Higher-order Melnikov functions for degenerate cubic hamiltonians

Advances in Differential Equations

Volumn 1, Issue 4, 1996, Pages 689-708

  Iliev, I D ^a  

^a INSTITUTE OF MATHEMATICS AND INFORMATICS   (Bulgaria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001722685     PISSN: 10799389     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (18)

1. A.A. Andronov, E.A. Leontovich, I.I. Gordon, and A. G. Maier, "Theory of Bifurcations of Dynamic Systems on a Plane," John Wiley and Sons, New York, 1973.
2. V.I. Arnol'd, A.N. Varchenko, and S.M. Gusein-Zade, "Singularities of Differentiable Maps II," Birkhäuser Verlag, Basel, 1988.
3. Bao-yi Li and Zhi-fen Zhang, A note of a G.S. Petrov's result about the weakened 16th Hilbert problem, Peking Univ. Research Rept., No. 35 (1993).
4. N. N. Bautin, On the number of limit cycles which appear with the variation of the coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Transl., 100 (1954), 1-19.
5. B. Drachman, S.A. van Gils and Zhang Zhi-fen, Abelian integrals for quadratic vector fields, J. reine angew. Math., 382 (1987), 165-180.
6. J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, preprint, October (1993) (to appear in Ergod. Theory & Dyn. Syst.).
7. J. Guckenheimer, R. Rand, and D. Schlomiuk, Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems, Nonlinearity, 2 (1989), 405-418.
8. E. Horozov and I.D. Iliev, On saddle-loop bifurcations of limit cycles in perturbations of quadratic Hamiltonian systems, J. Differential Equations, 113 (1994), 84-105.
9. E. Horozov and I.D. Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. Lond. Math. Soc., 69 (1994), 198-224.
10. I.D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differential Equations (to appear).
11. w where R(z, w) is a polynomial, Matem. Sb., 78 (1969), 260-273 (in Russian).
12. P. Mardešić, The number of limit cycles of polynomial perturbations of a Hamiltonian vector field, Ergod. Theory & Dyn. Syst., 10 (1990), 523-529.
13. G. S. Petrov, Number of zeros of complete elliptic integrals, Funct. Anal. Appl., 18 (1984), 148-149.
14. R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Bras. Mat., 17 (1986), 67-101.
15. Yue He, and Chengzhi Li, On the number of limit cycles arising from perturbations of a homoclinic loop of quadratic integrable systems, preprint, Peking University (1995).
16. Zhang Zhi-Fen, and Li Bao-Yi, High order Melnikov functions, and the problem of uniformity in global bifurcations, Ann. Mat. Pura ed Appl. CLXI (1992), 181-212.
17. H. Žola̧dek, Quadratic systems with center, and their perturbations, J. Differential Equations, 109 (1994), 223-273.
18. H. Žola̧dek, The cyclicity of triangles, and segments in quadratic systems, preprint.