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

Gavrilov, Lubomir

doi:10.5802/aif.1684

Cited by 31 publications

(33 citation statements)

References 26 publications

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“…If M 1 (h) ≡ 0 then Theorem 1 is proved in [7,Theorem 3]. Suppose that M 1 (h) ≡ 0, but M 2 (h) ≡ 0.…”

Section: Theorem 4 If the Abelian Integral J (H)mentioning

confidence: 99%

“…We use them in §3 to derive, by the Françoise recursive procedure [3], an appropriate formula for the second variation M 2 (h) of the Poincaré return map. In §4 we estimate, following [7], the zeros of M 2 and the limit cycles of (1) provided M 2 (h) ≡ 0. Then the result from Theorem 2 is a consequence of the fact that in the quadratic case n = 2, the second variation of the Poincaré map in the considered case suffices to determine the limit cycles in the whole plane [14].…”

Section: Theoremmentioning

confidence: 99%

“…We establish that the number of isolated zeros of M 2 (h) in a suitable complex domain does not exceed 2n − 2, which in particular yields for the quadratic case n = 2 that any small quadratic perturbation of X H , H ∈ À can produce no more than two limit cycles. Using the notion of a Chebyshev system, this means that the (2n − 1)-dimensional space of second-order Poincaré-Pontryagin functions corresponding to nth-degree polynomial perturbations of X H , H ∈ À, forms a Chebyshev system for any n. We note that according to a recent result in [7], the first-order Poincaré-Pontryagin functions M 1 (h) also belong to a Chebyshev system (of dimension n).…”

Section: Introductionmentioning

confidence: 99%

“…Let Á 1 : È n → Å n 1 be the linear mapping defined by ω → M ω 1 (h). Then Á 1 is an isomorphism of modules (see [5,7] where Á 1 is studied in detail). This implies that dim Å n 1 = n and…”

mentioning

confidence: 99%

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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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“…If M 1 (h) ≡ 0 then Theorem 1 is proved in [7,Theorem 3]. Suppose that M 1 (h) ≡ 0, but M 2 (h) ≡ 0.…”

Section: Theorem 4 If the Abelian Integral J (H)mentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Gavrilov

Iliev

2000

Ergod. Th. Dynam. Sys.

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“…It was proved for instance that in several cases, the vector space A H,d of Abelian integrals of degree d polynomials along the ovals of H, obeys the so-called Chebyshev property (the number of the zeros of each integral is smaller than the dimension of the vector space A H,d ); see [22], [15], [14], [11]. In this relation Arnold asked in [8, the 7th problem] whether the g-dimensional vector space of Abelian integrals (deg P − 1)] > 1, is Chebyshev.…”

Section: Take Real Polynomials H F G ∈ R[x Y] and Let δ(H) ⊂ {(X mentioning

confidence: 99%

Complete hyperelliptic integrals of the first kind and their non-oscillation

Gavrilov

Iliev

2003

Trans. Amer. Math. Soc.

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Redundant Picard–Fuchs System for Abelian Integrals

Novikov¹,

Yakovenko²

2001

Journal of Differential Equations

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We derive an explicit system of Picard Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants appears only in dimension approximately two times greater than the standard Picard Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were able to majorize the number of all (real and complex) zeros. In the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and non-homogeneity of uni-and bivariate complex polynomials are studied. 2001Elsevier Science

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

Cited by 31 publications

References 26 publications

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Complete hyperelliptic integrals of the first kind and their non-oscillation

Redundant Picard–Fuchs System for Abelian Integrals

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