Rocky Mountain Journal of Mathematics

Integrability of reversible and equivariant quadratic polynomial differential systems in the plane

Jaume Llibre and Claudia Valls

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the existence of first integrals for the class of reversible and equivariant quadratic polynomial differential systems in the plane. We put special emphasis in the study of the analytic first integrals.

Note

The first author was partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación, grant No. MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca, grant No. 2017SGR1617 and the H2020 European Research Council, grant No. MSCA-RISE-2017-777911. The second author was partially supported by FCT/Portugal through UID/MAT/04459/2013.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 2 (2019), 579-591.

Dates
Received: 10 July 2014
Revised: 26 September 2018
First available in Project Euclid: 23 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1561318394

Digital Object Identifier
doi:10.1216/RMJ-2019-49-2-579

Mathematical Reviews number (MathSciNet)
MR3973241

Zentralblatt MATH identifier
07079985

Subjects
Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.) 37J35: Completely integrable systems, topological structure of phase space, integration methods

Keywords
reversible equivariant quadratic planar polynomial equations analytic first integral

Citation

Llibre, Jaume; Valls, Claudia. Integrability of reversible and equivariant quadratic polynomial differential systems in the plane. Rocky Mountain J. Math. 49 (2019), no. 2, 579--591. doi:10.1216/RMJ-2019-49-2-579. https://projecteuclid.org/euclid.rmjm/1561318394


Export citation

References

  • M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Appl. Math. 55, Washington, DC, 1964.
  • L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous systems, I, Nonlin. Anal. 29 (1997), 783–811.
  • J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Eqs. 157 (1999), 163–182.
  • S.D. Furta, On non-integrability of general systems of differential equations, Z. Angew. Math. Phys. 47 (1996), 112–131.
  • W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms, Z. Angew. Math. Phys. 54 (2003), 1–21.
  • J. Llibre and J.C. Medrado, Darboux integrability and reversible quadratic vector fields, Rocky Mountain J. Math. 35 (2005), 1999–2057.
  • H. Poincaré, Mémoire sur les courbes définies par les équations différentielles (Oeuvreus de Henri Poincaré, I), Gauthiers–Villars, Paris, 1951.
  • J.W. Reyn, A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, Delf University of Technology, http://ta.twi.tudelft.nl/DV/Staff/J.W.Reyn.html, 1997.
  • Y. Ye, Qualitative theory of polynomial differential systems, Shanghai Scientific & Technical Publishers, Shanghai, 1995.
  • Y. Ye, et al., Theory of limit cycles, Transl. Math. Monogr. 66 (1994).