Perturbations of quadratic hamiltonian systems with symmetry

Horozov, Emil Ivanov; Iliev, Iliya Dimov

Annales de l'I.H.P. Analyse non linéaire, Tome 13 (1996) no. 1, pp. 17-56.

[1] N. A'Campo, Le groupe de monodromie du déploiement des singularités isolées de courbes planes I, II. I: Math. Annalen, Vol. 213, 1975, No. 1, pp. 1-32; II: In Proc. Int. Congr. Math., (Vancouver, 1974), Vol. 1, 1975, pp. 395-404. | EuDML | MR | Zbl

[2] V.I. Arnol'D, Geometrical methods in the theory of ordinary differential equations, Springer Verlag, Berlin, Heidelberg, New York, 1988. | MR | Zbl

[3] V.I. Arnol'D, A.N. Varchenko and S.M. Gusein-Zade, Singularities of differentiable maps II, Birkhäuser Verlag, Basel, 1988. | MR

[4] R. Bamón, A class of planar quadratic vector fields with a limit cycle surrounded by a saddle loop, Proceedings of the AMS, Vol. 288, 1983, pp. 719-724. | MR | Zbl

[5] 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., Vol. 100, 1954, pp. 1-19. | MR | Zbl

[6] R.I. Bogdanov, Bifurcation of the limit cycle of a family of planar vector fields, Selecta Math. Soviet., Vol. 1, 1981, pp. 373-387, Russian original Trudy Sem. I. G. Petrovskogo, 1976. | MR | Zbl

[7] W.A. Coppel, A survey of quadratic systems, J. Differential Equations, Vol. 2, 1966, pp. 293-304. | MR | Zbl

[8] W.A. Coppel and L. Gavrilov, The period function of a Hamiltonian quadratic system, Diff. Int. Equations, Vol. 6, 1993, No. 6, pp. 1357-1365. | MR | Zbl

[9] B. Drachman, S.A. Van Gils and Zhang Zhi-Fen, Abelian integrals for quadratic vector fields, J. reine angew. Math., Vol. 382, 1987, pp. 165-180. | EuDML | MR | Zbl

[10] W.F. Freiberger (Editor), The International Dictionary of Applied Mathematics, D. van Nostrand Company Inc., Princeton, New Jersey, 1960. | MR | Zbl

[11] L. Gavrilov and E. Horozov, Limit cycles and zeroes of Abelian integrals satisfying third order Picard-Fuchs equations, In "Bifurcations of Planar Vector Fields", Lect. Notes in Math., 1455, J.-P. Françoise, R. Roussarie, eds., pp 160-186. | MR | Zbl

[12] L. Gavrilov and E. Horozov, Limit cycles of perturbations of quadratic Hamiltonian vector fields, J. Math. Pures Appl., Vol. 72, 1993, pp. 213-238. | MR | Zbl

[13] S.M. Gusein-Zade, Intersection matrices for certain singularities of functions of two variables, Funct. Anal. Appl., Vol. 8, 1974, pp. 10-13. | MR | Zbl

[14] S.M. Gusein-Zade, Dynkin diagrams for singularities of functions of two variables, Funct. Anal. Appl., Vol. 8, 1974, pp. 295-300. | MR | Zbl

[15] E. Horozov and I.D. Iliev, On saddle-loop bifurcations of limit cycles in perturbations of quadratic Hamiltonian systems, J. Differential Equations, Vol. 113, 1994, No. 1, pp. 84-105. | MR | Zbl

[16] E. Horozov and I.D. Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. Lond. Math. Soc., Vol. 69, 1994, No. 1, pp. 198-224. | MR | Zbl

[17] E. Horozov and I.D. Iliev, Hilbert-Arnold problem for cubic Hamiltonians and limit cycles, In: Proc. Fourth Intern. Coll. Diff. Eqs, VSP Intern. Sci. Publs, Utrecht, 1994, pp. 115-124. | MR | Zbl

[18] Yu S. Il'Yashenko, Number of zeros of certain Abelian integrals in a real domain, Funct. Anal. Appl., Vol. 11, 1978, pp. 78-79. | Zbl

[19] Yu S. Il'Yashenko, S. Yakovenko, Double exponential estimate for the number of zeros of complete Abelian integrals, Preprint, June 1994.

[20] A. Khovanski, Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., Vol. 18, 1984, pp. 119-128. | Zbl

[21] Li Jibin and Ou Yuehua, Global bifurcations and chaotic behaviour in a disturbed quadratic system with two centers (in Chinese), Acta Math. Appl. Sinica, Vol. 11, 1988, No. 3, pp. 312-323. | Zbl

[22] G.S. Petrov, Nonoscillations of elliptic integrals, Funct. Anal. Appl., Vol. 24, 1990, No. 3, pp. 205-210. | MR | Zbl

[23] L.S. Pontrjagin, Über Autoschwingungssysteme, die den Hamiltonischen nahe liegen, Zeitr. der Sowjetunion, Vol. 6, 1934, pp. 25-28. | Zbl

[24] R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Bras. Mat., Vol. 617, 1986, pp. 67-101. | MR | Zbl

[25] A.N. Varchenko, Estimate of the number of zeros of Abelian integrals depending on parameters and limit cycles, Funct. Anal. Appl., Vol. 18, 1984, pp. 98-108. | MR | Zbl

[26] Ye Yan-Qian et al., Theory of limit cycles, Translation of Mathematical Monographs, Vol. 66, AMS, Providence, 1986. | MR | Zbl

[27] Zhi-Fen Zhang and Chengzhi Li, On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations, Res. Rept Peking Univ., No. 33, 1993.

[28] H. Żoładek, Quadratic systems with center and their perturbations, J. Differential Equations, Vol. 109, 1994, No. 2, pp. 223-273. | MR | Zbl

[29] A.B. Givental, Sturm's theorem for hyperelliptic integrals, Leningrad Math. J., Vol. 1, 1990. | MR | Zbl