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Limit cycles and zeroes of Abelian integrals satisfying third order picard — Fuchs equations

Part of the Lecture Notes in Mathematics book series (LNM,volume 1455)

Keywords

  • Meromorphic Function
  • Elliptic Curve
  • Riccati Equation
  • Real Coefficient
  • Quadratic System

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References

  1. V.I. Arnol’d, A.N. Varchenko, S.M. Gusein-Zade, Singularities of differentiable maps, Vol.2, Basel, Birkhauser Verlag, 1988.

    CrossRef  Google Scholar 

  2. J. Carr, S.-N. Chow, J.K. Hale, Abelian integrals and bifurcation theory, Journal Diff. Eqns. 59 (1985), 413–437.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. J. Carr, S.A. van Gils, J.A. Sanders, Nonresonant bifurcations with symmetry, SIAM J. Math. Anal., 18 (1987), No 3, 579–591.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. C.H. Jr. Clemens, Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities, Trans. Am. Math. Soc. 136 (1969) 93–108.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. R. Cushman, J.A. Sanders, A codimension two bifurcation with a third order Picard-Fuchs equation, Journal Diff. Eqns, 59 (1985), 243–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. S.N. Chow, J.A. Sanders, On the number of the critical points of the period, Journal. Diff. Eqns, 64 (1986) 51–66.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. B. Drachman, S.A. van Gils, Zhang Zhi-fen, Abelian integrals for quadratic vector fields, J. reine angew. Math. 382 (1987), 165–180.

    MathSciNet  MATH  Google Scholar 

  8. H. Dulac, Sur les cycles limites, Bull. Soc. Math. France, 51 (1923), 45–188.

    MathSciNet  MATH  Google Scholar 

  9. J. Ecalle, J. Martinet, R. Moussu, J.P. Ramis, Compt. Rendu de l’Acad. sci. (Paris), 304, Série 1, (1987), 375–378, 431–434.

    MathSciNet  Google Scholar 

  10. S.A. van Gils, E. Horozov, Uniqueness of limit cycles in planar vector fields which leave the axes invariant, Contemporary Mathematics 56 (1986), 117–129.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. A. Grothendieck, On the de Rham cohomology of algebraic varieties, Publ. Math. Inst. HES, 29 (1966), 351–359.

    MathSciNet  MATH  Google Scholar 

  12. Yu. Il’yashenko, Dulac’s memoir "Sur les cycles limites" and related problems of the local theory of differential equations, Russian Math. Surveys, 40 (1985), 1–49.

    CrossRef  MATH  Google Scholar 

  13. Yu. Il’yashenko, Zeroes of special Abelian integrals in a real domain, Funct. Anal. Appl. 11 (1977), 309–311.

    CrossRef  MATH  Google Scholar 

  14. A.G. Khovansky, Cycles of dynamical systems on a plane and Rolle’s theorem, Siberia Math. Journal, 25 (1984), no3, 198–203.

    Google Scholar 

  15. A.G. Khovansky, Real analytic manifolds with finiteness properties and complex abelian integrals, Funct. Anal. Appl. 18 (1984), 119–128.

    CrossRef  Google Scholar 

  16. G.S. Petrov, Number of zeroes of complete elliptic integrals Funct. Anal. Appl. 18 (1984), 148–150.

    CrossRef  MATH  Google Scholar 

  17. G.S. Petrov, Elliptic integrals and their nonoscillations, Funct. Anal. Appl. 20 (1986), 37–40.

    CrossRef  MATH  Google Scholar 

  18. C. Rousseau, Bifurcation methods in quadratic systems, Canadian Math. Soc., Conference Proc., 8 (1987), 637–653.

    MathSciNet  MATH  Google Scholar 

  19. D. Schlomiuk, J. Guckenheimer, R. Rand, Integrability of plane quadratic vector fields, to appear.

    Google Scholar 

  20. A.N. Varchenko, Estimate of the number of zeroes of abelian integrals depending on parameters and limit cycles, Funct. Anal. Appl., 18 (1984), 98–108.

    CrossRef  MATH  Google Scholar 

  21. Yu. Il’yashenko, Finiteness theorems for limit cycles, Russian Math. Surveys, 42 (1987), No. 3, p. 223.

    Google Scholar 

  22. W. A. Coppel, A survey of quadratic systems, J. Differential Equations, 2 (1966), 293–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. W. A. Coppel, Some quadratic systems with at most one limit cycle, Dynamics Reported, Vol. 2 (1989), 61–88. Ed.U.Kirchgraber & H.O.Walther, John Wiley & Sons Ltd. and B.G. Teubner.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Yu. Il’yashenko, Finiteness theorems for limit cycles, Russian Math. Surveys, 45 (1990), No. 2, p. 143–200.

    MathSciNet  MATH  Google Scholar 

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© 1990 Springer-Verlag

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Gavrilov, L., Horozov, E. (1990). Limit cycles and zeroes of Abelian integrals satisfying third order picard — Fuchs equations. In: Françoise, JP., Roussarie, R. (eds) Bifurcations of Planar Vector Fields. Lecture Notes in Mathematics, vol 1455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085392

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  • DOI: https://doi.org/10.1007/BFb0085392

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