Petrov modules and zeros of Abelian integrals

Gavrilov, Lubomir

doi:10.1016/s0007-4497(99)80004-9

Cited by 84 publications

(90 citation statements)

References 11 publications

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“…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…”

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confidence: 99%

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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confidence: 99%

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Gavrilov

Iliev

2000

Ergod. Th. Dynam. Sys.

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“…-The results by L. Gavrielov [7] show that a similar system of first order equations can be derived also for a general polynomial H provided that its principal homogeneous part is generic. However, the TRAJECTORIES OF POLYNOMIAL VECTOR FIELDS 601 resulting system will not be explicit, and there are almost no chances that the height of the right hand side would admit an upper bound uniform over all generic H.…”

Section: Appendix B Linear Systemsmentioning

confidence: 94%

“…The precise formulation follows. Take an arbitrary seed polynomial po e ^ and the sequence {p/c}^=i of its derivatives obtained by iterating L, (1)(2)(3)(4)(5)(6)(7) pk+i=Lpk, k =0,1,2,...…”

Section: Chains Of Polynomial Ideals: Effective Noetherianitymentioning

confidence: 99%