Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T04:36:56.724Z Has data issue: false hasContentIssue false
  • English
  • Français

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

Published online by Cambridge University Press:  20 November 2018

Jaume Llibre  and
Dana Schlomiuk
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona Spain e-mail: jllibre@mat.uab.es
Dana Schlomiuk
Affiliation:
Département de Mathématiques, et Statistique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, H3C 3J7 email: dasch@dms.umontreal.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

Keywords

34C4051F1414D0514D25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 310 - 343
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Andronov, A. A., Leontoovich, E. A., Gordon, I. I. and Maier, A. L., Qualitative theory of second-order dynamic systems. John Wiley and Sons, New York, 1973.Google Scholar
[2] Andronova, E. A., Decomposition of the parameter space of a quadratic equation with a singular point of center type and topological structure with limit cycles. Ph. D. Thesis, The Gorky Institute of Water Transport Engineers, Gorky, Russia, 1988 (Russian).Google Scholar
[3] Artés, J. C., Sistemes quadràtics amb un focus feble de tercer ordre. Master's Thesis, Universitat Autónoma de Barcelona, 1984 (Catalan).Google Scholar
[4] Art és, J. C. and Llibre, J., Quadratic Hamiltonian vector fields. J. Differential Equations 107(1994), 80–95; 129(1996), 559–560.Google Scholar
[5] Art és, J. C. and Llibre, J., Quadratic vector fields with a weak focus of third order. Publ. Mat. 41(1997), 7–39.Google Scholar
[6] Bautin, N. N., On the number of limit cycles which appear with the variation of coefficients from an equilibrium point of focus or center type. Transl. Amer.Math. Soc. 1(1962), 396–413.Google Scholar
[7] Briot, Ch. A. A. et Bouquet, J. C., Recherches sur les fonctions définies par les équations différentielles. J. École Polytechnique XXI(1856), 134–198.Google Scholar
[8] Cairó, L. and Llibre, J., Phase portraits of planar semi-homogeneous systems I. Nonlinear Anal. 29(1997), 783–811.Google Scholar
[9] Chen, L. S. and Wang, M. S., The relative position and number of limit cycles of the quadratic differential systems. Acta Math. Sinica 22(1979), 751–758.Google Scholar
[10] Coll, B., Gasull, A. and Llibre, J., Some theorems on the existence, uniqueness and non existence of limit cycles for quadratic systems. J. Differential Equations 67(1987), 372–399.Google Scholar
[11] Coppel, W. A., A survey of quadratic systems. J. Differential Equations 2(1966), 293–304.Google Scholar
[12] Darboux, G., Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. (Mélanges) Bull. Sci. Math. (2) 2(1878), 60–96; 123–144; 151–200.Google Scholar
[13] Dulac, H., Sur les cycles limites. Bull. Soc. Math. France 51(1923), 45–188.Google Scholar
[14] Dumortier, F. and Fiddelaers, P., Qualitative models for generic local 3-parameter bifurcations on the plane. Trans. Amer.Math. Soc. 326(1991), 101–126.Google Scholar
[15] Dumortier, F., Roussarie, R. and Rousseau, C., Hilbert's 16-th problem for quadratic vector fields. J. Differential Equations 110(1994), 66–133.Google Scholar
[16] Dumortier, F., Roussarie, R. and Rousseau, C., Elementary graphics of cyclicity 1 and 2 . Nonlinearity 7(1994), 1001–1043.Google Scholar
[17] Ecalle, J., Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Hermann, 1992.Google Scholar
[18] Fulton, W., Algebraic curves. An introduction to Algebraic Geometry. W. A. Benjamin, Inc., New York, 1969.Google Scholar
[19] Velasco, E. A. González, Generic properties of polynomial vector fields at infinity. Trans. Amer. Math. Soc. 143(1969), 201–222.Google Scholar
[20] Hartshorne, R., Albegraic geometry. Graduate Texts in Math. 52, Springer, 1977.Google Scholar
[21] Hilbert, D., Mathematische Probleme. Lecture at the Second International Congress of Mathematicians, Paris 1900; reprinted in Mathematical Developments Arising from Hilbert Problems, (ed. Browder, F. E.), Proc. Symp. Pure Math. 28, Amer.Math. Soc., Providence, RI, 1976, 1–34.Google Scholar
[22] Il'yashenko, Yu., Finiteness Theorems for Limit Cycles. Transl. Math. Monogr. 94, Amer. Math. Soc., 1991.Google Scholar
[23] Li, Chengzhi, Two problems of planar quadratic systems. Sci. Sinica 26(1983), 471–481.Google Scholar
[24] Li, Chengzhi, Non-existence of limit cycles around a weak focus of order three for any quadratic system. Chinese Ann. Math. Ser. B 7(1986), 174–190.Google Scholar
[25] Li, Chengzhi, Llibre, J. and Zhang, Zhifen, Weak focus, limit cycles and bifurcations for bounded quadratic systems. J. Differential Equations 115(1995), 193–223.Google Scholar
