Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems

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.

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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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Correspondence to Joan Torregrosa.

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Gasull, A., Lázaro, J.T. & Torregrosa, J. Rational Parameterizations Approach for Solving Equations in Some Dynamical Systems Problems. Qual. Theory Dyn. Syst. 18, 583–602 (2019). https://doi.org/10.1007/s12346-018-0300-5

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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