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Unfolding of the hamiltonian triangle vector field

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Abstract

In this paper, we study unfoldings of the Hamiltonian triangle vector field within quadratic vector fields. We complete and correct some previous results of Żołądek [17].

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References

  1. V. I. Arnol’d, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II. Monodromy and asymptotics of integrals. Monogr. Math. 83, Birkhäuser Boston, Inc., Boston, MA (1988).

  2. N. Bautin, On the number of limit cycles which appear by variations of coefficients from an equilibrium position of focus or center type. Trans. Amer. Math. Soc. 100, (1954), 181–196.

    Google Scholar 

  3. H. Dulac, Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull. Sci. Math. 32 (1908), No. 2, 230–252.

    Google Scholar 

  4. L. Gavrilov, Higher-order Poincaré–Pontryagin functions and iterated path integrals. Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), No. 4, 663–682.

    MathSciNet  MATH  Google Scholar 

  5. L. Gavrilov and I. D. Iliev, The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields. Amer. J. Math. 127 (2005), No. 6, 1153–1190.

    Article  MathSciNet  MATH  Google Scholar 

  6. I. D. Iliev, Higher-order Melnikov functions for degenerate cubic Hamiltonians. Adv. Differ. Equations 1 (1996), No. 4, 689–708.

    MathSciNet  MATH  Google Scholar 

  7. _______, The cyclicity of the period annulus of the quadratic Hamiltonian triangle. J. Differ. Equations 128 (1996), 309–326.

  8. _______, Perturbations of quadratic centers. Bull. Sci. Math. 122 (1998), No. 2, 107–161.

  9. P. Joyal, Un théorème de préparation pour fonctions á développement Tchébychévien. Ergodic Theory Dynam. Systems 14 (1994), No. 2, 305–329.

    Article  MathSciNet  MATH  Google Scholar 

  10. W. Kapteyn, Over de middelpunten de integral krommer van differentiaalvergelijkingen van de eerste orde en den eersten graad, Koninkl. Nederland. Akad. 19 (1911), 1446–1456.

    Google Scholar 

  11. _______, Niew onderzoec omtrent de middelpunten der integralen van differentiaalvergelijkingen van de eerste orde en den eersten graad. Koninkl. Nederland. Akad. 20 (1912), 1354–1365; 21 (1912), 27–33.

  12. C. Li, P.Mardešić, and R. Roussarie, Perturbations of symmetric elliptic Hamiltonians of degree four. J. Differ. Equations 231 (2006), No. 1, 78–91.

    Article  MATH  Google Scholar 

  13. P. Mardešić, Chebyshev systems and the versal unfolding of the cusps of order n. Hermann, Paris (1998).

    MATH  Google Scholar 

  14. M. Uribe, Principal Poincaré–Pontryagin function of polynomial perturbations of the Hamiltonian triangle. J. Dynam. Control Systems 12 (2006), No. 1, 109–134.

    Article  MathSciNet  MATH  Google Scholar 

  15. _______, Principal Poincaré–Pontryagin function associated to polynomial perturbations of a product of (d+1) straight lines. J. Differ. Equations 246 (2009), No. 4, 1313–1341.

  16. H. Żołądek, Bifurcations of certain family of planar vector fields tangent to axes. J. Differ. Equations 67 (1987), No. 1, 1–55.

    Article  MATH  Google Scholar 

  17. _______, Quadratic systems with center and their perturbations. J. Differ. Equations 109 (1994), 223–273.

  18. _______, The cyclicity of triangles and segments in quadratic systems. J. Differ. Equations 122 (1995), No. 1, 137–159.

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Correspondence to P. Mardešić.

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This work was partially supported by the Fondecyt Project 1080288. The first author thanks M. Saavedra and the Universidad de Concepcion for hospitality and support during the preparation of this work.

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Mardešić, P., Saavedra, M., Uribe, M. et al. Unfolding of the hamiltonian triangle vector field. J Dyn Control Syst 17, 291–310 (2011). https://doi.org/10.1007/s10883-011-9120-5

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  • DOI: https://doi.org/10.1007/s10883-011-9120-5

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