Perturbations of quadratic centers of genus one

Gautier, Sebastien; Gavrilov, Lubomir; Iliev, I.

doi:10.48550/arxiv.0705.1609

Cited by 2 publications

(6 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our last examples of application come from the paper of Gautier, Gavrilov and Iliev [8], where a program for finding the cyclicity of the period annuli of quadratic systems with centers of genus one is presented. They give a list of the essential perturbations of these centers (i.e., the one-parameter perturbations that produce the maximal number of limit cycles), together with the corresponding generating function of limit cycles (i.e., the Poincaré-Pontryagin-Melnikov function).…”

Section: Results On the Program Of Gautier Gavrilov And Ilievmentioning

confidence: 99%

“…In other papers (e.g. [8,12,13]), the authors use complex analysis and algebraic topology (analytic continuation, argument principle, monodromy, Picard-Lefschetz formula, . .…”

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

“…We are also convinced that this criterion will be useful to obtain new results on the issue. In this direction we tackle the program posed by Gautier, Gavrilov and Iliev [8] and we prove their conjecture in four new cases (see Subsection 4.1).…”

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

“…Section 4 is devoted to illustrate the application of our criteria. To this end, in Examples 4.2, 4.3 and 4.4 we reprove the results of Iliev and Perko [8], Zhao, Liang and Lu [24] and Peng [21], respectively. Apart from showing the simplicity in the application of the criteria, our aim with these examples is twofold.…”

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

“…Second, to show that it is possible to reformulate the problem in such a way it suffices to check that some polynomials do not vanish. In Section 4 we also present some new results concerning the program of Gautier, Gavrilov and Iliev [8]. Finally in the Appendix we give some details about the tools that are used in Section 4, namely, the notion of resultant between two polynomials and Sturm's Theorem.…”

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

See 4 more Smart Citations

A Chebyshev criterion for Abelian integrals

Grau

Mañosas²,

Villadelprat

2011

Trans. Amer. Math. Soc.

143

View full text Add to dashboard Cite

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.

show abstract

Section: Results On the Program Of Gautier Gavrilov And Ilievmentioning

confidence: 99%

“…In other papers (e.g. [8,12,13]), the authors use complex analysis and algebraic topology (analytic continuation, argument principle, monodromy, Picard-Lefschetz formula, . .…”

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

Section: Theorem B Let Us Consider the Abelian Integralsmentioning

confidence: 99%

See 3 more Smart Citations

A Chebyshev criterion for Abelian integrals

Grau

Mañosas²,

Villadelprat

2011

Trans. Amer. Math. Soc.

143

View full text Add to dashboard Cite

show abstract

On the number of zeros of Abelian integrals for a kind of quadratic reversible centers

Wang

Zhang

Cao

2023

MATH

View full text Add to dashboard Cite

<abstract><p>Hilbert$ ' $s 16th problem is extensively studied in mathematics and its applications. Arnold proposed a weakened version focusing on differential equations. While significant progress has been made for Hamiltonian systems, less attention has been given to integrable non-Hamiltonian systems. In recent years, investigating quadratic reversible systems in integrable non-Hamiltonian systems has gained widespread attention and shown promising advancements. In this academic context, our study is based on qualitative analysis theory. It explores the upper bound of the number of zeros of Abelian integrals for a specific class of quadratic reversible systems under perturbations with polynomial degrees of n. The Picard-Fuchs equation method and the Riccati equation method are employed in our investigation. The research findings indicate that when the degree of the perturbing polynomial is n ($ n\geq5 $), the upper bound for the number of zeros of Abelian integrals is determined to be $ 7n-12 $. To achieve this, we first numerically transform the Hamiltonian function of the quadratic reversible system into a standard form. By applying a combination of the Picard-Fuchs equation method and the Riccati equation method, we derive the representation of the Abelian integrals. Using relevant theorems, we estimate the upper bound for the number of zeros of the Abelian integrals, which consequently provides an upper bound for the number of limit cycles in the system. The research results demonstrate that when the perturbation polynomial degree is high or equal to n, the Picard-Fuchs equation method and the Riccati equation method can be applied to estimate the upper bound of the number of zeros of the Abelian integrals.</p></abstract>

show abstract

Perturbations of quadratic centers of genus one

Cited by 2 publications

References 0 publications

A Chebyshev criterion for Abelian integrals

A Chebyshev criterion for Abelian integrals

On the number of zeros of Abelian integrals for a kind of quadratic reversible centers

Contact Info

Product

Resources

About