Bounding the Number of Zeros of Abelian Integral for a Class of Integrable non-Hamilton System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750196 ◽  
Author(s):  
Shiyou Sui ◽  
Baoyi Li
Keyword(s):  
Upper Bound ◽  
Abelian Integral ◽  
Differential Systems ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Hamilton System

This paper investigates the planar differential systems [Formula: see text], [Formula: see text] under the perturbations of polynomials of [Formula: see text] with degree [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text]. The upper bound for the number of zeros of Abelian integral from the period annulus around the center is obtained.

ON THE NUMBER OF ZEROS OF ABELIAN INTEGRAL FOR A CUBIC ISOCHRONOUS CENTER

2012 ◽  
Vol 22 (01) ◽  
pp. 1250016 ◽  
Author(s):  
KUILIN WU ◽  
YUNLIN ZHAO
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integral ◽  
Isochronous Center ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study the number of limit cycles that bifurcate from the periodic orbits of a cubic reversible isochronous center under cubic perturbations. It is proved that in this situation the least upper bound for the number of zeros (taking into account the multiplicity) of the Abelian integral associated with the system is equal to four. Moreover, for each k = 0, 1, …, 4, there are perturbations that give rise to exactly k limit cycles bifurcating from the period annulus.

On the Distribution of Limit Cycles in a Liénard System with a Nilpotent Center and a Nilpotent Saddle

2016 ◽  
Vol 26 (02) ◽  
pp. 1650025 ◽  
Author(s):  
R. Asheghi ◽  
A. Bakhshalizadeh
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Integral Formula ◽  
Abelian Integral ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Period Annulus ◽  
Lienard System ◽  
Nilpotent Center ◽  
Nilpotent Saddle

In this work, we study the Abelian integral [Formula: see text] corresponding to the following Liénard system, [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are real bounded parameters. By using the expansion of [Formula: see text] and a new algebraic criterion developed in [Grau et al., 2011], it will be shown that the sharp upper bound of the maximal number of isolated zeros of [Formula: see text] is 4. Hence, the above system can have at most four limit cycles bifurcating from the corresponding period annulus. Moreover, the configuration (distribution) of the limit cycles is also determined. The results obtained are new for this kind of Liénard system.

scholarly journals Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Guoping Pang
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Sharp Bound ◽  
Abelian Integral ◽  
Heteroclinic Loop ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

scholarly journals ON THE NUMBER OF LIMIT CYCLES IN PERTURBATIONS OF A QUADRATIC REVERSIBLE CENTER

2012 ◽  
Vol 92 (3) ◽  
pp. 409-423
Author(s):  
JUANJUAN WU ◽  
LINPING PENG ◽  
CUIPING LI
Keyword(s):  
Limit Cycles ◽  
Abelian Integral ◽  
Reversible System ◽  
Linear Estimate ◽  
Bifurcation Of Limit Cycles ◽  
Arbitrary Degree ◽  
Number Of Zeros ◽  
Period Annulus

AbstractThis paper is concerned with the bifurcation of limit cycles from a quadratic reversible system under polynomial perturbations. It is proved that the cyclicity of the period annulus is two, and also a linear estimate of the number of zeros of the Abelian integral for the system under polynomial perturbations of arbitrary degreenis given.

Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

2018 ◽  
Vol 28 (05) ◽  
pp. 1850063 ◽  
Author(s):  
Shiyou Sui ◽  
Liqin Zhao
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Abelian Integral ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Cubic Polynomial

In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.

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.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

2021 ◽  
Vol 31 (09) ◽  
pp. 2150123
Author(s):  
Xiaoyan Chen ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

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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