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Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing One Finite Saddle-Node and a Separatrix Connection

    1. [1] Universitat Autònoma de Barcelona
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
  • Idioma: inglés
  • Enlaces
  • Resumen

    • This paper is part of a series of works whose ultimate goal is the complete classification of phase portraits of quadratic differential systems in the plane modulo limit cycles.

      It is estimated that the total number may be around 2000, so the work to find them all must be split in different papers in a systematic way so to assure the completeness of the study and also the non intersection among them. In this paper we classify the family of phase portraits possessing one finite saddle-node and a separatrix connection and determine that there are a minimum of 77 topologically different phase portraits plus at most 16 other phase portraits which we conjecture to be impossible. Along this paper we also deploy a mistake in the book (Artés et al. in Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018) linked to a mistake in Reyn and Huang (Separatrix configuration of quadratic systems with finite multiplicity three and a M0 1,1 type of critical point at infinity. Report Technische Universiteit Delft, pp 95–115, 1995).

  • Referencias bibliográficas
    • 1. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A. G.:Qualitative theory of second-order dynamic systems, translated from Russian...
    • 2. Artés, J. C., Kooij, R., Llibre, J.: Structurally stable quadratic vector fields, Memoires of the American Mathematical Society, vol 134,...
    • 3. Artés, J.C., Llibre, J.: Quadratic vector fields with a weak focus of third order. Publ. Mat. 41, 7–39 (1997)
    • 4. Artés, J.C., Llibre, J.: Statistical measure of quadratic vector fields. Resenhas 6, 85–97 (2003)
    • 5. Artés, J.C., Llibre, J., Medrado, J.C.: Nonexistence of limit cycles for a class of structurally stable quadratic vector fields. Discrete...
    • 6. Artés, J.C., Llibre, J., Rezende, A.C.: Structurally Unstable Quadratic Vector Fields of Codimension One. Birkhäuser/Springer, Cham (2018)
    • 7. Artés, J.C., Llibre, J., Schlomiuk, D.: The geometry of quadratic differential systems with a weak focus of second order. Int. J. Bifurc....
    • 8. Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.I.: Global Topological Configurations of Singularities for the Whole Family of Quadratic...
    • 9. Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Geometric configurations of singularities of planar polynomial differential systems—a...
    • 10. Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Codimension in planar polynomial differential systems. Preprint (2023)
    • 11. Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Multiple connections of separatrices on quadratic differential systems. Preprint (2023)
    • 12. Artés, J.C., Mota, M.C., Rezende, A.C.: Quadratic differential systems with a finite saddle-node and an infinite saddle-node (1, 1)SN...
    • 13. Artés, J.C., Mota, M.C., Rezende, A.C.: Quadratic differential systems with a finite saddle-node and an infinite saddle-node (1, 1)SN...
    • 14. Artés, J.C., Mota, M.C., Rezende, A.C.: Structurally unstable quadratic vector fields of codimension two: families possessing a finite...
    • 15. Artés, J.C., Oliveira, R.D.S., Rezende, A.C.: Structurally unstable quadratic vector fields of codimension two: families possessing either...
    • 16. Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: Global phase portraits of quadratic polynomial differential systems with a semi-elemental...
    • 17. Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: The geometry of quadratic polynomial differential systems with a finite and an infinite...
    • 18. Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: The geometry of quadratic polynomial differential systems with a finite and an infinite...
    • 19. Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: Topological classification of quadratic polynomial differential systems with a finite semi...
    • 20. Artés, J.C., Trullas, C.: Quadratic differential systems with a weak focus of first order and a finite saddle-node. Preprint (2023)
    • 21. Bamon, R.: Quadratic vector fields in the plane have a finite number of limit cycles. Inst. Hautes Études Sci. Publ. Math. 64, 111–142...
    • 22. Li, Chengzhi, Liu, Changjian, Yang, Jiazhong: A cubic system with thirteen limit cycles. J. Differ. Equ. 246, 3609–3619 (2009)
    • 23. Coll, B., Llibre, J.: Limit cycles for a quadratic system with an invariant straight line and some evolution of phase portraits. Colloq....
    • 24. Coppel, W.A.: A survey of quadratic systems. J. Differ. Equ. 2, 293–304 (1966)
    • 25. Coppel, W.A.: Some quadratic systems with at most one limit cycle. In: Kirchgraber, Y., Walther, H.O. (eds.) Dynamics Reported: A Series...
    • 26. Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Universitext, p. 298. Springer, New York (2008)
    • 27. Dumortier, F., Roussarie, R., Rousseau, C.: Hilbert’s 16th problem for quadratic vector fields. J. Differ. Equ. 110(1), 86–133 (1994)
    • 28. Ecalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture du Dulac, Actualités Mathématiques, p. 337....
    • 29. González Velasco, E.A.: Generic properties of polynomial vector fields at infinity. Trans. Am. Math. Soc. 143, 201–222 (1969)
    • 30. Hilbert, D.: Mathematische probleme. In: Nachr. Ges. Wiss., editor Second Internat. Congress Math. Paris, 1900, 253–297. Göttingen Math....
    • 31. Hilbert, D.: Mathematical problems. Bull. Am. Mat. Soc. 8, 437–479 (1902) (reprinted in Bull. Am. Math. 37, 407–436 (2000))
    • 32. Hirsch, M.: Differential Topology. Springer, New York (1976)
    • 33. Ilyashenko, Y.: Finiteness Theorems for Limit Cycles, Translation of Mathematical Monographs, vol. 94, p. 288. American Mathematical Society,...
    • 34. Jiang, Q., Llibre, J.: Qualitative classification of singular points. Qual. Theory Dyn. Syst. 6(1), 87–167 (2005)
    • 35. Neumann, D.: Classification of continuous flows on 2-manifolds. Proc. Am. Math. Soc. 48, 73–81 (1975)
    • 36. Peixoto, M.M.: Structural stability on two-dimensional manifolds. Topology 1, 101–120 (1962)
    • 37. Poincaré, H.: Mémoire sur les courbes définies par une équation differentielle. Journal de Mathématiques, 3e série, t. 7, 375–422 (1881)...
    • 38. Reyn, J.W.: A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, 3rd edn. Faculty of...
    • 39. Reyn, J.W.: Phase portraits of quadratic systems without finite critical points. Nonlinear Anal. 27, 207–222 (1996)
    • 40. Reyn, J.W.: Phase portraits of quadratic systems with finite multiplicity one. Nonlinear Anal. 28, 755–778 (1997)
    • 41. Reyn, J.W.: Phase Portraits of Planar Quadratic Systems. Mathematics and its Applications, vol. 583. Springer, New York (2007)
    • 42. Reyn, J.W., Huang, X.: Separatrix configuration of quadratic systems with finite multiplicity three and a M0 1,1 type of critical point...
    • 43. Reyn, J.W., Huang, X.: Phase portraits of quadratic systems with finite multiplicity three and a degenerate critical point at infinity....
    • 44. Reyn, J.W., Kooij, R.E.: Phase portraits of non-degenerate quadratic systems with finite multiplicity two. Differ. Equ. Dyn. Syst. 5,...
    • 45. Shi, S.: A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sin. 23(2), 153–158 (1980)
    • 46. Sotomayor, J.: Curvas definidas por equações diferenciais no plano. Instituto de Matemática Pura e Aplicada, Rio de Janeiro (1979)
    • 47. Vulpe, N.I.: Affine-invariant conditions for the topological distinction of quadratic systems with a center. Differentsial’nye Uravneniya...
    • 48. Zegeling, A.: Nests of limit cycles in quadratic systems. Preprint (2023)
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