About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 482305, 10 pages
http://dx.doi.org/10.1155/2013/482305
Research Article

On the Formal Integrability Problem for Planar Differential Systems

1Departamento de Matemàticas, Facultad de Ciencias, Avda. Tres de Marzo s/n, 21071 Huelva, Spain
2Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Catalonia, Spain

Received 15 November 2012; Accepted 28 January 2013

Academic Editor: Sung Guen Kim

Copyright © 2013 Antonio Algaba et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Algaba, E. Gamero, and C. García, “The integrability problem for a class of planar systems,” Nonlinearity, vol. 22, no. 2, pp. 395–420, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Algaba, E. Freire, E. Gamero, and C. García, “Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1726–1736, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Algaba, N. Fuentes, and C. García, “Centers of quasi-homogeneous polynomial planar systems,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 419–431, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Algaba, C. García, and M. Reyes, “Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems,” Chaos, Solitons & Fractals, vol. 45, no. 6, pp. 869–878, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  5. J. Chavarriga, H. Giacomini, J. Giné, and J. Llibre, “On the integrability of two-dimensional flows,” Journal of Differential Equations, vol. 157, no. 1, pp. 163–182, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Żoładek, “The problem of center for resonant singular points of polynomial vector fields,” Journal of Differential Equations, vol. 137, no. 1, pp. 94–118, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  7. X. Chen, J. Giné, V. G. Romanovski, and D. S. Shafer, “The 1: -q resonant center problem for certain cubic Lotka-Volterra systems,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11620–11633, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  8. C. Christopher, P. Mardešić, and C. Rousseau, “Normalizable, integrable, and linearizable saddle points for complex quadratic systems in 2,” Journal of Dynamical and Control Systems, vol. 9, no. 3, pp. 311–363, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. C. Christopher and C. Rousseau, “Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system,” Qualitative Theory of Dynamical Systems, vol. 5, no. 1, pp. 11–61, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. Ferčec, X. Chen, and V. G. Romanovski, “Integrability conditions for complex systems with homogeneous quintic nonlinearities,” The Journal of Applied Analysis and Computation, vol. 1, no. 1, pp. 9–20, 2011. View at MathSciNet
  11. B. Fercec, J. Gine, Y. Liu, and V. G. Romanovski, “Integrability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities,” Acta Applicandae Mathematicae, vol. 124, no. 1, pp. 107–122, 2013. View at Publisher · View at Google Scholar
  12. A. Fronville, A. P. Sadovski, and H. Żołpolhk adek, “Solution of the 1: −2 resonant center problem in the quadratic case,” Fundamenta Mathematicae, vol. 157, no. 2-3, pp. 191–207, 1998. View at Zentralblatt MATH · View at MathSciNet
  13. J. Giné, Z. Kadyrsizova, Y. Liu, and V. G. Romanovski, “Linearizability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1190–1201, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. Giné and V. G. Romanovski, “Linearizability conditions for Lotka-Volterra planar complex cubic systems,” Journal of Physics, vol. 42, no. 22, Article ID 225206, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. Giné and V. G. Romanovski, “Integrability conditions for Lotka-Volterra planar complex quintic systems,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 2100–2105, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. Hu, V. G. Romanovski, and D. S. Shafer, “1: −3 resonant centers on 2 with homogeneous cubic nonlinearities,” Computers & Mathematics with Applications, vol. 56, no. 8, pp. 1927–1940, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. V. G. Romanovski, X. Chen, and Z. Hu, “Linearizability of linear systems perturbed by fifth degree homogeneous polynomials,” Journal of Physics, vol. 40, no. 22, pp. 5905–5919, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser, Boston, Mass, USA, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. Algaba, C. García, and M. Reyes, “Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 314–323, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Algaba, C. García, and M. Reyes, “Rational integrability of two-dimensional quasi-homogeneous polynomial differential systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 5, pp. 1318–1327, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. Algaba, C. García, and M. Reyes, “Integrability of two dimensional quasi-homogeneous polynomial differential systems,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 1, pp. 1–22, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Algaba, C. Garcia, and M. Reyes, “A note on analytic integrability of planar vector fields,” European Journal of Applied Mathematics, vol. 23, pp. 555–562, 2012.
  23. J. Giné and X. Santallusia, “Essential variables in the integrability problem of planar vector fields,” Physics Letters A, vol. 375, no. 3, pp. 291–297, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. S. Gravel and P. Thibault, “Integrability and linearizability of the Lotka-Volterra system with a saddle point with rational hyperbolicity ratio,” Journal of Differential Equations, vol. 184, no. 1, pp. 20–47, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. C. Liu, G. Chen, and G. Chen, “Integrability of Lotka-Volterra type systems of degree 4,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1107–1116, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. C. Liu, G. Chen, and C. Li, “Integrability and linearizability of the Lotka-Volterra systems,” Journal of Differential Equations, vol. 198, no. 2, pp. 301–320, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J.-F. Mattei and R. Moussu, “Holonomie et intégrales premières,” Annales Scientifiques de l'École Normale Supérieure, vol. 13, no. 4, pp. 469–523, 1980. View at Zentralblatt MATH · View at MathSciNet
  28. J. Chavarriga, H. Giacomin, J. Giné, and J. Llibre, “Local analytic integrability for nilpotent centers,” Ergodic Theory and Dynamical Systems, vol. 23, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Giné, “On the number of algebraically independent Poincaré-Liapunov constants,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1870–1877, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. Giné, “Higher order limit cycle bifurcations from non-degenerate centers,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8853–8860, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  31. J. Giné and J. Mallol, “Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e132–e137, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. A. Liapunov, Problème Général de la Stabilité du Mouvement, vol. 17 of Annals of Mathematics Studies, Pricenton University Press, 1947.
  33. H. Poincaré, “Mémoire sur les courbes définies par les équations différentielles,” Journal de Mathematiques, vol. 37, pp. 375–422, 1881, vol. 8, pp. 251–296, 1882, Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, pp. 3–84, 1951.
  34. H. Dulac, “Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre,” Bulletin des Sciences Mathématiques, vol. 32, no. 2, pp. 230–252, 1908.
  35. A. D. Bruno, Local Methods in Nonlinear Differential Equations, Springer, Berlin, Germany, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  36. A. D. Bruno, Power Geometry in Algebraic and Differential Equations, vol. 57 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, The Netherlands, 2000. View at MathSciNet
  37. S. S. Abhyankar and T. T. Moh, “Embeddings of the line in the plane,” Journal für die Reine und Angewandte Mathematik, vol. 276, pp. 148–166, 1975. View at Zentralblatt MATH · View at MathSciNet