Bifurcation of limit cycles for the quadratic differential system (III)l=n=0

1999 ◽  
Vol 15 (4) ◽  
pp. 368-373
Author(s):  
Zhang Xiang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Bifurcation Of Limit Cycles ◽  
Quadratic Differential System

SOME BIFURCATION DIAGRAMS FOR LIMIT CYCLES OF QUADRATIC DIFFERENTIAL SYSTEMS

2001 ◽  
Vol 11 (01) ◽  
pp. 197-206 ◽  
Author(s):  
H. S. Y. CHAN ◽  
K. W. CHUNG ◽  
DONGWEN QI
Keyword(s):  
Singular Point ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Bifurcation Diagrams ◽  
Differential Systems ◽  
Numerical Examples ◽  
Quadratic Differential System ◽  
Realistic Parameter ◽  
Parameter Values

Concrete numerical examples of quadratic differential systems having three limit cycles surrounding one singular point are shown. In case another finite singular point also exists, a (3, 1) distribution of limit cycles is also obtained. This is the highest number of limit cycles known to occur in a quadratic differential system so far. Representative bifurcation diagrams are drawn for realistic parameter values.

scholarly journals The acyclicity of a quadratic differential system

2021 ◽  
pp. 32-41
Author(s):  
Адам Дамирович Ушхо ◽  
Вячеслав Бесланович Тлячев ◽  
Дамир Салихович Ушхо
Keyword(s):  
Differential Equations ◽  
Differential System ◽  
Limit Cycles ◽  
Quadratic Differential ◽  
Qualitative Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Differential System ◽  
Straight Line

Дан краткий обзор некоторых основных публикаций, посвященных исследованию вопроса о предельных циклах и сепаратрисах квадратичных дифференциальных систем. Рассмотрено наличие замкнутых траекторий для определенного класса автономных квадратичных систем на плоскости. Доказательство основано на применении теории прямых изоклин, признаков Дюлака и Бендиксона качественной теории дифференциальных уравнений. Предложенное доказательство покрывает результаты известной работы Л.А. Черкаса и Л.С. Жилевич. We now give a brief overview of some of the main publications devoted to the study of the question of limit cycles and separatrices of quadratic differential systems. In this paper, we consider the existence of closed trajectories for a certain class of autonomous quadratic systems on the plane. The proof is based on the application of the theory of straight line isoclines, Dulac and Bendixon criteria of the qualitative theory of differential equations. The proposed proof covers the results of the well-known work of L.A. Cherkas and L.S. Zhilevich.

Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center

2015 ◽  
Vol 25 (01) ◽  
pp. 1550007 ◽  
Author(s):  
Fengjie Geng ◽  
Hairong Lian
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Bifurcation Of Limit Cycles ◽  
Period Annulus ◽  
Real Polynomials ◽  
Positive Integers

In this paper, we deal with the following differential system [Formula: see text] where p, q are positive integers, and P(x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n - 1)p1 + (t + 1)q - 1 + 2rp1q1(q + 3) + 2tqrp1q1, where t = [n/2q] + 2, (p, q) = r(p1, q1), p1 and q1 are coprime.

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