Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Algebraic invariant curves and first integrals for Riccati polynomial differential systems


Authors: Jaume Llibre and Clàudia Valls
Journal: Proc. Amer. Math. Soc. 142 (2014), 3533-3543
MSC (2010): Primary 34C05, 34A34, 34C14
DOI: https://doi.org/10.1090/S0002-9939-2014-12085-5
Published electronically: June 23, 2014
MathSciNet review: 3238428
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the algebraic invariant curves and first integrals for the Riccati polynomial differential systems of the form $ x'=1$, $ y' =a(x) y^2 +b(x)y +c(x)$, where $ a(x)$, $ b(x)$ and $ c(x)$ are polynomials. We characterize them when $ c(x)=\kappa (b(x)-\kappa a(x))$ for some $ \kappa \in \mathbb{C}$. We conjecture that algebraic invariant curves and first integrals for these Riccati polynomial differential systems only exist if $ c(x)=\kappa (b(x)-\kappa a(x))$ for some $ \kappa \in \mathbb{C}$.


References [Enhancements On Off] (What's this?)

  • [1] F. Baldassarri and B. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math. 101 (1979), no. 1, 42-76. MR 527825 (81d:34002), https://doi.org/10.2307/2373938
  • [2] Colin Christopher and Jaume Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations 16 (2000), no. 1, 5-19. MR 1768817 (2001g:34001)
  • [3] Colin Christopher, Liouvillian first integrals of second order polynomial differential equations, Electron. J. Differential Equations (1999), No. 49, 7 pp. (electronic). MR 1729833 (2000i:34017)
  • [4] Colin Christopher, Jaume Llibre, and Jorge Vitório Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 (2007), no. 1, 63-117. MR 2276503 (2008f:34065), https://doi.org/10.2140/pjm.2007.229.63
  • [5] L. Fuchs, Über die linearen Differentialgleichungen zweiter Ordnung welche algebraische Integrale besitzen, zweiter Abhandlung, J. für Math. 85 (1878).
  • [6] Alain Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys. 37 (1996), no. 4, 1871-1893. MR 1380879 (97b:70021), https://doi.org/10.1063/1.531484
  • [7] Einar Hille, Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1976. MR 0499382 (58 #17266)
  • [8] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757 (6,65f)
  • [9] E. Kamke, Differentialgleichungen ``losungsmethoden und losungen'', Col. Mathematik und ihre anwendungen vol. 18, Akademische Verlagsgesellschaft Becker und Erler Kom-Ges., Leipzig, 1943.
  • [10] Jerald J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), no. 1, 3-43. MR 839134 (88c:12011), https://doi.org/10.1016/S0747-7171(86)80010-4
  • [11] J. Liouville, Sur la détermination des intégrales dont la valeur est algébrique, J. de l'École Polytechnique 22 (1833).
  • [12] Jaume Llibre and Clàudia Valls, Integrability of the Bianchi IX system, J. Math. Phys. 46 (2005), no. 7, 072901, 13. MR 2153554 (2006f:37087), https://doi.org/10.1063/1.1955453
  • [13] George M. Murphy, Ordinary differential equations and their solutions, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0114953 (22 #5762)
  • [14] Zeev Nehari, Conformal mapping, Dover Publications Inc., New York, 1975. Reprinting of the 1952 edition. MR 0377031 (51 #13206)
  • [15] Andrei D. Polyanin and Valentin F. Zaitsev, Handbook of exact solutions for ordinary differential equations, 2nd ed., Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 2001201 (2004g:34001)
  • [16] P.Th. Pépin, Méthode pour obtenir les intégrales algébriques des équations différentielles linéaires du second ordre, Atti. dell' Accad. Pont. de Nuovi Lincei XXXIV (1881), 243-388.
  • [17] Michael F. Singer, Liouvillian solutions of $ n$th order homogeneous linear differential equations, Amer. J. Math. 103 (1981), no. 4, 661-682. MR 623132 (82i:12028), https://doi.org/10.2307/2374045
  • [18] Felix Ulmer and Jacques-Arthur Weil, Note on Kovacic's algorithm, J. Symbolic Comput. 22 (1996), no. 2, 179-200. MR 1422145 (97j:12006), https://doi.org/10.1006/jsco.1996.0047
  • [19] William T. Reid, Riccati differential equations, Mathematics in Science and Engineering, vol. 86, Academic Press, New York, 1972. MR 0357936 (50 #10401)
  • [20] Henryk Żoładek, Polynomial Riccati equations with algebraic solutions, Differential Galois theory (Bedlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci., Warsaw, 2002, pp. 219-231. MR 1972457 (2004j:34200), https://doi.org/10.4064/bc58-0-17

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 34C05, 34A34, 34C14

Retrieve articles in all journals with MSC (2010): 34C05, 34A34, 34C14


Additional Information

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Catalonia, Spain
Email: jllibre@mat.uab.cat

Clàudia Valls
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
Email: cvalls@math.ist.utl.pt

DOI: https://doi.org/10.1090/S0002-9939-2014-12085-5
Keywords: Algebraic first integrals, algebraic invariant curves, Riccati polynomial differential equations.
Received by editor(s): November 18, 2011
Received by editor(s) in revised form: December 13, 2011, May 1, 2012, September 6, 2012, and October 28, 2012
Published electronically: June 23, 2014
Additional Notes: The first author was partially supported by the MICINN/FEDER grant MTM2008–03437, AGAUR grant 2009SGR-410, ICREA Academia and FP7-PEOPLE-2012-IRSES-316338
The second author was partially supported by the FCT through CAMGDS, Lisbon
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society