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ABSTRACT: We provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of any homogeneous and
quasi-homogeneous center, which can be obtained using the Abelian integral method of first order. We show that these bounds
are the best possible using the Abelian integral method of first order. We note that these centers are in general non-Hamiltonian.
As a consequence of our study we provide the biggest known number of limit cycles surrounding a unique singular point in terms
of the degree n of the system for arbitrary large n.
Journal of Dynamics and Differential Equations 04/2012; 21(1):133-152. · 0.73 Impact Factor
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ABSTRACT: We study the number of zeros of Abelian integrals for the quadratic centers having almost all their orbits formed by cubics,
when we perturb such systems inside the class of all polynomial systems of degreen
Science in China Series A Mathematics 04/2012; 45(8):964-974. · 0.70 Impact Factor
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Journal of Dynamics and Differential Equations 11/2008; 20(4):735-736. · 0.73 Impact Factor
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ABSTRACT: In this paper, we give an upper bound on the number of zeros of Abelian integrals for the quadratic centres having almost all their orbits formed by quartics, under polynomial perturbations of arbitrary degree n. The bound is linearly dependent on n.
Nonlinearity 04/2002; 15(3):863. · 1.39 Impact Factor
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ABSTRACT: We consider three classes of polynomial differential equations of the form ẋ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.
Proceedings of the Edinburgh Mathematical Society 09/2000; 43(03):529 - 543. · 0.41 Impact Factor
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ABSTRACT: The main objective of this paper is to provide an explicit and fairly accurate upper bound for the number of zeros of Abelian integrals defined by quadratic isochronous centres when we perturb them inside the class of all polynomial systems of degree n.
Nonlinearity 07/2000; 13(5):1775. · 1.39 Impact Factor
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ABSTRACT: The conjectureE(k)⩽k is proved to be true if and only ifk=1, 2, 3, whereE(k) is the cyclicity of condimensionk generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles except ensembles with
“lips” is ⩽6. By the way, the methods usually used in the study of cyclicity of polycycles such as derivation division algorithm,
Khovanskii procedure and the method of critical point analysis are introduced.
Chinese Science Bulletin 10/1998; 43(22):1849-1864. · 1.32 Impact Factor
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ABSTRACT: In this paper we provide the greatest lower bound about the number of (non-infinitesimal) limit cycles surrounding a unique singular point for a planar polynomial differential system of arbitrary degree.
Extracta mathematicae, ISSN 0213-8743, Vol. 16, Nº 3, 2001, pags. 441-448.
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ABSTRACT: 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.
Journal of Differential Equations.
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ABSTRACT: We consider the class of polynomial differential equations ˙ x = −y + P n (x, y), ˙ y = x + Q n (x, y), where P n and Q n are homogeneous polynomials of degree n. Inside this class we identify a new subclass of systems having a center at the origin. We show that this subclass contains at least two subfamilies of isochronous centers. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centers when we perturb them inside the class of all polynomial differential systems of the above form. In particular, we present a function whose simple zeros correspond to the limit cycles which bifurcate from the periodic orbits of Hamiltonian systems.
