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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

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

Notes

Acknowledgements

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).

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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

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