×

Limit cycles bifurcated from a reversible quadratic center. (English) Zbl 1142.34019

The main result of the paper is that under quadratic perturbations the exact upper bound of the number of limit cycles produced by the period annulus of the system
\[ \dot{z}=-iz(1+A\bar{z}) \]
with a nonzero complex number \(A\) is two. It follows basically from a careful estimate of the number of zeros of the associate Abelian integrals. Since there is no Picard-Fuchs equation in such case, the author uses different technique for this purpose.

MSC:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] V. I. Arnold,Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, New York, 1983. · Zbl 0507.34003
[2] V. I. Arnold,Ten problems, Adv. Soviet Math.,1 (1990), 1–8.
[3] J. C. Artés, J. Llibre and D. Schlomiuk,The geometry of quadratic differential systems with a weak focus of second order, Inter. J. Bifur. & Chaos, to appear. · Zbl 1124.34014
[4] N. N. Bautin,On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb.30(72) (1952), 181–196 (Russian); Transl. Amer. Math. Soc.100(1) (1954), 397–413. · Zbl 0059.08201
[5] G. Chen, C. Li, C. Liu andJ. Llibre,The cyclicity of period annuli of some classes of reversible quadratic systems, Disc. & Contin. Dyn. Sys.16(2006), 157–177. · Zbl 1119.34028
[6] F. Chen, C. Li, J. Llibre andZ. Zhang,A uniform proof on the weak Hilbert’s 16th problem for n=2. J. Diff. Eqns.221 (2006), 309–342. · Zbl 1098.34024
[7] L. Gavrilov,The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math.,143 (2001), 449–497. · Zbl 0979.34024
[8] A. Garijo, A. Gasull andX. Jarque,On the period function for a family of complex differential equations, J. Diff. Eqns.224 (2006), 314–331. · Zbl 1103.34026
[9] E. Horozov andI. D. Iliev,On the number of limit cycles in perturbation of quadratic Hamiltonian systems, Proc. Lond. Math. Soc.69 (1994), 198–224. · Zbl 0802.58046
[10] I. D. Iliev,Higher order Melnikov functions for degenerate cubic hamiltonians, Adv. Differential Equations,1 (1996), 689–708. · Zbl 0851.34042
[11] I. D. Iliev,Perturbations of quadratic centers, Bull. Sci. Math.122 (1998), 107–161. · Zbl 0920.34037
[12] I. D. Iliev, C. Li, andJ. Yu,Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two unbounded heteroclinic loops. Nonlinearity18 (2005), 305–330. · Zbl 1077.34035
[13] A. G. Khovansky,Real analytic manifold with finiteness properties and complex Abelian integrals, Funct. Anal. Appl.18 (1984), 119–128. · Zbl 0584.32016
[14] C. Li andZ. Zhang,A criterion for determining the monotonicity of ratio of two Abelian integrals, J. Diff. Eqns.124 (1996), 407–424. · Zbl 0849.34022
[15] C. Li andZ. Zhang,Remarks on 16th weak Hilbert problem for n=2, Nonlinearity15 (2002) 1975–1992. · Zbl 1219.34042
[16] D. Schlomiuk,Algebraic particular integrals, integrability and the problem of the centers Trans. Amer. Math. Soc.338 (1993), 799–841. · Zbl 0777.58028
[17] A. N. Varchenko,Estimation of the number of zeros of an Abelian integral depending on parameters and limit cycles, Funct. Anal. Appl.18 (1984), 98–108. · Zbl 0578.58035
[18] Z. Zhang andC. Li,On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations. Advance in Math.26 (1997), 445–460. · Zbl 0948.34020
[19] H. \.Zoladek,Quadratic systems with center and their perturbations, J. Diff. Eqs.109 (1994), 223–273. · Zbl 0797.34044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.