polish Acad. Sci., Warsaw, (2002), 219-231. doi: 10.4064/bc58-0-17.  Google Scholar

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