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

Lubomir Gavrilov (1999)

Annales de l'institut Fourier

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 arbitrary cubic polynomial. Finally we...

An attraction result and an index theorem for continuous flows on ℝ n × [ 0 , ∞ )

Klaudiusz Wójcik (1997)

Annales Polonici Mathematici

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on E = ℝ n + 1 for which ∂ E = ℝ n × 0 is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ℝ n × [ 0 , ∞ ) .

Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system.

Li Jibin, Liu Zhenrong (1991)

Publicacions Matemàtiques

In this paper we consider a class of perturbation of a Hamiltonian cubic system with 9 finite critical points. Using detection functions, we present explicit formulas for the global and local bifurcations of the flow. We exhibit various patterns of compound eyes of limit cycles. These results are concerned with the weakened Hilbert's 16th problem posed by V. I. Arnold in 1977.

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

Yulin Zhao, Zhifen Zhang (2000)

Publicacions Matemàtiques

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)yis 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.

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