Rocky Mountain Journal of Mathematics

Darboux Integrability and Reversible Quadratic Vector Fields

Jaume Llibre and João Carlos Medrado

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Rocky Mountain J. Math., Volume 35, Number 6 (2005), 1999-2057.

First available in Project Euclid: 5 June 2007

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Primary: 34C05: Location of integral curves, singular points, limit cycles 58F14

Darboux integrability reversible vector fields


Llibre, Jaume; Medrado, João Carlos. Darboux Integrability and Reversible Quadratic Vector Fields. Rocky Mountain J. Math. 35 (2005), no. 6, 1999--2057. doi:10.1216/rmjm/1181069627.

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  • A.F. Andreev, Investigation of the behavior of the integral curves of a system of two differential equations in the neighborhood of a singular point, Trans. Amer. Math. Soc. 8 (1958), 183-207.
  • A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.L. Maier, Qualitative theory of second-order dynamic systems, Wiley, New York, 1973.
  • V.I. Arnold and Y. S. Ilyashenko, Dynamical systems - I, Ordinary differential equations, Encyclopaedia Math. Sci., Vols. 1-2, Springer, Heidelberg, 1988.
  • L. Cairó, M. R. Feix and J. Llibre, Integrability and algebraic solutions for planar polynomial differential systems with emphasis on the quadratic systems, Resenhas da Universidade de São Paulo 4 (1999), 127-161.
  • L. Cairó and J. Llibre, Darboux first integrals and invariants for real quadratic systems having an invariant conic, J. Phys. Math. Gen. 35 (2002), 589-608.
  • J. Chavarriga, J. Llibre and J. Sotomayor, Algebraic solutions for polynomial vector fields with emphasis in the quadratic case, Exposition. Math. 15 (1997), 161-173.
  • C. Christopher, Invariant algebraic curves and conditions for a center, Proc. Roy. Soc. Edinburgh 124 Sect. A (1994), 1209-1229.
  • C. Christopher and J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations 16 (2000), 5-19.
  • --------, Algebraic aspects of integrability for polynomial systems, Qual. Theory Dynam. Syst. 1 (1999), 71-95.
  • C. Christopher, J. Llibre and J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, to appear.
  • G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. 2ème série 2 (1878), 60-96; 123-144; 151-200.
  • E.A.V. Gonzales, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201-222.
  • C. Gutierrez and J. Llibre, Darboux integrability for polynomial vector fields on the $2$-dimensional sphere, Extracta Math. 17 (2002), 289-301.
  • K. Jänich, Topology, Undergrad. Texts Math., Springer-Verlag, New York, 1984.
  • Qibao Jiang and J. Llibre, Qualitative classification of singular points, preprint 319, Centre de Recerca Matemàtica, 1996.
  • J.P. Jouanolou, Equations de Pfaff algébriques, Lectures Notes in Math. 708, Springer-Verlag, New York, 1979.
  • J. Llibre and G. Rodríguez, Invariant hyperplanes and Darbox integrability for $d$-dimensional polynomial differential systems, Bull. Sci. Math. 124 (2000), 1-21.
  • --------, Darboux integrability of polynomial vector fields on $2$-dimensional surfaces, Inter. J. Bifurcations Chaos 12 (2002), 2821-2833.
  • J. Llibre and X. Zhang, Polynomial first integrals of quadratic systems, Rocky Mountain J. Math. 31 (2002), 1317-1371.
  • L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc. 76 (1954), 127-148.
  • J.C.R. Medrado and M.A. Teixeira, Symmetric singularities of reversible vector fields in dimension three, Physica 112 (1998), 122-131.
  • --------, Codimension-two singularities of reversible vector fields in 3 D, Qual. Theory Dynam. Sys. 2 (2001), 399-428.
  • D. Neumann, Classification of continuous flows on $2$-manifolds, Proc. Amer. Math. Soc. 48 (1975), 73-81.
  • H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del Circolo Matematico di Palermo 5 (1891), 161-191; 11 (1897), 193-239.
  • J.W. Reyn, A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, 3rd ed., Delft Univ. of Technology, Faculty of Tech. Math. and Informatics, Report, 1994; see also
  • R. Roussarie, Bifurcation of planar vector fields and Hilbert's sixteenth problem, Progr. Math., vol. 164, Birkhäuser Verlag, Basel, 1998.
  • D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799-841.
  • M.A. Teixeira, Singularities of reversible vector fields, Physica D 100 (1997), 101-118.
  • J.A. Weil, Constant et polynómes de Darboux en algèbre différentielle: applications aux systèmes différentiels linéaires, Ph.D. Thesis, École Polytechnique, 1995, 673-688.
  • Ye Yanqian, Qualitative theory of polynomial differential systems, Shanghai Scientific & Technical Publ., Shanghai, 1995, in Chinese.
  • Ye Yanqian et al., Theory of limit cycles, Amer. Math. Soc., Providence, RI, 1984.
  • Zhang Zhifen, Ding Tongren, Huang Wenzao and Dong Zhenxi, Qualitative theory of differential equations, Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1992.