Abelian integrals related to morse polynomials and perturbations of plane Hamiltonian vector fields

Annales de l'Institut Fourier

Volumn 49, Issue 2, 1999, Pages 611-652

  Gavrilov, Lubomir ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Abelian integrals; Limit cycles

Indexed keywords

EID: 0033440641     PISSN: 03730956     EISSN: None     Source Type: Journal    
DOI: 10.5802/aif.1684     Document Type: Article

References (30)

1. N. A'CAMPO, Le groupe de monodromie du déploiment des singularités isolées de courbes planes, I, Math. Ann., 213 (1975), 1-32.
2. V.I. ARNOLD, S.M. GUSEIN-ZADE, A.N. VARCHENKO, Singularities of Differentiable Maps, vols. 1 and 2, Monographs in mathematics, Birkhäuser, Boston, 1985 and 1988.
3. V.I. ARNOLD, Yu. S. IL'YASHENKO, Ordinary Differential Equations, in 'Dynamical Systems, I', Encyclopaedia of Math. Sci., vol. 1, Springer, Berlin, 1988.
4. V.I. ARNOLD, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, New York, 1988.
5. E. BRIESKORN, Die Monodromie der isolierten Singularitäten von Hyperfläschen, Manuscripta Math., 2 (1970), 103-161.
6. L. GAVRILOV, Isochronicity of plane polynomial Hamiltonian systems, Nonlinearity, 10 (1997), 433-448.
7. L. GAVRILOV, Petrov modules and zeros of Abelian integrals, Bull. Sci. Math., 122 (1998), 571-584.
8. L. GAVRILOV, Nonoscillation of elliptic integrals related to cubic polynomials of order three, Bull. London Math. Soc., 30 (1998), 267-273.
9. L. GAVRILOV, Modules of Abelian integrals, Proc. of the IVth Catalan days of applied mathematics, p. 35-45, Tarragona, Spain, 1998.
10. L. GAVRILOV, E. HOROZOV, Limit cycles of perturbations of quadratic vector fields, J. Math. Pures Appl., 72 (1993), 213-238.
11. P.A. GRIFFITHS, J. HARRIS, Principles of Algebraic Geometry, John Wiley and Sons, 1978.
12. E. HOROZOV, I. ILIEV, Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians, Nonlinearity, 11 (1998), 1521-1537.
13. E. HOROZOV, I.D. ILIEV, On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. London Math. Soc., 69 (1994), 198-224.
14. S.M. HUSEIN-ZADE, Dynkin digrams of singularities of functions of two variables, Functional Anal. Appl., 8 (1974), 10-13, 295-300.
15. I.D. ILIEV, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.
16. I.D. ILIEV, Higher-order Melnikov functions for degenerate cubic Hamiltonians, Adv. Diff. Equations, 1 (1996), 689-708.
17. Yu. IL'YASHENKO, Dulac's memoir "On Limit Cycles" and related problems of the local theory of differential equations, Russian Math. Surveys, 40 (1985), 1-49.
18. B. MALGRANGE, Intégrales asymptotiques et monodromie, Ann. scient. Ec. Norm. Sup., 7 (1974), 405-430.
19. P. MARDEŠIĆ, The number of limit cycles of polynomial deformations of a Hamiltonian vector field, Ergod. Th. and Dynam. Sys., 10 (1990), 523-529.
20. P. MARDEŠIĆ, Chebishev systems and the versai unfolding of the cusp of order n, Hermann, collection Travaux en Cours, 1998.
21. W.D. NEUMANN, Complex algebraic curves via their links at infinity, Inv. Math., 98 (1989), 445-489.
22. G.S. PETROV, Number of zeros of complete elliptic integrals, Funct. Anal. Appl., 18 (1984), 73-74.
23. G.S. PETROV, Elliptic integrals and their nonoscillation, Funct. Anal. Appl., 20 (1986), 37-40.
24. G.S. PETROV, Nonoscillation of elliptic integrals, Funct. Anal. Appl., 24 (1990), 45-50.
25. L.S. PONTRYAGIN, On dynamic systems close to Hamiltonian systems, Zh. Eksp. Teor. Fiz., 4 (1934), 234-238, in russian.
26. R. ROUSSARIE, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Bras. Math., (2), vol.17 (1986), 67-101.
27. C. ROUSSEAU, H. ŻOLADEK, Zeros of complete elliptic integrals for 1: 2 resonance, J. Diff. Equations, 94 (1991), 41-54.
28. M. SEBASTIANI, Preuve d'une conjecture de Brieskorn, Manuscripta Math., 2 (1970), 301-308.
29. H. ŻOLADEK, Abelian integrals in unfolding of codimension 3 singular planar vector firlds, in 'Bifurcations of Planar Vector Fields', Lecture Notes in Math., vol. 1480, Springer (1991).
30. H. ŻOLADEK, Quadratic systems with center and their perturbations, J. Diff. Equations, 109 (1994), 223-273.