Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Algebraic particular integrals, integrability and the problem of the center

Author: Dana Schlomiuk
Journal: Trans. Amer. Math. Soc. 338 (1993), 799-841
MSC: Primary 34C05; Secondary 58F14
MathSciNet review: 1106193
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III). We show that a quadratic system with a weak focus $ F$, possessing algebraic particular integrals not passing through $ F$ of the following types, satisfies in some coordinate axes the condition (I), (II) or (III) respectively and hence has a center at $ F$: either a parabola and an irreducible cubic particular integral having only one point at infinity, coinciding with the one of the parabola; or a straight line and an irreducible conic curve; or distinct straight lines (possibly with complex coefficients). We show that each one of these geometric properties is generic for systems satisfying the corresponding algebraic condition for the center. Another version of this result in terms of real algebraic curves is given. These results make clear the many facets of the problem of the center in the quadratic case, in particular the question of integrability and form a basis for analogous investigations for the general problem of the center for cubic systems.

References [Enhancements On Off] (What's this?)

  • [1] D. V. Anosov and V. I. Arnold (Editors), Dynamical systems, vol. I, Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin and New York, 1988, p. 233. MR 970793 (89g:58060)
  • [2] V. I. Arnold, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, "Mir," Moscow, 1980, p. 323. (Translation of the Russian 1978 edition published by Nauka.)
  • [3] N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, Mat. Sb. 30 (72) (1952), 181-196; English transl. in Math. USSR-Sb. 100 (1954), 397-413. MR 0045893 (13:652a)
  • [4] V. Berlinskii, On the behaviour of the integral curves of a differential equation, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (15) (1960), 3-18. (Russian) MR 0132249 (24:A2095)
  • [5] F. Browder (Editor), Mathematical developments arrising from Hilbert's problems, Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976, pp. 50-51. MR 0419125 (54:7158)
  • [6] Yuanshun Chin (Yuanshun Qin), On surfaces defined by ordinary differential equations, Lecture Notes in Math., vol. 1151, Springer-Verlag, Berlin and New York, 1985, pp. 115-131.
  • [7] G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. (1878), 60-96, 123-144, 151-200.
  • [8] T. D. Davies and E. M. James, Nonlinear differential equations, Addison-Wesley, Reading, Mass., 1966, p. 191. MR 0199482 (33:7626)
  • [9] H. Dulac, Détermination et integration d'une certaine classe d'èquations différentielle ayant pour point singulier un centre, Bull. Sci. Math Sér. (2) 32 (1908), no. 1, 230-252.
  • [10] M. Frommer, Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmheitsstellen, Math. Ann. 109 (1934), 395-424. MR 1512902
  • [11] A. Giovini and G. Niesi, Cocoa user's manual, Department of Mathematics, University of Genova, 1989.
  • [12] F. Göbber and K. D. Willamowski, Ljapunov approach to multiple Hopf bifurcation, J. Math. Anal. Appl. 71 (1979), 333-350. MR 548769 (82b:34052)
  • [13] E. A. González Velasco, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201-222. MR 0252788 (40:6005)
  • [14] J. Guckenheimer, R. Rand, and D. Schlomiuk, Degenerate homoclinic cycles in perturbations of quadratic hamiltonian systems, Nonlinearity 2 (1989), 405-418. MR 1005056 (90g:58104)
  • [15] D. Hilbert, Mathematische Probleme (lecture), Second Internat. Congress Math. Paris, 1900, Nachr. Ges. Wiss. Gottingen Math.-Phys. Kl. 1900, pp. 253-297; reprinted in [5, p. 134].
  • [16] Ju. S. Il'yashenko, Perturbations of the polynomial Hamiltonian equations in the real and complex domains, Sympos. Methods of the Qualitative Theory and the Theory of Bifurcations, Gorky, 1988, pp. 63-67.
  • [17] P. de. Jager, Phase portraits of quadratic systems--higher order singularities and separatrix cycles, Doctoral Thesis, May 1989, Technische Universiteit Delft.
  • [18] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland. (1911), 1446-1457. (Dutch)
  • [19] -, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912), 1354-1365; 21, 27-33. (Dutch)
  • [20] I. S. Kukles and M. Khasanova, O povedenii harakteristik odnovo diferentialnovo uravneniia v kruge Poincare, Dokl. Akad. Nauk Tadzhik SSR 7 (1964), no. 12, 3-6.
