Quadratic Systems with Center and Their Perturbations

Żołądek, Henryk

doi:10.1006/jdeq.1994.1049

Cited by 226 publications

(121 citation statements)

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“…Denote I = col(R, M, X). Then I(h) satisfies a Picard-Fuchs system of the form I = (Ah + B)I (22) and the validity of the result in the proposition depends on whether the matrix A is lowertriangular. To check this, we can use the system (1.5) derived in [15].…”

Section: L(h) = Ax(h) + B(h − H L )R(h) + Cm(h) (A B C = Constants)mentioning

confidence: 98%

“…The order of the Poincaré-Pontryagin function M k (h) (also called Melnikov integral), giving the possible maximum number of zeros, is known only in the quadratic case and in the symmetric cubic case (when the perturbed field possesses central symmetry). Thus, the cyclicity under arbitrary quadratic perturbations of the period annulus of a reversible quadratic Hamiltonian vector field is determined by the second Poincaré-Pontryagin integral, except for the Hamiltonian triangle, whose cyclicity is determined by the thirdorder variation [14,22]. In general, the order of the Poincaré-Pontryagin integral which generates a module of Abelian integrals of a maximal possible dimension is unknown, and its determination appears to be a difficult task involving the solution of the corresponding center-focus problem and viewing the structure of the related center manifold.…”

Section: Introductionmentioning

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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Section: L(h) = Ax(h) + B(h − H L )R(h) + Cm(h) (A B C = Constants)mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Gavrilov

Iliev

2000

Ergod. Th. Dynam. Sys.

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“…systems which have at least two real invariant lines) which have a center, we do not do this here, not to conflict with the terminology in the literature and in particular with [76]. We stress that, as already indicated above, in this work we use the notation L-V-C to denote the family of systems which can be reduced by affine transformations and time rescaling to the Kapteyn-Bautin normal form with a = 0 = b + d, analogously to the terminology in [76].…”

Section: Normal Forms and Bifurcation Diagramsmentioning

confidence: 95%

“…This terminology, was introduced in [76] for the family of systems which using our coordinates is defined by the equations a = 0 and b+d = 0. In general in the literature, the term Lotka-Volterra was used for quadratic systems which have at least two real invariant lines.…”

Section: Normal Forms and Bifurcation Diagramsmentioning

confidence: 99%

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

Schlomiuk¹

2014

Publicacions Matemàtiques

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Abstract:We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress.

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“…The quadratic polynomial vector fields with a center can be divided in four dosses: Q^, Qf, Q § and Q^ (Zoladek [30]), called Lotka-Volterra case, Hamiltonian case, reversible case and codimension 4 case respectively. In this section we shall study the Poincare-Pontryagin functions…”

Section: Abelian Integrals Which Arise In Polynomial Perturbations Ofmentioning

confidence: 99%

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

Gavrilov¹

1999

Annales De L’institut Fourier

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Quadratic Systems with Center and Their Perturbations

Cited by 226 publications

References 0 publications

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

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

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