[26] Li, Weigu, Llibre, J., Nicolau, M. and Zhang, Xiang, On the differentiability of first integrals of two dimensional flows. Proc. Amer.Math. Soc. 130(2002), 2079–2088.Google Scholar
[27] Lunkevich, V. A. and Sibirskii, K. S., Integrals of general quadratic differential systems in cases of a center. Differential Equations 8(1982), 563–568.Google Scholar
[28] Markus, L., Global structure of ordinary differential equations in the plane. Trans. Amer.Math. Soc. 76(1954), 127–148.Google Scholar
[29] Mattei, J. F. and Moussu, R., Holonomie et intğrales premières. Ann. Sci. École Norm. Sup. (4) 13(1980), 469–523.Google Scholar
[30] Milnor, J., Topology from the differential viewpoint. The University Press of Virginia, Charlottesville, 1972.Google Scholar
[31] Newman, D. A., Classification of continuous flows on 2-manifolds. Proc. Amer. Math. Soc. 48(1975), 73–81.Google Scholar
[32] Nikolaev, I. and Vulpe, N., Topological classification of quadratic systems at infinity. J. LondonMath. Soc. 55(1997), 473–488.Google Scholar
[33] Pal, J. and Schlomiuk, D., Summing up the dynamics of quadratic Hamiltonian systems with a center. Canad. J. Math. 49(1997), 583–599.Google Scholar
[34] Pal, J. and Schlomiuk, D., Intersection multiplicity and limit cycles in quadratic differential systems with a weak focus. Preprint, September 1999, 40 pages.Google Scholar
[35] Poincaré, H., Mémoire sur les courbes définies par les équations différentielles. J.Math. Pures Appl. (4) 1(1885), 167–244; Oeuvres de Henri Poincaré 1, Gauthier-Villars, Paris, 1951, 95–114.Google Scholar
[36] Roussarie, R., A note on finite cyclicity and Hilbert's 16-th problem. Springer Lecture Notes in Math. 1331(1988), 161–168.Google Scholar
[37] Roussarie, R., Smoothness property for bifurcation diagrams. Publ. Mat. 41(1997), 243–268.Google Scholar
[38] Roussarie, R. and Schlomiuk, D., On the geometric structure of the class of planar quadratic differential systems. Qualitative Theory Systems 3(2002), 93–122.Google Scholar
[39] Schlomiuk, D., Algebraic particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc. 338(1993), 799–841.Google Scholar
[40] Schlomiuk, D., Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields. In: Bifurcations and Periodic Orbits of Vector Fields, (ed., Schlomiuk, D.), 1993, 429–467.Google Scholar
[41] Schlomiuk, D., Basic algebro-geometric concepts in the study of planar polynomial vector fields. Publ. Mat. 41(1997), 269–295.Google Scholar
[42] Schlomiuk, D. and Pal, J., On the geometry in the neighborhood of Infinity of Quadratic Differential Systems with a Weak Focus. Qualitative Theory Systems 2(2001), 1–43.Google Scholar
[43] Schlomiuk, D. and Vulpe, N., Geometry of quadratic differential systems in the neighborhood of the infinity. CRM-Report, CRM-2831, Universit é de Montréal, December 2001, 41pp.Google Scholar
[44] Songling, Shi, A concrete example of the existence of four limit cycles for planar quadratic systems. Sci. Sinica 23(1980), 153–158.Google Scholar
[45] Songling, Shi, A method of constructing cycles without contact around a weak focus. J. Differential Equations 41(1981), 301–312.Google Scholar
[46] Songling, Shi, On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields. J. Differential Equations 52(1984), 52–57.Google Scholar
[47] Sibirsky, K. S., Introduction to the algebraic theory of invariants of differential equations. Nonlinear Science: Theory and Applications. Manchester University Press, Manchester, 1988.Google Scholar
[48] Sotomayor, J., Liçóes de equaçóes diferenciais ordinárias. Projecto Euclides, IMPA, Rio de Janeiro, 1979.Google Scholar
[49] Sotomayor, J. and Paterlini, R., Quadratic vector fields with finitely many periodic orbits. Springer Lecture Notes in Math. 1007(1983), 753–766.Google Scholar
[50] Vulpe, N. I., Affine-invariant conditions for the topological discrimination of quadratic systems with center. Differential Equations 19(1983), 273–280.Google Scholar
[51] Yanqian, Ye and Others, Theory of Limit Cycles. Transl. Math. Monogr. 66, Amer. Math. Soc., 1986.Google Scholar
[52] Pingguang, Zhang, On the distribution and number of limit cycles for quadratic systems with two foci. (chinese), Acta Math. Sinica 44(2001), 37–44.Google Scholar
[53] Pingguang, Zhang, On the distribution and number of limit cycles for quadratic systems with two foci. Qualitative Theory Systems 3(2002), 1–28.Google Scholar
[54] Żołądek, H., Quadratic Systems with Center and Their Perturbations. J. Differential Equations 109(1994), 223–273.Google Scholar