The Cyclicity of the Period Annulus of the Quadratic Hamiltonian Systems with Non-Morsean Point

Zhao, Yulin; Zhaojun, Liang; Lü, Gang

doi:10.1006/jdeq.1999.3704

Cited by 35 publications

(23 citation statements)

References 13 publications

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“…As the period function TðhÞ is given by Abelian integral (1.2), it is natural that the initial problem can be reduced to estimate the number of zeros of Abelian integral dTðhÞ=dh: A simple but important fact is that dTðhÞ=dh can be expressed as a linear combination of two basic integrals which satisfy a Picard-Fuchs equation. Some results concerned with the study of the number of zeros of Abelian integrals can be found in [CsH,F1,Gl3,I1,I2,NY,P,RZh,Zh,ZLL,ZS,ZZ2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

The Monotonicity of Period Function for Codimension Four Quadratic System Q4

Zhao¹

2002

Journal of Differential Equations

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Section: Introductionmentioning

confidence: 99%

The Monotonicity of Period Function for Codimension Four Quadratic System Q4

Zhao¹

2002

Journal of Differential Equations

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“…Accordingly, the result in [24] will follow once we show that I 0 , I 1 , I 2 is an ECTsystem on (0, 1). By applying Lemma 4.1, the same straightforward manipulation as before shows that I i (h) = …”

Section: Lemma 41 Let γ H Be An Oval Inside the Level Curve {A(x) +mentioning

confidence: 82%

“…In the literature there are many papers dealing with zeros of Abelian integrals (see for instance [5,6,10,11,12,23,24] and the references therein). In many cases, it is essential to show that a collection of Abelian integrals has some kind of Chebyshev property.…”

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 [7], Zhao, Liang and Lu [24] and Peng [22], 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%

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A Chebyshev criterion for Abelian integrals

Grau

Mañosas²,

Villadelprat

2011

Trans. Amer. Math. Soc.

143

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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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“…In the paper [I3], Iliev gave the formula of higher-order Melnikov function for quadratic perturbations of non-generic quadratic integrable system. By the study of the number of zeros of higher Melnikov function, we know that the cyclicity of period annulus of non-generic quadratic Hamiltonian systems under quadratic perturbations is 3 for the Hamiltonian triangle case, and 2 for other cases (see [CLY,GI,I1,ZLL,ZZh2]). …”

Section: The Tangential Hilbert 16th Problemmentioning

confidence: 99%

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Zhao

2005

Journal of Differential Equations

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Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k + n, under polynomial perturbation of arbitrary degree.

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The Cyclicity of the Period Annulus of the Quadratic Hamiltonian Systems with Non-Morsean Point

Cited by 35 publications

References 13 publications

The Monotonicity of Period Function for Codimension Four Quadratic System Q4

The Monotonicity of Period Function for Codimension Four Quadratic System Q4

A Chebyshev criterion for Abelian integrals

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

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