On the number of zeros of Abelian integrals for a kind of quartic Hamiltonians

2014 ◽  
Vol 228 ◽  
pp. 329-335 ◽  
Author(s):  
Juanjuan Wu ◽  
Yongkang Zhang ◽  
Cuiping Li
Keyword(s):  
Abelian Integrals ◽  
Number Of Zeros

QUADRATIC PERTURBATIONS OF A CLASS OF QUADRATIC REVERSIBLE LOTKA–VOLTERRA SYSTEMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1350137
Author(s):  
YI SHAO ◽  
A. CHUNXIANG
Keyword(s):  
Limit Cycles ◽  
Positive Answer ◽  
Abelian Integrals ◽  
Volterra System ◽  
Bifurcation Of Limit Cycles ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Volterra Systems

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system [Formula: see text] with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

scholarly journals On the number of zeros of Abelian integrals

2010 ◽  
Vol 181 (2) ◽  
pp. 227-289 ◽  
Author(s):  
Gal Binyamini ◽  
Dmitry Novikov ◽  
Sergei Yakovenko
Keyword(s):  
Abelian Integrals ◽  
Number Of Zeros

scholarly journals ON THE NUMBER OF ZEROS OF THE ABELIAN INTEGRALS FOR A CLASS OF PERTURBED LIÉNARD SYSTEMS

2007 ◽  
Vol 17 (09) ◽  
pp. 3281-3287
Author(s):  
TONGHUA ZHANG ◽  
YU-CHU TIAN ◽  
MOSES O. TADÉ
Keyword(s):  
Algebraic Structure ◽  
Upper Bound ◽  
Abelian Integrals ◽  
Hilbert’S 16Th Problem ◽  
Number Of Zeros ◽  
Liénard Systems ◽  
Hilbert's 16Th Problem

Addressing the weakened Hilbert's 16th problem or the Hilbert–Arnold problem, this paper gives an upper bound B(n) ≤ 7n + 5 for the number of zeros of the Abelian integrals for a class of Liénard systems. We proved the main result using the Picard–Fuchs equations and the algebraic structure of the integrals.

scholarly journals An explicit linear estimate for the number of zeros of Abelian integrals

Nonlinearity ◽  
2012 ◽  
Vol 25 (6) ◽  
pp. 1931-1946 ◽  
Author(s):  
Gal Binyamini ◽  
Gal Dor
Keyword(s):  
Abelian Integrals ◽  
Linear Estimate ◽  
Number Of Zeros

THE ABELIAN INTEGRALS OF A ONE-PARAMETER HAMILTONIAN SYSTEM UNDER POLYNOMIAL PERTURBATIONS

2004 ◽  
Vol 14 (07) ◽  
pp. 2449-2456 ◽  
Author(s):  
TONGHUA ZHANG ◽  
WENCHENG CHEN ◽  
MAOAN HAN ◽  
HONG ZANG
Keyword(s):  
Hamiltonian System ◽  
Upper Bound ◽  
Algebraic Curve ◽  
Open Interval ◽  
Abelian Integrals ◽  
Number Of Zeros

An upper bound Z(2,n)≤[n/2]+6[(n-1)/2]+5 is derived for the number of zeros of Abelian integrals Ĩ(h)=∮γ(h)g(x,y)dx-f(x,y)dy on the open interval (0,1/6), where γ(h) is an oval lying on the algebraic curve Hλ=(1/3)x3+(1/2)x2+λy3-y2=h, f(x,y), g(x,y) are polynomials of x and y, and n= max { deg f(x,y), deg g(x,y)}. The proof exploits the expansion of [Formula: see text] near λ=0.

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