Perturbations of quadratic centers

Bulletin des Sciences Mathematiques

Volumn 122, Issue 2, 1998, Pages 107-161

  Iliev, Iliya D ^a  

^a INSTITUTE OF MATHEMATICS AND INFORMATICS   (Bulgaria)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032020006     PISSN: 00074497     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0007-4497(98)80080-8     Document Type: Article

References (32)

1. [1] BAUTIN (N. N.). - On the number of limit cycles which appear with the variations of the coefficients from an equilibrium point of focus or center type, in: AMS Translations-Series 1, 5, 1962, p. 396-413 [Russian original: Math. Sbornik, 30, 1952, p. 181-196].
2. [2] CHICONE (C.). - On bifurcation of limit cycles from centers, in: Bifurcations of planar vector fields, J.-P. FRANÇOISE et R. ROUSSARIE (eds.). - Berlin, Springer, 1990 (Lect. Notes in Mathematics, 1455, p. 20-43).
3. [3] CHICONE (C.) and JACOBS (M.). - Bifurcations of limit cycles from quadratic isochrones, J. Differential Equations, 91, 1991, p. 268-326.
4. [4] COPPEL (W. A.). - Some quadratic systems with at most one limit cycle, Dynamics Reported, 2, 1989, p. 61-88.
5. [5] FRANÇOISE (J.-P.). - Successive derivatives of a first return map, application to the study of quadratic vector fields (Preprint, October 1993), Ergod. Theory & Dyn. Syst., 16, 1996, p. 87-96.
6. [6] GAVRILOV (L.) and HOROZOV (E.) - Limit cycles and zeroes of Abelian integrals satisfying third order Picard-Fuchs equations, in Bifurcations of planar vector fields, J.-P. FRANÇOISE and R. ROUSSARIE (eds.). - Berlin, Springer, 1990 (Lect. Notes in Mathematics, 1455, p. 160-186).
7. [7] HOROZOV (E.) and ILIEV (I. D.). - On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. Lond. Math. Soc., 69, 1994, p. 198-224.
8. [8] HOROZOV (E.) and ILIEV (I. D.). - Perturbations of quadratic Hamiltonian systems with symmetry, Ann. Inst. Henri Poincaré, Analyse non linéaire, 13, 1996, p. 17-56.
9. [9] ILIEV (I. D.). - Higher order Melnikov functions for degenerate cubic Hamiltonians (preprint 1994), Adv. Differential Equations, 1, 1996, p. 689-708.
10. [10] ILIEV (I. D.). - The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differential equations, 128, 1996, p. 309-326.
11. [11] IL'YASHENKO (Yu. S.). - Number of zeros of certain Abelian integrals in a real domain, Funct. Anal. Appl., 11, 1978, p. 78-79.
12. [12] KAPTEJN (W.). - New investigations on the midpoints of integrals of differential equations of the first order and the first degree, Koninkl. Nederl. Wetensch. Verslagen, 20, 1912, p. 1354-1465, 21, p. 27-33 (in Dutch).
13. [12] KAPTEJN (W.). - New investigations on the midpoints of integrals of differential equations of the first order and the first degree, Koninkl. Nederl. Wetensch. Verslagen, 20, 1912, p. 1354-1465, 21, p. 27-33 (in Dutch).
14. [13] LI (C.) and ROUSSEAU (C.). - A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4, J. Differential Equations, 79, 1989, p. 132-167.
15. [14] LOUD (W. S.). - Behaviour of the periods of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations, 3, 1964, p. 21-36.
16. [15] MARDEŠIĆ (P.). - The number of limit cycles of polynomial perturbations of a Hamiltonian vector field, Ergod. Th. & Dynam. Sys., 10, 1990, p. 523-529.
17. [16] PETROV (G. S.). - On the number of zeros of complete elliptic integrals, Funct. Anal. Appl., 18, 1984, p. 73-74.
18. [17] PETROV (G. S.). - Elliptic integrals and their non-oscillatorness, Funct. Anal. Appl., 20, 1986, p. 46-49.
19. 6 y)∂/∂y, in Bifurcations of planar vector fields, J.-P. FRANÇOISE and R. ROUSSARIE (eds.). - Berlin, Springer, 1990 (Lect. Notes in Mathematics, 1455, p. 334-355).
20. [19] ROUSSEAU (C.) and ZOLADEK (H.). - Zeros of complete elliptic integrals for 1:2 resonance, J. Differential Equations, 94, 1991, p. 41-54.
21. [20] SHAFER (D. S.) and ZEGELING (A.). - Bifurcation of limit cycles from quadratic centers, J. Differential Equations, 122, 1995, p. 48-70.
22. [21] SIBIRSKII (K. S.). - Introduction to the algebraic theory of invariants of differential equations. - Manchester Univ. Press, 1988 [Russian Original: Stiintsa, Kishinev, 1976].
23. [22] VAN GILS (S. A.). - An inhomogeneous Picard-Fuchs equation, in Dynamics, Bifurcation and Symmetry, P. CHOSSAT ed., p. 333-341. - Kluwer, 1994.
24. [23] VAN HORSSEN (W. T.) and KOOU (R. E.). - Bifurcations of limit cycles in a particular class of quadratic systems with two centres, J. Differential Equations, 114, 1994, p. 538-569.
25. [24] YAKOVENKO (S.). - A geometric proof of the Bautin theorem, in: Concerning the Hilbert 16th problem, AMS Translations-Series, 2, 165, 1995.
26. [25] YUE HE and CHENGZHI LI. - On the number of limit cycles arising from perturbation of homoclinic loops of quadratic integrable systems (Preprint Peking Univ., 1995).
27. [26] ZHI-FEN ZHANG and BAO-YI LI. - A note on a result of G. S. Petrov about the weakened 16th Hilbert problem, J. Math. Anal. Appl., 190, 1995, p. 489-516.
28. [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.
29. [28] ZOŁA̧DEK (H.). - Quadratic systems with center and their perturbations, J. Differential Equations, 109, 1994, p. 223-273.
30. [29] ZOłA̧DEK (H.). - Eleven small limit cycles in a cubic vector field, Nonlinearity, 8, 1995, p. 843-860.
31. [30] ZOŁA̧DEK (H.). - The cyclicity of triangles and segments in quadratic systems, J. Differential Equations, 122, 1995, p. 137-159.
32. [31] ZOŁA̧DEK (H.). - Asymptotic properties of Abelian integrals arising in quadratic, systems, (Preprint, Univ. of Warsaw, 1994).