Advertisement

Qualitative Theory of Dynamical Systems

, Volume 5, Issue 1, pp 135–194 | Cite as

Planar quadratic vector fields with invariant lines of total multiplicity at least five

  • Dana Schlomiuk
  • Nicolae Vulpe
Article

Abstract

In this article we consider the action of the real affine group and time rescaling on real planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of real quadratic differential systems with at least five invariant lines. These conditions are such that no matter how a system may be presented, one can verify by using them whether the system has or does not have at least five invariant lines and to check to which orbit (or family of orbits) it belongs.

Key Words

quadratic differential system Poincaré compactification algebraic invariant curve algebraic affine invariant configuration of invariant lines 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Artes, J. Llibre,On the number of slopes of invariant straight lines for polynomial differential systems, J. of Nanjing University13 (1996), 143–149.MATHMathSciNetGoogle Scholar
  2. 2.
    J. Artes, B. Grünbaum, J. Llibre,On the number of invariant straight lines for polynomial differential systems, Pacific Journal of Mathematics184, (1998), 207–230.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Christopher, J. Llibre,Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential equations16 (2000), no. 1, 5–19.MATHMathSciNetGoogle Scholar
  4. 4.
    C. Christopher, J. Llibre, J. V. Pereira,Multiplicity of invariant algebraic curves, Preprint (2002).Google Scholar
  5. 5.
    C. Christopher, J. Llibre, C. Pantazi, X. Zhang,Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Phys.A 35 (2002), no. 10, 2457–2476.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    G. Darboux,Mémoire sur les équations différentielles du premier ordre et du premier degré, Bulletin de Sciences Mathématiques, 2me série,2 (1) (1878), 60–96; 123–144; 151–200.Google Scholar
  7. 7.
    T. A. Druzhkova,Quadratic differential systems with algebraic integrals, Qualitative theory of differential equations, Gorky Universitet2 (1975), 34–42 (Russian).Google Scholar
  8. 8.
    W. Fulton,Algebraic curves. An introduction to Algebraic Geometry, W.A. Benjamin, Inc., New York (1969).Google Scholar
  9. 9.
    J. H. Grace, A. Young,The algebra of invariants, New York: Stechert, 1941.Google Scholar
  10. 10.
    J. Llibre, D. Schlomiuk,The geometry of quadratic systems with a weak focus of third order, Canad. J. Math. Vol. 56 (2) (2004), 310–343MATHMathSciNetGoogle Scholar
  11. 11.
    J. Llibre, N. Vulpe,Planar cubic polynomial differential systems with the maximum number of invariant straight lines, Report, núm. 34 (2002), Universitat Autònoma de Barcelona, 54 pp.Google Scholar
  12. 12.
    R. A. Lyubimova,On some differential equation which possesses invariant lines, Differential and integral eequations, Gorky Universitet,1 (1977) (Russian).Google Scholar
  13. 13.
    R. A. Lyubimova,On some differential equation which possesses invariant lines, Differential and integral eequations, Gorky Universitet,21 (1984) (Russian).Google Scholar
  14. 14.
    P. J. Olver,Classical Invariant Theory, London Mathematical society student texts:44, Cambridge University Press (1999).Google Scholar
  15. 15.
    H. Poincaré,Sur l'intégration algébrique des équations différentielles, C. R. Acad. Sci. Paris,112 (1891), 761–764.Google Scholar
  16. 16.
    H. Poincaré,Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré, I. Rend. Circ. Mat. Palermo5 (1891), 169–191.Google Scholar
  17. 17.
    H. Poincaré,Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré, II. Rend. Circ. Mat. Palermo11 (1897), 169–193–239.CrossRefGoogle Scholar
  18. 18.
    M. N. Popa,Application of invariant processes to the study of homogeneous linear particular integrals of a differential system, Dokl. Akad. Nauk SSSR,317, no. 4 (1991) (Russian); translation in Soviet Math. Dokl.43 (1991), no. 2.Google Scholar
  19. 19.
    M. N. Popa,Aplications of algebras to differential systems, Academy of Science of Moldova (2001) (Russian).Google Scholar
  20. 20.
    M. N. Popa,Aplications of algebraic methods to differential systems, Romania, Piteshty Univers., The Flower Power Edit. (2004) (Romanian).Google Scholar
  21. 21.
    M. N. Popa and K. S. Sibirskii,Conditions for the existence of a homogeneous linear partial integral of a differential system, Differentsial'nye Uravneniya,23, no. 8 (1987) (Russian).Google Scholar
  22. 22.
    D. Schlomiuk,Elementary first integrals and algebraic invariant curves of differential equations, Expo. Math.11 (1993), 433–454.MATHMathSciNetGoogle Scholar
  23. 23.
    D. Schlomiuk,Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields, In Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk (ed.) (1993), 429–467.Google Scholar
  24. 24.
    D. Schlomiuk, N. Vulpe,Planar quadratic differential systems with invariant straight lines of at least five total multiplicity, CRM Report no. 2922, Université de Montréal (2003), 42 pp.Google Scholar
  25. 25.
    D. Schlomiuk, N. Vulpe,Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity, CRM Report no. 3181, Université de Montréal (2005), 39 pp.Google Scholar
  26. 26.
    D. Schlomiuk, N. Vulpe,The full study of planar quadratic differential systems possessing exactly one line of singularities, finite ore infinite, CRM Report no. 3183, Université de Montréal (2005), 40 pp.Google Scholar
  27. 27.
    K. S. Sibirskii,Introduction to the algebraic theory of invariants of differential equations, Translated from the Russian. Nonlinear Science: Theory and Applications. Manchester University Press, Manchester (1988).Google Scholar
  28. 28.
    K. S. Sibirskii,Method of invariants in the qualitative theory of differential equations, Kishinev: RIO AN Moldavian SSR (1968).Google Scholar
  29. 29.
    K. S. Sibirskii,Conditions for the presence of a straight integral line of a quadratic differential system in the case of a center or a focus, Mat. Issled. No. 106, Differ. Uravneniya i Mat. Fizika (1989) (Russian).Google Scholar
  30. 30.
    J. Sokulski,On the number of invariant lines for polynomial vector fields, Nonlinearity,9 (1996).Google Scholar
  31. 31.
    N. I. Vulpe,Polynomial bases of comitants of differential systems and their applications in qualitative theory, (Russian) “Shtiintsa”, Kishiney (1986) (Russian).Google Scholar
  32. 32.
    R. J. Walker,Algebraic Curves, Dover Publications, Inc., New York (1962).MATHGoogle Scholar
  33. 33.
    Zhang Xiang,Number of integral lines of polynomial systems of degree three and four, J. of Nanjing University, Math. Biquarterly10 (1993), 209–212.Google Scholar

Copyright information

© Birkhäuser-Verlag 2004

Authors and Affiliations

  1. 1.Département de Mathématiques et de Statistiques Université de MontréalMontrealCanada
  2. 2.Institute of Mathematics and Computer Science Academy of Science of MoldovaMoldovaRussian

Personalised recommendations