Limit cycles near an eye-figure loop in some polynomial Lienard systems

被引：4

作者：

Bakhshalizadeh, A. ^{[1
]}

Asheghi, R. ^{[1
]}

Zangeneh, H. R. Z. ^{[1
]}

Gashti, M. Ezatpanah ^{[1
]}

机构：

[1] Isfahan Univ Technol, Dept Math Sci, Esfahan 8415683111, Iran

来源：

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | 2017年 / 455卷 / 01期

关键词：

Lienard system; Eye-figure loop; Limit cycle; Abelian integral; HAMILTONIAN-SYSTEMS; CUSPIDAL LOOP; BIFURCATIONS; NUMBER;

D O I：

10.1016/j.jmaa.2017.05.064

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper, we study the number of limit cycles in the family dH - epsilon omega = 0, where H = y(2)/2 - integral(x)(0) g(u)du, omega = yf(x)dx, with g(x) = x(x(2) - 1)(x(2) - 1/4)(2), and f (x) an even polynomial of degree 10. We will consider mainly the bifurcation of limit cycles near the eye-figure loop and the center of dH = 0. Our investigation focuses on the lower bound of the maximal number of limit cycles for these systems. In particular, we show that the perturbed system can have at least 8 limit cycles when deg(f(x)) = 10. (C) 2017 Elsevier Inc. All rights reserved.

引用

页码：500 / 515

页数：16

共 18 条

[1]

[Anonymous], 1928, REV GENERALE LELECTR, DOI DOI 10.1016/J.JWEIA.2013.09.002

[2]

[Anonymous], 2006, J SHANGHAI NORM U NA

[3] Limit cycles in a Lienard system with a cusp and a nilpotent saddle of order 7 [J].

Asheghi, R. ;

Bakhshalizadeh, A. .

CHAOS SOLITONS & FRACTALS, 2015, 73 :120-128

[4] THE NUMBER OF SMALL-AMPLITUDE LIMIT-CYCLES OF LIENARD EQUATIONS [J].

BLOWS, TR ;

LLOYD, NG .

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 1984, 95 (MAR) :359-366

[5] Small-amplitude limit cycle bifurcations for Lienard systems with quadratic or cubic damping or restoring forces [J].

Christopher, C ;

Lynch, S .

NONLINEARITY, 1999, 12 (04) :1099-1112

[6]

Han M., 1999, Ann. Differ. Equ., V15, P113

[7] Limit Cycles Near Homoclinic and Heteroclinic Loops [J].

Han, Maoan ;

Yang, Junmin ;

Tarta, Alexandrina -Alina ;

Gao, Yang .

JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 2008, 20 (04) :923-944

[8] On the number of limit cycles of polynomial Lienard systems [J].

Han, Maoan ;

Romanovski, Valery G. .

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS, 2013, 14 (03) :1655-1668

[9] HOPF BIFURCATIONS FOR NEAR-HAMILTONIAN SYSTEMS [J].

Han, Maoan ;

Yang, Junmin ;

Yu, Pei .

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 2009, 19 (12) :4117-4130

[10] Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system [J].

Han, Maoan ;

Zang, Hong ;

Yang, Junmin .

JOURNAL OF DIFFERENTIAL EQUATIONS, 2009, 246 (01) :129-163

← 1 2 →