Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

Nonlinearity

Volumn 11, Issue 6, 1998, Pages 1521-1537

  Horozov, Emil ^a   Iliev, Iliya D ^b  

^a UNIVERSITY OF SOFIA   (Bulgaria)

^b INSTITUTE OF MATHEMATICS AND INFORMATICS   (Bulgaria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0347466739     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/11/6/006     Document Type: Article

References (22)

1. Arnol'd V I 1977 Loss of stability of self-oscillations close to resonance and versai deformations of equivariant vector fields Funct. Anal. Appl. 11 85-92
2. Françoise J-P 1996 Successive derivatives of a first return map, application to the study of quadratic vector fields Ergod. Theor. Dynam. Syst. 16 87-96
3. Gavrilov L Petrov modules and zeros of Abelian integrals Bull. Sci. Math. to appear
4. Gavrilov L 1998 Nonoscillation of elliptic integrals related to cubic polynomials with symmetry of order three Bull. Lond. Math. Soc. 30 267-73
5. Gavrilov L and Horozov E 1990 Limit cycles and zeroes of Abelian integrals satisfying third order Picard-Fuchs equations Bifurcations of Planar Vector Fields Lecture Notes in Mathematics 1455 ed J-P Françoise and R Roussarie (Berlin: Springer) pp 160-86
6. Griffiths P and Harris J 1978 Principles of Algebraic Geometry (New York: Wiley)
7. Horozov E and Iliev I D 1994 On the number of limit cycles in perturbations of quadratic Hamiltonian systems Proc. Lond. Math. Soc. 69 198-224
8. Iliev I D 1998 Perturbations of quadratic centers Bull. Sci. Math. 122 107-61
9. w where R(z, w) is a polynomial Mat. Sbornik 78 360-73
10. Il'yashenko Yu S and Yakovenko S 1995 Double exponential estimate for the number of zeros of complete Abelian integrals Inv. Math. 121 613-50
11. Il'yashenko Yu S and Yakovenko S 1996 Counting real zeros of analytic functions satisfying linear ordinary differential equations J. Diff. Eqns 126 87-105
12. Khovanskii A 1984 Real analytic manifolds with finiteness properties and complex Abelian integrals Funct. Anal. Appl. 18 119-28
13. Mardešić P 1990 The number of limit cycles of polynomial perturbations of a Hamiltonian vector field Ergod. Theor. Dynam Syst. 10 523-9
14. Novikov D and Yakovenko S 1995 Simple exponential estimate for the number of real zeros of complete Abelian integrals Ann. Inst. Fourier 45 897-927
15. Petrov G S 1986 Elliptic integrals and their non-oscillatorness Funct. Anal. Appl. 20 (1) 46-9
16. Petrov G S 1987 Complex zeros of an elliptic integral Funct. Anal. Appl. 21 (3) 87-8
17. Petrov G S 1990 Nonoscillatorness of elliptic integrals Funct. Anal. Appl. 24 (3) 45-50
18. Rousseau C and Żoładek H 1991 Zeros of complete elliptic integrals for 1:2 resonance J. Diff. Eqns 94 41-54
19. Stoev N 1994 MSc Thesis University of Sofia
20. Varchenko A N 1984 Estimate of the number of zeros of Abelian integrals depending on parameters and limit cycles Funct. Anal. Appl. 18 98-108
21. Yakovenko S 1994 Complete Abelian integrals as rational envelopes Nonlinearity 7 1237-50
22. Petrov G S 1984 On the number of zeros of complete elliptic integrals Funct. Anal Appl. 18 (2) 73-4