A quadratic system with two parallel straight-line-isoclines

被引：4
作者：
Gaiko, Valery A. [1 ]
机构：
[1] Belarusian State Univ Informat & Radioelect, Dept Math, Minsk 220040, BELARUS
来源：
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS | 2009年 / 71卷 / 11期
关键词：
Planar quadratic dynamical system; Isocline; Field rotation parameter; Bifurcation; Limit cycle; Wintner-Perko termination principle;
D O I：
10.1016/j.na.2009.05.010
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
In this paper, a quadratic system with two parallel straight-line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian [Ye Yanqian et al.,Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, American Mathematical Society, Providence, R1, 1986]. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight-line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner-Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in [V.A. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem, Kluwer, Boston, 2003]. (C) 2009 Published by Elsevier Ltd
引用
收藏
页码：5860 / 5865
页数：6
相关论文
共 12 条
[1]  
Bautin N.N., 1990, METHODS WAYS QUALITA
[2]   LIMIT-CYCLES AND ROTATED VECTOR FIELDS [J].
DUFF, GFD .
ANNALS OF MATHEMATICS, 1953, 57 (01) :15-31
[3]  
Erugin N.P., 1950, PRIKL MATEM MEKH, V14, P459
[4]   Qualitative theory of two-dimensional polynomial dynamical systems: Problems, approaches, conjectures [J].
Gaiko, V .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1997, 30 (03) :1385-1394
[5]  
Gaiko V. A., 2003, Nonlinear Phenomena in Complex Systems, V6, P734
[6]  
Gaiko V.A., 2003, GLOBAL BIFURCATION T
[7]   Hilbert's Sixteenth Problem and global bifurcations of limit cycles [J].
Gaiko, VA .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2001, 47 (07) :4455-4466
[8]   Limit cycles of quadratic systems [J].
Gaiko, Valery A. .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2008, 69 (07) :2150-2157
[9]   Centennial history of Hilbert's 16th problem [J].
Ilyashenko, Y .
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 2002, 39 (03) :301-354
[10]  
Perko L., 2002, Differential Equations and Dynamical Systems
12