Markov, Yavor (1996)
Serdica Mathematical Journal
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We prove that in quadratic perturbations of generic Hamiltonian vector fields with two saddle points and one center there can appear at most two limit cycles. This bound is exact.
Chengzhi Li, Jaume Llibre, Zhi-fen Zhang (1995)
Publicacions Matemàtiques
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We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.
Yulin Zhao, Zhifen Zhang (2000)
Publicacions Matemàtiques
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It is proved in this paper that the maximum number of limit cycles of system ⎧ dx/dt = y ⎨ ⎩ dy/dt = kx - (k + 1)x2 + x3 + ε(α + βx + γx2)y is equal to two in the finite plane, where k > (11 + √33) / 4 , 0 < |ε| << 1, |α| + |β| + |γ| ≠ 0. This is partial answer to the seventh question in [2], posed by Arnold.
Tigan, Gheorghe (2006)
Applied Mathematics E-Notes [electronic only]
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Christov, Ognyan, Dragnev, Dragomir (1995)
Serdica Mathematical Journal
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The present paper deals with the KAM-theory conditions for systems describing the motion of a particle in central field.
Antonio Ambrosetti, Vittorio Coti Zelati (1987)
Annales de l'I.H.P. Analyse non linéaire
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Antonio Giorgilli (1988)
Annales de l'I.H.P. Physique théorique
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Javier Chavarriga, Héctor Giacomini, Jaume Giné (1997)
Publicacions Matemàtiques
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Let (P,Q) be a C vector field defined in a open subset U ⊂ R. We call a null divergence factor a C solution V (x, y) of the equation P ∂V/∂x + Q ∂V/ ∂y = ( ∂P/∂x + ∂Q/∂y ) V. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux...