The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-Morsean point. (English) Zbl 0960.37027
This paper deals with the number of limit cycles for small quadratic perturbations of quadratic Hamiltonian systems with non-Morsean point, that is
\[
\begin{aligned} \dot x&= 2xy+ \varepsilon \Bigl( \sum_{i+j\leq 2} a_{ij} (\varepsilon) x^i y^j\Bigr),\\ \dot y&= 6x+ 6x^2- y^2+ \varepsilon \Bigl( \sum_{i+j\leq 2} b_{ij} (\varepsilon) x^i y^j \Bigr), \end{aligned} \tag \(1_\varepsilon\)
\]
where \(a_{ij} (\varepsilon)\) and \(b_{ij} (\varepsilon)\) depend analytically on the small parameter \(\varepsilon\). The authors prove that, for small \(\varepsilon\), the maximum number of limit cycles in \((1)_\varepsilon\) which emerge from the period annulus is equal to two.
Reviewer: Messoud Efendiev (Berlin)
MSC:
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
References:
| [1] | Arnold, V. I., Geometrical Methods in the Theory of Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0507.34003 |
| [2] | Drachman, B.; Van Gils, S. A.; Zhifen, Zhang, Abelian integrals for quadratic vector fields, J. Reine Angew. Math., 382, 165-180 (1987) · Zbl 0621.58033 |
| [3] | Dumortier, F.; Li, C.; Zhang, Z., Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Differential Equations, 139, 146-193 (1997) · Zbl 0883.34035 |
| [4] | Francoise, J. P., Successive derivative of first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. System, 16, 87-96 (1996) · Zbl 0852.34008 |
| [5] | W. Gao, Analytic Theory of Ordinary Differential Equations, to appear. [In Chinese]; W. Gao, Analytic Theory of Ordinary Differential Equations, to appear. [In Chinese] |
| [6] | Gavrilov, L.; Horozov, E., Limit cycles of perturbations of quadratic Hamiltonian vector fields, J. Math. Pures Appl., 72, 213-238 (1993) · Zbl 0829.58034 |
| [7] | Hartman, P., Ordinary Differential Equations (1982), Birkhäuser · Zbl 0125.32102 |
| [8] | Horozov, E.; Iliev, I. D., On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. London Math. Soc., 69, 198-224 (1994) · Zbl 0802.58046 |
| [9] | Horozov, E.; Iliev, I. D., Perturbations of quadratic Hamiltonian systems with symmetry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13, 17-56 (1996) · Zbl 0854.34035 |
| [10] | Iliev, I. D., Higher order Melnikov functions for degenerate cubic Hamiltonians, Adv. Differential Equations, 1, 689-708 (1996) · Zbl 0851.34042 |
| [11] | Iliev, I. D., The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differential Equations, 128, 309-326 (1996) · Zbl 0853.58084 |
| [12] | Iliev, I. D., Perturbations of quadratic centers, Bull. Sci. Math., 122, 107-161 (1998) · Zbl 0920.34037 |
| [13] | Li, B.; Zhang, Z., A note on a result of G. S. Petrov about the weakened 16th Hilbert problem, J. Math. Anal. Appl., 190, 489-516 (1995) · Zbl 0829.34022 |
| [14] | Li, C.; Llibre, J.; Zhang, Z., Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line, Publ. Mat., 39, 355-356 (1995) · Zbl 0856.58033 |
| [15] | W. Li, Normal Form Theory and their Application, to appear in High Educations Press. [In Chinese]; W. Li, Normal Form Theory and their Application, to appear in High Educations Press. [In Chinese] |
| [16] | Zhang, Z., Qualitative Theory of Differential Equations. Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101 (1992), AMS: AMS Providence · Zbl 0779.34001 |
| [17] | Zhang, Z.; Li, B., Higher order Melnikov functions and the problem of uniformity in global bifurcations, Ann. Mat. Pura Appl. (4), CLXI, 181-212 (1992) · Zbl 0768.34023 |
| [18] | Zoladek, H., Quadratic systems with centers and their perturbations, J. Differential Equations, 109, 223-273 (1994) · Zbl 0797.34044 |
| [19] | Zoladek, H., The cyclicity of triangles and segments in quadratic systems, J. Differential Equations, 122, 137-159 (1995) · Zbl 0840.34031 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
