On the Number of Limit Cycles in Perturbations of Quadratic Hamiltonian Systems

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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“…First we recall the normal form for all cubic Hamiltonians having a center. As in [10,11], introduce the following basic one-forms…”

Section: The Relative Cohomology Decomposition Of Polynomial One-formsmentioning

confidence: 99%

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Gavrilov

Iliev

2000

Ergod. Th. Dynam. Sys.

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“…Finally, the main results of this paper are proved in section 5. Some techniques in section 4 and section 5 are borrowed from [4].…”

Section: Either I(h) Vanishes Identically or Its Lowest Upper Boundmentioning

confidence: 99%

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

Zhao¹,

Zhang²

2000

Publicacions Matemàtiques

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“…For instance, in some papers (e.g. [4,14,22]) the authors study the geometrical properties of the so-called centroid curve using the fact that it verifies a Riccati equation (which is itself deduced from a Picard-Fuchs system). In other papers (e.g.…”

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

A Chebyshev criterion for Abelian integrals

Grau

Mañosas²,

Villadelprat

2011

Trans. Amer. Math. Soc.

136

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Abstract. We present a criterion that provides an easy sufficient condition in order for a collection of Abelian integrals to have the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.

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On the Number of Limit Cycles in Perturbations of Quadratic Hamiltonian Systems

Abstract: We prove that in quadratic perturbations of generic quadratic Hamiltonian vector fields with three saddle points and one centre there can appear at most two limit cycles. This bound is exact.

Cited by 100 publications

References 0 publications

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

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

A Chebyshev criterion for Abelian integrals

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