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Phase portraits of the quadratic vector fields with a polynomial first integral

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Abstract

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral.

IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.

Since we observe that all the phase portraits of the linear polynomial differential systems having a polynomial first integral are also realizable by Hamiltonian systems of degree 1, an open question appears: Are all the phase portraits of polynomial differential systems of degreen having a polynomial first integral realizable by Hamiltonian systems of degreen?

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References

  1. A. F. Andreev,Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Translation of Amer. Math. Soc.,8 (1958), 183–207.

    MATH  Google Scholar 

  2. A. Andronov, E. A. Leontovich, I. I. Gordon and A. L. Maier,Qualitative Theory of Second-Order Dynamic Systems, Wiley, New York, 1973.

    MATH  Google Scholar 

  3. J. C. Artés, J. Llibre,Quadratic hamiltonian vector field, Journal of Differential Equations,107 (1994), 80–95.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. C. Artés, J. Llibre,A Correction to the paper “Quadratic hamiltonian vector field”, Journal of Differential Equations,129 (1996), 559–560.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Cairó, H. Giacomini, J. Llibre,Liouvillian first integrals for the planar Lotka-Volterra systems, Ren. Circ. Mat. Palermo,52 (2003), 389–418.

    MATH  Google Scholar 

  6. L. Cairó, J. Llibre,Integrability of the 2-dimentional Lotka-Volterra system via polynomial (invers) integrating factors, J. Phys. A,33 (2000), 2407–2417.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Chavarriga, B. García, J. Llibre, J. S. Pérez del Río, J. A. Rodríguez,Polynomial first integrals of quadratic vector fields, J. Differential Equations,230 (2006), 393–421.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Christopher,Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburgh,124 A (1994), 1209–1229.

    MathSciNet  Google Scholar 

  9. C. Christopher, J. Llibre,Algebraic aspects of integrability for polynomial systems, Qualitative Theory of Dynamical Systems,1 (1999), 71–95.

    MathSciNet  Google Scholar 

  10. C. Christopher, J. Llibre,Integrability via invariant algebraic curves for planar polynomial differential systems, Annals of Differential Equations,16 (2000), 5–19.

    MathSciNet  MATH  Google Scholar 

  11. A. Cima, J. Llibre,Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. of Math. Anal. and Appl.,147 (1990), 420–448.

    Article  MathSciNet  MATH  Google Scholar 

  12. G. Darboux,Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème série,2 (1878), 60–96; 123–144; 151–200.

    Google Scholar 

  13. E. A. González,Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc.,143 (1969), 201–222.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. P. Jouanolou,Equations de Pfaff algébriques, in “Lectures Notes in Mathematics”,708, Springer-Verlag, New York/Berlin, 1979.

    MATH  Google Scholar 

  15. S. Labrunie,On the polynomial first integrals of the (abc) Lotka-Volterra system, J. Math. Phys.,37 (1996), 5539–5550.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Llibre,Integrability of polynomial differential systems, Handbook of differential equations, Ordinary Differential Equations, volume 1, Elsevier(2004).

  17. J. Moulin-Ollagnier,Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math.,121 (1997), 463–476.

    MathSciNet  MATH  Google Scholar 

  18. H. Poincaré,Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II, Rend. Circ. Mat. Palermo,5 (1891), 161–191;11 (1897), 193–239.

    Article  Google Scholar 

  19. M. J. Prelle, M. F. Singer,Elementary first integrals of differential equations, Trans. Amer. Math. Soc.,279 (1983), 613–636.

    Article  MathSciNet  Google Scholar 

  20. J. W. Reyn,A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, Delf University of Technology, 1997.

  21. M.F. Singer,Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc.,333 (1992), 673–688.

    Article  MathSciNet  MATH  Google Scholar 

  22. Ye Yanqian, Others,Theory of Limit Cycles, Transl. Math. Monographs,66, Amer. Math. Soc., Providence, R. P., 198

  23. Ye Yanqian, Others,Qualitative Theory of Polynomial Differential Systems, Shangai Scientific Technical Publishers, Shangai 1995, (in Chinese).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Oviedo, Avda Calvo Sotelo s/n, 33007, Oviedo, Spain

    Belén García & Jesús S. Pérez Del Río

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain

    Jaume Llibre

Authors
  1. Belén García
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  2. Jesús S. Pérez Del Río
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  3. Jaume Llibre
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Corresponding author

Correspondence to Belén García.

Additional information

The first and third authors are supported by a MEC grant number MTM 2005-02094. The second author is partially supported by a DGICYT grant number MTM2005-06098-C02-01 and by a CICYT grant number 2005SGR 00550.

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García, B., Pérez Del Río, J.S. & Llibre, J. Phase portraits of the quadratic vector fields with a polynomial first integral. Rend. Circ. Mat. Palermo 55, 420–440 (2006). https://doi.org/10.1007/BF02874780

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