Displaying similar documents to “Limit Cycles of Perturbations of a Class of Quadratic Hamiltonian Vector Fields”

Estimate for the Number of Zeros of Abelian Integrals on Elliptic Curves

Mihajlova, Ana (2004)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: Primary 34C07, secondary 34C08. We obtain an upper bound for the number of zeros of the Abelian integral. The work was partially supported by contract No 15/09.05.2002 with the Shoumen University “K. Preslavski”, Shoumen, Bulgaria.

Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop.

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.

Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

Lubomir Gavrilov (1999)

Annales de l'institut Fourier

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Let 𝒜 be the real vector space of Abelian integrals I ( h ) = ∫ ∫ { H ≤ h } R ( x , y ) d x ∧ d y , h ∈ [ 0 , h ˜ ] where H ( x , y ) = ( x 2 + y 2 ) / 2 + ... is a fixed real polynomial, R ( x , y ) is an arbitrary real polynomial and { H ≤ h } , h ∈ [ 0 , h ˜ ] , is the interior of the oval of H which surrounds the origin and tends to it as h → 0 . We prove that if H ( x , y ) is a semiweighted homogeneous polynomial with only Morse critical points, then 𝒜 is a free finitely generated module over the ring of real polynomials ℝ [ h ] , and compute its rank. We find the generators of 𝒜 in the case when H is an...