  • [21] E. Landis and I. Petrovski, Letter to the editor, Mat. Sb. 73 (1967), 160.
  • [22] Kh. R. Latipov, O Razredelenii osobih tochek uravneniya Frommer na vsei ploskosti, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (44) (1965), 96-104. MR 0173041 (30:3256)
  • [23] Kh. R. Latipov and I. I. Shirov, Investigations of differential equations, Izv. Akad. Nauk UzSSR. Sec. Fiz.-Mat. Nauk (1963), 117-131. (Russian)
  • [24] Y. C. Lu, Singularity theory and an introduction to catastrophe theory, Springer-Verlag, Berlin and New York, 1976, p. 199. MR 0461562 (57:1547)
  • [25] N. A. Lukashevich, Integral curves of a certain differential equation, Differentsial'nye Uravneniya 1 (1965), no. 1, 82-95. MR 0188540 (32:5978)
  • [26] V. A. Lunkevich and K. S. Sibirskii, Integrals of a general quadratic differential system in cases of a center, Differential Equations 18 (1982), no. 5, 786-792. MR 661356 (83i:34005)
  • [27] M. A. Lyapunov, Problème général de la stabilité de mouvement, Princeton Univ. Press, Princeton, N.J., 1947, pp. 1-474. (First printing 1892, Harkov)
  • [28] I. Petrovski and E. Landis, On the number of limit cycles of the equation $ dx/dy = P(x,y)/Q(x,y)$ where $ P$ and $ Q$ are polynomials of the second degree, Mat. Sb. 37 (1955), 209-250; Amer. Math. Soc. Transl. (2) 10 (1958), 177-221.
  • [29] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. de Math. (3) 37 (1881), 375-422; 8 (1882), 251-296; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp. 3-84.
  • [30] -, Mémoire sur les courbes définies par les équations différentielles, J. Math. Pures Appl. (4) 1 (1885), 167-244; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, 1951, Paris, pp. 95-114.
  • [31] L. S. Pontrjagin, Über Autoschwingungssysteme, die den hamiltonschen nahe liegen, Physikalische Zeitschrift der Sowjetunion, Band 6, Heft 1-2, 1934, pp. 25-28.
  • [32] D. Schlomiuk, The "center-space" of plane quadratic systems and its bifurcation diagram, Rapport de Recherche du Département de Mathématiques et de Statistique, D.M.S. No 88-18, L'Université de Montréal, 1988.
  • [33] -, Invariant reducible cubics and the conditions for the center, Rapport de Recherche du Département de Mathématiques et de Statistique, D.M.S. No 89-11, 1989.
  • [34] -, Invariant conics of quadratic systems with a weak focus, Rapport de Recherche du Département de Mathématiques et de Statistique, D.M.S. No 89-19, Univérsité de Montréal, 1989.
  • [35] -, Une caractérisation géométrique générique des champs de vecteurs quadratiques avec un centre, C. R. Acad. Sci. Paris Ser. I Math. 310 (1990), 723-726. MR 1055236 (91d:58188)
  • [36] -, Algebraic integrals of quadratic systems with a weak focus, Proc. Conf. on Bifurcations of Planar Vector Fields, Luminy, 1989, Lecture Notes in Math., vol. 1455, Springer-Verlag, Berlin and New York, 1991, pp. 373-384. MR 1094389 (92b:34040)
  • [37] D. Schlomiuk, J. Guckenheimer, and R. Rand, Integrability of plane quadratic vector fields, Exposition. Math. 8 (1990), 3-25. MR 1042200 (91b:58200)
  • [38] Shi Songling, A counterexample to Chin's proposed solution to Hilbert's $ 16$th problem, Bull. London Math. Soc. 20 (1988), 597-599. MR 980762 (90k:34031a)
  • [39] -, A method of constructing cycles without contact around a weak focus, J. Differential Equations 41 (1981), 301-312. MR 633819 (83e:58074)
  • [40] -, Hilbert's sixteenth problem (the second part): its present state, Ann. Sci. Math. Québec 14 (1990), 193-206. MR 1085355 (93a:58145)
  • [41] M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673-688. MR 1062869 (92m:12014)
  • [42] M. Stillman, M. Stillman, and D. Bayer Macauley user manual, preprint, 1989.
  • [43] Date Tsutomu, Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems, J. Differential Equations 32 (1979), 311-334. MR 535166 (80f:34038)
  • [44] N. I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, transl. from Differentsial'nye Uravneniya 19 (1983), no. 3, 371-379. MR 696089 (85b:58106)
  • [45] Yan-Qian Ye et al., The theory of limit cycles, Transl. Math. Monographs, vol. 66, Amer. Math. Soc., Providence, R.I., 1984.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34C05, 58F14

Retrieve articles in all journals with MSC: 34C05, 58F14

Additional Information

Keywords: Algebraic particular integral, constant of motion, center, bifurcation, polynomial system, quadratic system
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society