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Normal Forms of Planar Polynomial Differential Systems

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Abstract

Using the invariant theory, we develop an algorithmic method, which is based on the construction of a matrix of linear transformation, to compute normal forms of planar polynomial differential systems. We illustrate our method in the case where the planar polynomial differential system is cubic.

Keywords

Polynomial differential system Invariant Linear transformation Normal form 

Mathematics Subject Classification

34C20 14L36 34G14 15A72 

References

  1. 1.
    Boularas, D.: A new classification of planar homogeneous quadratic systems. Qual. Theory Dyn. Syst. 2(1), 93–110 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boularas, D., Dali, D.: Sur les bases de concomitants centro-affines des systèmes différentiels quadratiques plans complets. Cahiers Mathématiques d’Oran 2, 25–30 (1987)MATHGoogle Scholar
  3. 3.
    Calin, I.: On rational bases of \(GL(2, {\mathbb{R}})\)-comitants of planar polynomial systems of differential equations. Bul. Acad. Stiinte Repub. Mold. Mat. 2, 69–86 (2003)MathSciNetMATHGoogle Scholar
  4. 4.
    Ciubotaru, S.: Rational bases of \(GL(2, {\mathbb{R}})\)-comitants and of \(GL(2, {\mathbb{R}})\)-invariants for the planar system of differential equations with nonlinearities of the fourth degree. Bul. Acad. Stiinte Repub. Mold. Mat. 3(79), 1434 (2015)MathSciNetGoogle Scholar
  5. 5.
    Cox, D., Little, J., O’Shea, D.: Ideals Varieties and Algorithms. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2007)Google Scholar
  6. 6.
    Dali D.: Gröbner bases of algebraic invariants in polynomial differential systems, pp. 16–21. In: Le Matematiche, LXIII-Fasc. II (2008)Google Scholar
  7. 7.
    Dali, D., Cheng, S.S.: Decomposition of centro-affine covariants of polynomial differential systems. Taiwan. J. Math. 14, 1903–1924 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Date, T.: Classification and analysis of two-dimensional real homogeneous quadratic differential equations systems. J. Differ. Equ. 32(3), 311–323 (1979)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dieudonné, J., Carrell, J.: Invariant theory, old and new. Adv. Math. 4, 1–80 (1970)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases. J. Pure Appl. Algebra. 139(1), 61–88 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gurevich, G.B.: Foundation of the Theory of Algebraic Invariants. GITTL, Moscow (1948)Google Scholar
  12. 12.
    Hilbert, D.: Invariant Theory. Cambridge University Press, Cambridge (1789)Google Scholar
  13. 13.
    Lazard, D.: Gröbner bases Gaussian elimination and resolution of systems of algebraic equations. In: Computer algebra, EUROCAL ’83, European Computer Algebra Conference, London, England, 28–30 March 1983Google Scholar
  14. 14.
    Llibre, J., Vulpe, N.I.: Planar cubic polynomial differential systems with the maximum number of invariant straight lines. Rocky Mt. J. Math. 38, 1301–1373 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Macari, P.M., Popa, M.N.: Typical representation of polynomial differential systems by means of first-order comitants. Izv. Akad. Nauk Respub. Moldova Mat. 3, 5260 (1996). (Russian)MathSciNetGoogle Scholar
  16. 16.
    Mott, J.L., Kandel, A., Baker, T.P.: Discrete Mathematics for Computer Scientists and Mathematicians. Prentice Hall, Englewood Cliffs (1986)MATHGoogle Scholar
  17. 17.
    Popa, M.N.: Applications of Algebras to Differential Systems. Academy of Science of Moldova, Chisinau (2001)Google Scholar
  18. 18.
    Popov, V.L.: Invariant theory. Am. Math. Soc. Trans. 148(2), 99–112 (1991)MathSciNetMATHGoogle Scholar
  19. 19.
    Schlomiuk, D., Vulpe, N.I.: Planar quadratic differential systems with invariant straight lines of total multiplicity four. Nonlinear Anal. 68(4), 681–715 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Schlomiuk, D., Vulpe, N.I.: Application of symbolic calculation and polynomial invariant to the classification of singularities of differential systems. Comput. Algebra Sci. Comput. 8136(1), 340–354 (2013)CrossRefMATHGoogle Scholar
  21. 21.
    Sibirskii, C.S.: Introduction to the Algebraic Theory of Invariants of Differential Equations, Nonlinear Science, Theory and Applications. Manchester University Press, Manchester (1988)Google Scholar
  22. 22.
    Vulpe, N.I.: Polynomial basis of centro-affine comitants of a differential system. Doklady Akademii Nauk SSSR, 250, 5, 1033-7Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of Sciences and Technology Houari BoumedieneBab EzzouarAlgeria

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