A Note on a Result of G. S. Petrov About the Weakened 16th Hilbert Problem

Li, Baoyi; Zhang, Zhifen

doi:10.1006/jmaa.1995.1088

Cited by 44 publications

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“…We recall that when a = 0 (the Bogdanov-Takens case), the analogue of Theorem 1 follows from the results of Petrov [18,19], Mardešić [17] and Li and Zhang [16]. It should also be noticed that the Chebyshev property does not hold for some of the reversible cases [11].…”

Section: Theoremmentioning

confidence: 89%

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Gavrilov

Iliev

2000

Ergod. Th. Dynam. Sys.

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Abstract. We study degree n polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré-Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is n − 1. In the present paper we prove that if the first Poincaré-Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is 2(n − 1). In the case when the perturbation is quadratic (n = 2) we obtain a complete result-there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.

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Section: Theoremmentioning

confidence: 89%

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Gavrilov

Iliev

2000

Ergod. Th. Dynam. Sys.

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“…The papers [6,8] show that two is the maximum number of limit cycles of system (1.1) = , if, in addition, (1.1) 0 has three saddle points and one center. Some results concerned with certain specific quadratic centers can be found in [2,3,4,9,13,16,17].…”

Section: Introductionmentioning

confidence: 99%

The Cyclicity of the Period Annulus of the Quadratic Hamiltonian Systems with Non-Morsean Point

Zhao

Zhaojun

Lü

2000

Journal of Differential Equations

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“…In what follows for n=2 or n=3 systems (1) are called quadratic or cubic systems, respectively. Many authors have studied the limit cycles which bifurcate from periodic orbits of a center for a quadratic system; see, for instance, [4,9,15,14,19,[22][23][24]26].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

Llibre

et al. 2002

Journal of Differential Equations

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This paper consists of two parts. In the first part we study the relationship between conic centers (all orbits near a singular point of center type are conics) and isochronous centers of polynomial systems. In the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics, when we perturb such systems inside the class of all polynomial systems of degree n. © 2002 Elsevier Science (USA)

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A Note on a Result of G. S. Petrov About the Weakened 16th Hilbert Problem

Cited by 44 publications

References 0 publications

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

The Cyclicity of the Period Annulus of the Quadratic Hamiltonian Systems with Non-Morsean Point

Linear Estimation of the Number of Zeros of Abelian Integrals for Some Cubic Isochronous Centers

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