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.

BIFURCATIONS OF LIMIT CYCLES FROM A QUINTIC HAMILTONIAN SYSTEM WITH A FIGURE DOUBLE-FISH

2013 ◽  
Vol 23 (07) ◽  
pp. 1350116 ◽  
Author(s):  
MINGHUI QI ◽  
LIQIN ZHAO
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integrals ◽  
Liénard Systems

In this paper, we consider Liénard systems of the form [Formula: see text] where 0 < ∣∊∣ ≪ 1 and (α, β, γ) ∈ ℝ3. We prove that the least upper bound of the number of isolated zeros of the related Abelian integrals [Formula: see text] is 4 (counting the multiplicity) and this upper bound is a sharp one.

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.

THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR A PERTURBATION OF HYPERELLIPTIC HAMILTONIAN SYSTEM WITH DEGENERATED POLYCYCLE

2013 ◽  
Vol 23 (03) ◽  
pp. 1350047 ◽  
Author(s):  
JIHUA WANG ◽  
DONGMEI XIAO ◽  
MAOAN HAN
Keyword(s):  
Asymptotic Expansion ◽  
Hamiltonian System ◽  
Upper Bound ◽  
Abelian Integral ◽  
Abelian Integrals ◽  
Exceptional Family ◽  
Level Curves ◽  
Number Of Zeros ◽  
Complete Study ◽  
Hyperbolic Saddle

In this paper, we provide a complete study of the zeros of Abelian integrals obtained by integrating the 1-form (α + βx + x2)ydx over the compact level curves of the hyperelliptic Hamiltonian [Formula: see text]. Such a family of compact level curves is bounded by a polycycle passing through a nilpotent cusp and a hyperbolic saddle of this hyperelliptic Hamiltonian system, which is not the exceptional family of ovals proposed by Gavrilov and Iliev. It is shown that the least upper bound for the number of zeros of the related hyperelliptic Abelian integral is two, and this least upper bound can be achieved for some values of parameters (α, β). This implies that the Abelian integral still has Chebyshev property for this nonexceptional family of ovals. Moreover, we derive the asymptotic expansion of Abelian integrals near a polycycle passing through a nilpotent cusp and a hyperbolic saddle in a general case.

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

2004 ◽  
Vol 14 (05) ◽  
pp. 1853-1862 ◽  
Author(s):  
TONGHUA ZHANG ◽  
WENCHENG CHEN ◽  
HONG ZANG
Keyword(s):  
Hamiltonian System ◽  
Upper Bound ◽  
Abelian Integral ◽  
Abelian Integrals ◽  
Number Of Zeros

In this paper, a one-parameter Hamiltonian system under cubic perturbations is investigated and the upper bound of the number of zeros of the Abelian integral is obtained by using Horozov and Iliev's method.

ABELIAN INTEGRALS FOR THE ONE-PARAMETER BOGDANOV–TAKENS SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2723-2727 ◽  
Author(s):  
YONGKANG ZHANG ◽  
BAOYI LI ◽  
CUIPING LI
Keyword(s):  
Upper Bound ◽  
Algebraic Curve ◽  
Open Interval ◽  
Melnikov Function ◽  
Abelian Integrals ◽  
First Order ◽  
Number Of Zeros ◽  
The One

An explicit upper bound Z(2, n) ≤ n + m - 1 is derived for the number of zeros of Abelian integrals M1(h) = ∮γ(h) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where γ(h) is an oval lying on the algebraic curve Hλ = (1/2)x2 + (1/2)y2 - (1/3)x3 - λy3 = h, P(x, y), Q(x, y) are polynomials of x and y, and max { deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M1(h, λ) near λ = 0 and assume (∂m/∂λm)M1(h, λ)|λ = 0 not vanish identically.

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.

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