Qualitative Theory of Dynamical Systems

, Volume 18, Issue 2, pp 583–602 | Cite as

Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

  • Armengol Gasull
  • J. Tomás Lázaro
  • Joan TorregrosaEmail author


We show how the use of rational parameterizations facilitates the study of the number of solutions of many systems of equations involving polynomials and square roots of polynomials. We illustrate the effectiveness of this approach, applying it to several problems appearing in the study of some dynamical systems. Our examples include Abelian integrals, Melnikov functions and a couple of questions in Celestial Mechanics: the computation of some relative equilibria and the study of some central configurations.


Bifurcation Resultant Rational parameterization Abelian integral Poincaré–Melnikov–Pontryagin function Relative equilibria Central configuration 

Mathematics Subject Classification

Primary: 34C23 Secondary: 13P15 14E05 34C08 37N05 



This work has received funding from the Ministerio de Economía, Industria y Competitividad - Agencia Estatal de Investigación (Grants MTM2015-65715-P and MTM2016-77278-P (FEDER)), the Agència de Gestió d’Ajuts Universitaris i de Recerca (Grants 2017 SGR 1049 and 2017 SGR 1617).


  1. 1.
    Abhyankar, S.S.: What is the difference between a parabola and a hyperbola? Math. Intel. 10(4), 36–43 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abhyankar, S.S., Bajaj, C.L.: Automatic parameterization of rational curves and surfaces. III. Algebraic plane curves. Comput. Aided Geom. Des. 5(4), 309–321 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alfaro, F., Pérez-Chavela, E.: Families of continua of central configurations in charged problems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 9(4), 463–475 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Álvarez-Ramírez, M., Llibre, J.: The symmetric central configurations of the 4-body problem with masses \(m_1=m_2\ne m_3=m_4\). Appl. Math. Comput. 219, 5996–6001 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Arnold, H.A.: The crossed ladders. Math. Mag. 29(3), 153–154 (1956)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Arredondo, J.A., Pérez-Chavela, E., Stoica, C.: Dynamics in the Schwarzschild isosceles three body problem. J. Nonlinear Sci. 24(6), 997–1032 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernat, J., Llibre, J., Pérez-Chavela, E.: On the planar central configurations of the 4-body problem with three equal masses. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 16(1), 1–13 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Christopher, C., Li, C.: Limit Cycles of Differential Equations. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2007)Google Scholar
  9. 9.
    Coll, B., Gasull, A., Prohens, R.: Periodic orbits for perturbed non-autonomous differential equations. Bull. Sci. Math. 136(7), 803–819 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cukierman, F.: Notas sobre integrales abelianas. Spanish, Notes of the course Geometría Algebraica, Escuela de Matemática de América Latina y el Caribe EMALCA, Perú. (2008). Accessed 22 Nov 2018
  11. 11.
    Ferragut, A., García-Saldaña, J.D., Gasull, A.: Detection of special curves via the double resultant. Qual. Theory Dyn. Syst. 16(1), 101–117 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    García-Saldaña, J.D., Gasull, A., Giacomini, H.: Bifurcation values for a family of planar vector fields of degree five. Discrete Contin. Dyn. Syst. 35(2), 669–701 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gardner, M.: Mathematical circus. MAA Spectrum. Mathematical Association of America, Washington, DC (1992). More puzzles, games, paradoxes, and other mathematical entertainments from Scientific American, Revised reprint of the 1981 edition, With a preface by Donald KnuthGoogle Scholar
  14. 14.
    Gasull, A., Lázaro, J.T., Torregrosa, J.: On the Chebyshev property for a new family of functions. J. Math. Anal. Appl. 387(2), 631–644 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gasull, A., Tomás Lázaro, J., Torregrosa, J.: Upper bounds for the number of zeroes for some Abelian integrals. Nonlinear Anal. 75(13), 5169–5179 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory and Applications. Birkhäuser Boston Inc, Boston (1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    Jeffrey, D.J., Rich, A.D.: Computer Algebra Systems: A Practical Guide, Chapter Simplifying Square Roots of Square Roots by Denesting, p. p. xvi+436. Wiley, Chichester (1999)Google Scholar
  18. 18.
    Leandro, E.S.G.: Finiteness and bifurcations of some symmetrical classes of central configurations. Arch. Ration. Mech. Anal. 167(2), 147–177 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Moeckel, R.: On central configurations. Math. Z. 205(4), 499–517 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schicho, J.: Rational parametrization of surfaces. J. Symb. Comput. 26(1), 1–29 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sendra, J.R., Winkler, F., Pérez-Díaz, S.: Rational Algebraic Curves, A Computer Algebra Approach. Algorithms and Computation in Mathematics, vol. vol, p. 22. Springer, Berlin (2008)Google Scholar
  22. 22.
    Shi, J., Xie, Z.: Classification of four-body central configurations with three equal masses. J. Math. Anal. Appl. 363(2), 512–524 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sturmfels, B.: Solving systems of polynomial equations, volume 97 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2002)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Departament de MatemàtiquesUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations